Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer

EXTENDING THE COINV ARIANT THEOREMS OF CHEV ALLEY, SHEPHARD– TODD, MITC HELL, AND SPRINGER ABRAHAM BROER, VICTOR REINER, LA R R Y SMITH, AND PETER WEBB Abstra ct. W e extend in sever al directions in v ariant theory results of Chev alley , Shephard and T odd, Mitchell and Springer. Their results compare the group algebra for a ﬁnite reﬂection group with its coin- v ariant algebra, and compare a group representation with its mo dule of relativ e coinv ariants. Our extensions apply to arbitrary ﬁnite groups in any characteristic. Contents 1. Statemen t of Results 4 1.1. Chev alley , S hephard –T od d , Mitc hell Typ e Results 4 1.2. Sprin ger-T yp e Resu lts 7 1.3. A F urth er Generaliza tion 8 2. Generalities for Rings, Modu les, and T or 10 2.1. Review of Graded Resol utions 10 2.2. The Group Acti on on T or 11 2.3. The Group Acti on on Resolutions 12 2.4. A Short Review of Brau er theory 15 2.5. The Case of Do mains with T rivial Group Actio n 16 2.6. Pro of of Theorem 1.1 .1 21 2.7. Remarks on Group Cohomology and other related Constructions 22 3. Pro of of Theorems 1. 2.1 and 1.3.1 24 3.1. Reduction 1: Remo vin g the Γ-acti on 25 3.2. Reduction 2: Replacing Sp eci al Fiber with General Fib er 26 3.3. Reduction 3: W orkin g Lo cally via Completions 28 3.4. (Semi-)lo cal Analysis of the Fib er s 29 3.5. Finishing the pro ofs: Incorp orating the C -Action 33 3.6. Indu ced Mo dules and the Pr o of of C orollary 1.2.2 37 4. Character V alues and Hilb ert ser ies 39 4.1. Molien’s theorem and Brauer c haracter v alues 39 Date : No vem b er 5, 2018. Key wor ds and phr ases. reﬂection group, in v ariant th eory , p olyn omial in v ariants, rela- tive in v ariants, coinv arian ts, Brauer isomorphism, cyclic sieving p henomenon. W ork of ﬁrst au t hor supp orted by NSERC . W ork of second auth or supported by NSF gran t DMS- 0245379 . W ork of fourth author supp orted by MSRI. 1 2 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB 4.2. A mo d ular v ersion 41 References 47 Let k b e an arbitrary ﬁeld, V an n -dimen sional k -vect or space, and G a ﬁnite su b group of GL ( V ). Then G acts by linear change of co ordin ates on the algebra k [ V ] of p olynomial functions on V . If x 1 , . . . , x n is a basis for the dual v ector spac e V ∗ w e ma y iden tify k [ V ] with the p olynomial algebra k [ x 1 , . . . , x n ] in x 1 , . . . , x n regarded as formal v ariables. The coinv arian t algebra is the quotie n t alge bra k [ V ] / ( k [ V ] G + ) ∼ = k [ V ] ⊗ k [ V ] G k where ( k [ V ] G + ) denotes the ideal of k [ V ] generated by th e elemen ts of the in v ariant s ubalgebra k [ V ] G of strictly p ositiv e degree, and k = k [ V ] G /k [ V ] G + is regarded as a trivial k [ V ] G -mo dule. The coin v ariant algebra is a ﬁn ite dimensional G -repr esentati on. Muc h of its signiﬁcance (see, e.g., [4], [15], [29]) derives fr om the fact that in fa v orable cases it pro vides a gr ade d v ersion of the regular repr esen tatio n k ( G ), where k ( G ) d enotes the group algebra, but regarded as a G -representat ion. This is made pr ecise in the follo win g result, due to Chev alley (and also Sh ephard and T od d) in c h aracteristic zero, and to M itc hell in p ositiv e c haracteristic. Theorem(Chev alley [8] , Shephard and T o dd [28] , Mit c hell [20] ). L et k b e an arbitr ary ﬁeld, V an n -dimensional k -ve ctor sp ac e, and G a ﬁnite sub gr oup of GL ( V ) . Supp ose that k [ V ] G is a p olynomial algebr a. Then as k ( G ) -mo dules, the c oinvariant algebr a k [ V ] ⊗ k [ V ] G k and the r e gular r epr e - sentation k ( G ) have the same c omp osition factors c ounting multiplicities. Something similar for relative in v ariants ma y b e dedu ced under su it- able hypotheses as w e explain next. Supp ose w e ha v e a second group Γ and a ( k (Γ) , k ( G ))-bimo du le U , i.e., a right k ( G )-mo dule U whic h has a comm u ting left action of k (Γ). W e c hose to us e the terminology of bimo d- ules here as an aid in keeping the diﬀerent actions d istinct. If ho w ev er no confusion can arise we frequentl y ident ify ( k (Γ) , k ( G ) )-bimo dules with left k (Γ × G )-mo du les; the left action of ( γ , g ) ∈ Γ × G on an element u ∈ U b eing giv en by γ ug − 1 . The mo du le M of U - relat iv e inv arian ts is deﬁned b y M := ( U ⊗ k k [ V ]) G where G acts on the tens or pr o duct diagonally , viz., g ( u ⊗ x ) = ug − 1 ⊗ g x . Note th at M h as the stru cture of a graded ( k (Γ) , k [ V ] G )-bimo dule, in whic h Γ ac ts trivially on V an d on k [ V ]. Corollary . L et k b e a ﬁeld, V = k n an n -dimensional ve ctor sp ac e over k , G a ﬁnite sub gr oup of GL ( V ) , Γ a se c ond ﬁnite gr oup, and U a ﬁnite- dimensional ( k (Γ) , k ( G )) -bimo dule. R e gar d the r e lative invariants M = ( U ⊗ k k [ V ]) G as a ( k (Γ) , k [ V ] G ) -bimo dule. If | G | is invertible in k and k [ V ] G is a p olynomial algebr a, then one has a k (Γ) -mo dule isomorphism M ⊗ k [ V ] G k ∼ = U. EXTENDING COINV ARIANT THEOREM S 3 Note that this corollary includes th e nonmo du lar version of the Chev al- ley , Shephard-T o d d , Mitc hell Theorem as the sp ecial case wh ere Γ = G and U = k ( G ) as a ( k ( G ) , k ( G ))-bimo dule. More generally , if w e let H b e any subgroup of G and Γ = N G ( H ), the normalizer of H in G , w e obtain a k -linear rep resen tation U = k ( H \ G ) from the p erm utation represent ation of G on th e s et H \ G of right cosets of H in G . R egarded as a ( k (Γ) , k ( G ))- bimo du le, the relativ e inv ariant s M = ( U ⊗ k k [ V ]) G b ecome the subalgebra of H -in v ariant p olynomials k [ V ] H regarded as a k ( N G ( H ))-mo du le (see § 3.6 b elo w). C er tain other cases of relativ e in v arian t m o dules M app ear fr e- quen tly in the literature, suc h as th e i th -exterior p ow er U = ∧ i ( V ∗ ) of V ∗ (resp. U = V itself ), and M is the mo dule of G -inv ariant diﬀer ential i - forms (resp. G -invariant ve ctor ﬁelds ) on V , (see e.g., [32], [22, § 6.1]), or, for a simple k ( G )-mod ule U where the Hilb ert ser ies of M ⊗ k [ V ] G k deﬁn es the fake de gr e es for U (see e.g., [12, § 1.6]). Our ﬁrs t m ain r esult, Theorem 1.1.1 b elo w, extends the Chev alley , Sheph ard–T o dd, Mitc hell resu lt and its corollary by remo ving the hyp othesis that k [ V ] G b e p olynomial and | G | lie in k × . Our other main r esults are inspired by a generalization of the Chev alley- Shephard -T o d d Th eorem d ue to Spr inger [33], whic h incorp orates the action of an extra cyclic group. W e recall this next. Giv en a ﬁnite sub group G ⊆ GL ( V ), say that v ∈ V is a regular vector if the orbit Gv is a regular orbit , meaning that the s tabilizer in G of v is 1, or equiv alen tly that th e orbit ac hiev es the maxim um card inalit y | Gv | = | G | . An elemen t c ∈ G is a regular element if it has a regular eigen vect or v ∈ V , after p ossibly extending the ﬁ eld k to include the corresp onding eigen v alue ζ ∈ k × . Letting C = h c i denote the cyclic sub group generated by c , the group algebra k ( G ) b ecomes a ( k ( G ) , k ( C ))-bimo dule in which ( g , c ) acts on the b asis elemen t t h of k ( G ) corresp onding to h in G via ( g , c ) · t h := t g hc . As b efore we ma y identify ( k ( G ) , k ( C ))-bimo dules with k ( G × C )-mo du les by letting a left action of c ∈ C corresp ond to a right action of c − 1 . W e also let C act on k [ V ] by the algebra automorphism s whic h are a scalar m ultiplication in eac h degree and determined by requirin g c j ( x i ) = ζ j x i for i = 1 , 2 , . . . , n . In th is w a y w e obtain the s tructure of a ( k ( G ) , k ( C ))-bimo d ule on k [ V ] as w ell as k [ V ] ⊗ k [ V ] G k . The follo wing th eorem wa s pr ov en by Spr inger in c h aracteristic zero, and extended to arbitrary ﬁ elds in [23]. Theorem(Springer [33] , Reiner-Stanton-W ebb [23] ). L et k b e a ﬁeld, V a ﬁnite- dimensional k -ve ctor sp ac e, G a ﬁnite sub gr oup G ⊂ GL ( V ) . Supp ose that k [ V ] G is a p olynomial algebr a, and c is a r e gular element of G . L et C = h c i as ab ove. Then one has the e quality h k [ V ] ⊗ k [ V ] G k i = [ k ( G ) ] . in R ( k ( G × C )) , wher e R ( k ( G × C )) denotes the Gr othendie ck ring of ﬁnite dimensional ( k ( G ) , k ( C )) -bimo dules. 4 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB W e are concerned with extensions of all these resu lts to arbitrary groups in any c h aracteristic. This w ill require signiﬁcant reformulation s in ce th e naiv e versions of these results would not b e correct. F or example, a simple consequence of th e Chev alley , Shephard –T od d, Mitc h ell result is that the coin v ariant algebra for G has dim en sion | G | when ev er k [ V ] G is a p olynomial algebra, bu t this fails w hen k [ V ] G is not p olynomial (see e.g., [30]). The authors would like to thank Radh a K essar, Gennady Lyub eznik, Ezra Miller, T onny S pringer and Dennis S tanton for h elpful con v ersations and stim ulating questions. 1. St a t e ment of Res u l ts In this section w e state our main resu lts and illustrate them with a simple example. F or bac kgroun d material on inv ariant theory s ee [4, 9, 29], on represent ation theory see [26], particularly P art I I I, and on reﬂection groups see [12, 15]. 1.1. C hev alley , Shephard–T o dd, Mitc hell Type Results. W e ﬁrst in- dicate h o w to remo v e the hypothesis that k [ V ] G b e p olynomial, and | G | lie in k × , f rom the Ch ev alley , Shephard –T o dd, Mitc hell Theorem and its corollary . O ur result, Theorem 1.1.1, compares the u ngraded k (Γ)-mo du le U with v arious graded k (Γ)-mo du les, sho wing how, in a s ense to b e m ade precise b elo w, (i) M ⊗ k [ V ] G k is an ov erestimate for U , (ii) it is th e ﬁrst in a sequence of alternating o v erestimates an d u n- derestimates, and (iii) th ese estimates conv erge to a s u itably d eﬁned limit. T o explain what this means, recall that k (Γ) is not in general semisimp le, so one comp ensates f or this by w orking with comp osition facto rs. A con v enient w a y to do this is to introd uce the Grothendiec k ring R ( k (Γ)) of ﬁnite- dimensional k (Γ)-modu les. T h is is deﬁned to b e the ring with one generator [ M ] f or eac h isomorphism class { M } of ﬁnite dimen sional k (Γ)-modu les, and one relation [ M ′ ] − [ M ] + [ M ′′ ] = 0 for eac h short exact s equ ence 0 − → M ′ − → M − → M ′′ − → 0 . Addition in R ( k (Γ)) is ind uced b y d irect sum and pro d uct b y tensor p ro du ct. There is also a partial ordering on R ( k (Γ)) deﬁned by requ ir ing for t wo elemen ts x, y ∈ R ( k (Γ)) th at x ≥ y if x − y = [ M ] for some (gen uine) k (Γ)-mo dule M . F or example, giv en t w o gen u ine mo d ules M 1 and M 2 , th e inequalit y [ M 1 ] ≥ [ M 2 ] m eans that for ev ery simple k (Γ)-mo du le S one has the inequalit y [ M 1 : S ] ≥ [ M 2 : S ], where [ M : S ] denotes the multiplic it y of S as a comp osition factor in M . Giv en a ﬁ nite group Γ, a (non-n egativ ely) graded k (Γ)-modu le is one with a d irect s u m decomp osition M = ⊕ d ≥ 0 M d in whic h eac h M d is a ﬁn ite- dimensional k (Γ)-mo du le. Such an M giv es rise to an elemen t [ M ]( t ) := EXTENDING COINV ARIANT THEOREM S 5 P d ≥ 0 [ M d ] t d in the formal p o w er series ring ov er the Grothendiec k rin g R ( k (Γ))[[ t ]] := Z [[ t ]] ⊗ Z R ( k (Γ)) . If one forgets the group action w e ob- tain the formal p o wer series P d ≥ 0 dim k ( M d ) t d ∈ Z [[ t ]] called the Hilb ert series of M in this manuscript. The motiv ating example for us of suc h a graded k (Γ)-mo d u le will b e ⊕ i ≥ 0 T or R i ( M , k ) in th e situation wh ere R is a ﬁnitely generated graded, connected, commutat iv e k -algebra with a grade-preserving action of Γ, and M is a ﬁ n itely generated grad ed R -mo du le with a compatible k (Γ)-mo dule structure (see Section 2). These h yp otheses imply that for eac h i ≥ 0 the k -mo dule T or R i ( M , k ) acquires a grading from R and M , and that eac h graded comp onen t T or R i ( M , k ) j is ﬁn ite-dimensional o ver k . F urther m ore, for eac h ﬁ xed i the ( i, j )-comp onen t is non-zero for only ﬁnitely many j and moreo v er all such j are greater than or equal to i . Lik ewise, f or eac h ﬁxed j the ( i, j )-comp onent is non-zero for only ﬁnitely many i . This means for i = 0 , 1 , . . . that P j ≥ 0 ( − 1) i [T or R i ( M , k ) j ] t j is in fact a p oly- nomial in t whic h is is divisible by t i and, that the doubly inﬁnite sum P i ≥ 0 ( − 1) i P j ≥ 0 [T or R i ( M , k ) j ] t i is a well d eﬁned elemen t of R ( k (Γ))[[ t ]]. This s um, w hic h we often d enote by P i ≥ 0 ( − 1) i [T or R i ( M , k )]( t ), generalizes the m ultiplicit y symb ol of S erre from the case with no group action (see e.g., [25] and [31, § 3]). Theorem 1.1.1. L et k b e a ﬁe ld, V a ﬁnite-dimensional k -ve ctor sp ac e, G a ﬁnite sub gr oup of GL ( V ) , and Γ a ﬁnite gr oup. L et U b e a ﬁnite- dimensional ( k (Γ) , k ( G )) -bimo dule, and let M := ( U ⊗ k k [ V ]) G , r e gar de d as a ( k (Γ) , k [ V ] G ) -bimo dule. Set K := k ( V ) G , the ﬁeld of G -invariant r ational functions on V (i.e., the ungr ade d ﬁeld of fr actions of k [ V ] G ). (i) In R ( k (Γ)) , one has the ine quality [ M ⊗ k [ V ] G k ] ≥ [ U ] . F urthermor e, one has e quality if and only i f M i s k [ V ] G -fr e e, in which c ase K ⊗ k U has a K ( Γ) -mo dule ﬁltr ation {F j } for which the factor F j / F j − 1 is isomorphic to the j th homo gene ous c omp onent ( M ⊗ k [ V ] G k ) j ⊗ k K . (ii) Mor e gener al ly, for any m ≥ 0 , in R ( k (Γ)) , one has the ine quality m X i =0 ( − 1) i X j ≥ 0 h T or k [ V ] G i ( M , k ) j i  ≥ [ U ] if m is even, ≤ [ U ] if m is o dd, with e quality if and only if T or k [ V ] G i ( M , k ) v anishes for i > m . (iii) The element ∞ X i =0 ( − 1) i ∞ X j =0 h T or k [ V ] G i ( M , k ) j i t i 6 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB of R ( k (Γ))[[ t ]] has the pr op erty that, for e ach simple mo dule S , the p ower series in t gi ving the c o eﬃcient of [ S ] lies i n Q ( t ) . F urther- mor e, t = 1 is a r e gular value for these r ational func tions, and the evaluation at t = 1 is X i ≥ 0 ( − 1) i X j ≥ 0 h T or k [ V ] G i ( M , k ) j i t i      t =1 = [ U ] in R ( k (Γ)) . Theorem 1.1.1 is pr o ven in Section 2.5, using a h omologica l strengthen- ing of C hev alley’s metho d fr om [8], relying u ltimately on the Normal Basis Theorem fr om Galois theory . W e illustr ate this with a simple examp le. Example 1.1.2 Let G b e the cyclic group Z / 2 of order 2 regarded as the subgroup of GL (2 , C ) generated by the scalar matrix g = − I 2 × 2 whic h is th e negativ e of the identit y . Note that the rin g of inv ariants is not a p olynomial algebra, rather one has R := C [ x, y ] Z / 2 = C [ x 2 , xy , y 2 ] . There are t wo simp le C ( Z / 2)-mod ules, U + and U − , b oth 1-dimens ional, with g acting by the scalar +1 , − 1 on U + , U − , r esp ectiv ely . Cho ose Γ to b e the trivial group { 1 } and regard it as a subgroup of Aut C ( Z / 2) ( U ± ). The Grothendiec k r ing R ( C (Γ)) is isomorphic to Z , with the isomorphism sending the class [ 1 ] of the trivial 1-dimensional C (Γ)- mo dule to the in teger 1. An y C -v ecto r space of dimension d then represents the elemen t [ C d ] = d [ 1 ] in R ( C (Γ)). In particular, [ U + ] = [ U − ] = [ 1 ] in R ( C (Γ)). One can easily c hec k that M + := ( U + ⊗ C C [ x, y ]) Z / 2 = C [ x, y ] Z / 2 = R M − := ( U − ⊗ C C [ x, y ]) G = R x + Ry . So M + is a free R -mo du le of rank 1, and all inequalities asserted in Theorem 1.1.1 b ecome trivial equalities. By con trast, M − has an interesting, inﬁ nite, 2-p erio dic 1 R -free resolution · · · d 4 − → R ( − 7) 2 d 3 − → R ( − 5) 2 d 2 − → R ( − 3) 2 d 1 − → R ( − 1) 2 d 0 − → M − → 0 , in whic h R ( − d ) d en otes a free R -mo dule of r ank 1 having a basis elemen t of d egree d . Here the diﬀerentia l d 0 maps the t w o b asis elemen ts of R ( − 1) 2 on to x, y in M − , w hile the diﬀerentials d i for i ≥ 1 can b e chose n as follo w s: d i =  x 2 xy xy y 2  for eve n i, d i =  y 2 − xy − xy x 2  for o d d i. 1 In fact, whenever R is a hyp ersurfac e algebr a , i.e., R = k [ V ] G ∼ = k [ f 1 , . . . , f n , f n +1 ] / ( h ) for a single homogeneous relation h among the f i ’s, there will alwa ys b e such an R -free resolution of M which is even tually 2-p erio dic (see e.g. , [36, § 6] or [11]). EXTENDING COINV ARIANT THEOREM S 7 F or any m ≥ 0, th is giv es the follo wing strict inequalities in R ( C (Γ)) ∼ = Z   m X i =0 ( − 1) i X j ≥ 0 T or R i ( M − , C ) j   = [ C 2 ]= 2[ 1 ] > [ 1 ]= [ U − ] if m is ev en, [0] = 0[ 1 ] < [ 1 ]= [ U − ] if m is o dd, as predicted by Th eorem 1.1.1 (i, ii). In the limit as m → ∞ , one mak es sense of this by noting that T or R i ( M − , C ) j v anishes u nless the in ternal degree j and h omologica l degree i s atisfy j = 2 i + 1. Hence one can calculate in R ( C (Γ))[[ t ]] ∼ = Z [[ t ]] th at X i ≥ 0 ( − 1) i  T or R i ( M − , C )  ( t ) = 2[ 1 ] t 1 − 2[ 1 ] t 3 + 2[ 1 ] t 5 − 2[ 1 ] t 7 + · · · = 2 t 1 + t 2 [ 1 ] . The co eﬃcien t of [ 1 ] ∈ R ( C (Γ)) is a rational fun ction of t w ith t = 1 as a regular v alue, and up on su b stituting t = 1, one obtains in R ( C (Γ)) X i ≥ 0 ( − 1) i  T or R i ( M − , C )  ( t )      t =1 = 2 t 1 + t 2 [ 1 ]      t =1 = 2 2 [ 1 ] = [ 1 ] = [ U − ] as predicted by Theorem 1.1.1(iii). 1.2. Springer-Typ e Results. Let k b e a ﬁ eld, V a ﬁnite-dimensional k -v ector sp ace, G a ﬁnite s u bgroup of GL ( V ). Supp ose that k [ V ] G is a p olynomial algebra, and c is a r egular elemen t of G . A s with the C hev al- ley , S hephard-T o d d , Mitc h ell T heorem, we wish to dedu ce a more general v ersion of Sp ringer’s T heorem that applies to an arbitrary ( k (Γ) , k ( G ))- bimo du le U for an y ﬁnite group Γ. The k (Γ)-mo d ule of relativ e in v arian ts M := ( U ⊗ k k [ V ]) G carries a commuting C -ac tion in w hic h C acts on k [ V ] b y scalars as b efore, and do es nothing to the factor of U in U ⊗ k k [ V ]. In this wa y M b ecomes a graded ( k (Γ) , k ( C )) -bimo dule. It is also p ossi- ble to view U itself as a ( k (Γ) , k ( C ))-bimo dule in a diﬀerent wa y , namely with the action of C coming as the restriction of the action of G , of wh ic h C is a subgroup . F rom th e theorem of Sprin ger, it is n ot hard to deduce that if k [ V ] G is a p olynomial algebra and b oth | G | , | Γ | lie in k × , then as ( k (Γ) , k ( C ))-bimo dules, M ⊗ k [ V ] G k ∼ = U. Our next main result shows h o w to remov e the h yp othesis that | G | , | Γ | lie in k × . Theorem 1.2.1. L et k b e any ﬁeld, V a ﬁnite-dimensional k -ve c tor sp ac e of k and G a ﬁnite sub g r oup of GL ( V ) with k [ V ] G a p olynomial algebr a. L et C = h c i b e the cyc lic sub gr oup gene r ate d by a r e gular element c in G with r e g ular e i genvalue ζ and U a ﬁnite-dimensional ( k (Γ) , k ( G )) -bimo dule for some ﬁnite gr oup Γ . R e gar d M := ( U ⊗ k k [ V ]) G as a k [ V ] G -mo dule and as a ( k (Γ) , k ( C )) -bimo dule, wher e C acts on k [ V ] via sc alar multiplic ation 8 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB determine d by c j ( x i ) = ζ j x i for i = 1 , 2 , . . . , n and x 1 , . . . , x n is a b asis f or V ∗ . Then X i ≥ 0 ( − 1) i X j ≥ 0 h T or k [ V ] G i ( M , k ) j i = [ U ] . in R ( k (Γ × C )) ; the sum b eing ﬁnite si nc e k [ V ] G is a p olynomial algebr a. Pro ving Th eorem 1.2.1 (in particular, with no h yp othesis on the ﬁeld k ) w as one of our original motiv ations. It easily implies ou r main application, Corollary 1.2.2 b elo w, whic h r esolv es in the aﬃrmativ e b oth C onjecture 3 and Question 4 in [23]. Corollary 1.2.2. L et k b e a ﬁeld, V a ﬁnite-dimensional k -ve ctor sp ac e over k and H ⊂ G two neste d ﬁnite sub gr oups of GL ( V ) . Assume k [ V ] G is a p olynomial algebr a, and let C = h c i b e the cyclic sub gr oup gener ate d by a r e gular element c in G , with eig e nvalue ζ on some r e gu lar ve ctor in V and ˆ ζ ∈ C a c omplex lift of ζ . L et Γ := N G ( H ) /H , wher e N G ( H ) denotes the normalizer of H i n G . Give k [ V ] H the structur e of a ( k (Γ) , k ( C ) ) -bimo dule in which c sc ales the variables x 1 , . . . , x n in V ∗ by ζ , and Γ acts by line ar substitutions. L et k ( H \ G ) b e the k - ve ctor sp ac e with b asis the right c osts of H in G r e gar de d as a ( k (Γ) , k ( C )) -bimo dule wher e γ · H g · c := γ H gc = H γ g c. Then in R ( k (Γ × C )) one has the e quality X i ≥ 0 ( − 1) i X j ≥ 0 h T or k [ V ] G i ( k [ V ] H , k ) j i = [ k ( H \ G )] . Ignoring the Γ -action, this i mplies that the quotient of Hilb ert series X ( t ) := [ k [ V ] H ]( t ) [ k [ V ] G ]( t ) , is a p olynomial in t , and to gether with the C -actio n on the set X = H \ G , gives a triple ( X, X ( t ) , C ) that exhibits the cyclic sieving phenomenon of [24] : Namely, for e ach element c j in C the c ar dinality of the ﬁxe d p oint set X c j ⊂ X is given by evaluating X ( t ) at the c ompl ex r o ot-of-unity ˆ ζ j of the same multiplic ative or der as c j . In other wor ds, | X c j | = [ X ( t )] t = ˆ ζ j . 1.3. A F urther Generalizat ion. Theorem 1.2.1 follo ws from the m ore general Th eorem 1.3.1, w h ic h do es not mak e the assump tion that k [ V ] G is p olynomial and has other applications. The guiding principle in this generalizat ion is to replace the condition that k [ V ] G b e a p olynomial algebra with the assump tion it con tains a No ether n ormalizatio n R ⊆ k [ V ] G fulﬁlling certain ke y tec hnical cond itions. Recall that a No ether normalization for k [ V ] G is a p olynomial su balgebra R ⊂ k [ V ] G o ver which k [ V ] G is a ﬁ nitely generated mo d ule. T o state our result requires some p r eliminaries. L et V b e a ( k ( G ) , k ( C ))- bimo du le with G and C ﬁnite groups, so that C acts on k [ V ] and k [ V ] G . EXTENDING COINV ARIANT THEOREM S 9 Supp ose there exists a homogeneous and C -stable No ether normalization R ⊂ k [ V ] G ⊂ k [ V ], w ith the add itional p rop erties that the ﬁb er Φ v := φ − 1 ( φ ( v )) ov er th e p oint φ ( v ) in the ramiﬁed cov ering V → S p ec( R ) has free (but n ot necessarily tr an s itiv e) G -actio n, and is stable un der C . T his means that C preserves the to w er of inclusions m φ ( v ) ⊂ R ⊂ k [ V ] G ⊂ k [ V ] , where m φ ( v ) is the (generally inhomogeneous) maximal ideal in R corre- sp ond ing to φ ( v ) in Sp ec ( R ). The qu otient rin g k [ V ] / m φ ( v ) k [ V ] =: A (Φ v ) can b e though t of as the co ordinate ring for the ﬁb er Φ v regarded as a (p os- sibly non-reduced) subscheme of the aﬃne space V . This ring A (Φ v ) carries an in teresting ( k ( G ) , k ( C ))-bimodu le structur e, w h ose precise d escription w e defer u n til § 3.5 wher e the extra generalit y is exploited. Ho wev er, it is worth mentioning here what this bimo d ule structure lo oks lik e u n der the h yp otheses of T h eorem 1.2.1, that is, if k [ V ] G is p olynomial, so that w e can c ho ose R = k [ V ] G , and where c is a regular elemen t of G with eigen v alue ζ on a regular eigen vecto r v . In this sp ecial case, if we tak e for C the group of scalar matrices in GL ( V ) generated by ζ times the iden tit y matrix, then A (Φ v ) ∼ = k ( G ) carries the same ( k ( G ) , k ( C ))-bimo dule structure as wa s describ ed on k ( G ) in Sp ringer’s theorem (Theorem 1.2.1). Bac k in the general setting, giv en a ( k (Γ) , k ( G ))-bimo dule U , one lets C act trivially on U and Γ act trivially on k [ V ]. In this w a y , the relativ e in v ariants M := ( U ⊗ k k [ V ]) G carry the stru cture of a graded ( k (Γ) , k ( C ))- bimo du le, compatible with its R -mo d ule structure. S imilarly ( U ⊗ k A (Φ v )) G carries the structur e of a ( k (Γ) , k ( C ))-bimo dule in w h ic h Γ acts only on the U factor, and C acts only on the A (Φ v ) factor. Theorem 1.3.1. L et k b e a ﬁeld, G, Γ , and C ﬁnite gr oups, and V a ﬁ- nite dimensional ( k ( G ) , k ( C )) -bimo dule on which G acts faithful ly (so G ⊂ GL ( V ) ). R e gar d V , k [ V ] and k [ V ] G as trivi al k (Γ) -mo dules. Supp ose ther e is a No ether normalization R ⊂ k [ V ] G that is stable u nder the action of C on k [ V ] . Supp ose further that ther e is a ve ctor v in V such that the ﬁb er Φ v := φ − 1 ( φ ( v )) c ontaining v for the map φ : V → Sp ec( R ) c arries b oth a fr e e (but not ne c essarily tr ansitive) G -action and is stable under C . Denote by m φ ( v ) the maximal i de al in R c orr esp onding to φ ( v ) in Sp ec( R ) and set A (Φ v ) = k [ V ] / m φ ( v ) k [ V ] which is a k ( C ) -mo dule. L et U b e a ﬁnite-dimen- sional ( k (Γ) , k ( G )) -bimo dule r e g ar de d as a trivial k ( C ) -mo dule. Then the r elative invariants M := ( U ⊗ k k [ V ]) G satisfy X i ≥ 0 ( − 1) i X j ≥ 0  T or R i ( M , k ) j  = h ( U ⊗ k A (Φ v )) G i in R ( k (Γ × C )) ; the sum b eing ﬁnite sinc e R is a p olynomial algebr a. F r om this we easily deduce Th eorem 1.2.1 in Section 3.5. Section 4 il- lustrates a diﬀerent t yp e of app lication of Theorem 1.3.1, namely to Brauer c h aracter v alues in the simple group of ord er 168. 10 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB 2. Generalities for Rings, M o dules, and T or W e record here some issues sur r ounding T or R ( M , k ) and the action of a group on R -resolutions of M . Without the group action m ost of these results can b e found in [31]. Because the group v aries in the app lications w e denote it by the new symbol G . It will alwa ys b e assumed throughout Section 2 that (i) R = ⊕ d ≥ 0 R d is a comm u tativ e, N -graded, connected ( R 0 = k ), No etherian k -al gebra (i.e., ﬁ nitely generated as an algebra o v er k ) and (ii) G is a group whic h acts on R b y graded k -algebra automorphisms . In addition w e will consider ﬁnitely generated R -mo dules with a compatible homogeneous action of G in a sense conv enient ly describ ed in terms of the sk ew group algebra R ⋊ G (see e.g., [2]). This is the free R -mo dule with element s { t g } g ∈G indexed by G as a basis, and w hose multiplica tion is determined by the rule r t g · st h = r · g ( s ) t g h and bilinearit y , for all r, s ∈ R and g , h ∈ G . W e pu t a grading on R ⋊ G b y requir ing that an elemen t r t g has the same degree as r . An R ⋊ G -mod u le is the same thing as an R -mo du le M with an action of G on M regarded as a graded ab elian group b y grading preserving group endomorph isms satisfying g ( r m ) = g ( r ) g ( m ) and g 1 ( g 2 m ) = ( g 1 g 2 ) m . This is what we mean by a compatible action of G . So the thir d standing assumption in this section is that (iii) M = ⊕ d ≥ 0 M d is an N -graded R ⋊ G -mo dule wh ic h is No etherian (i.e., ﬁ nitely generated) as an R -mo dule. The kind of graded R ⋊ G -modu les w e will consider arises, for example in the situation discussed in the in tro d uction: W e hav e a ( k ( G ) , k ( C ))- bimo du le V , so that C acts on k [ V ] , k [ V ] G and p ossibly also on a No ether normalization R ⊂ k [ V ] G , as well as compatibly on the R -mo d ule M := ( U ⊗ k k [ V ]) G for an y ( k (Γ) , k ( G ) )-bimo dule U . In fact p utting G = Γ × C , M b ecomes an R ⋊ G -mo du le. 2.1. Review of Graded Resolutions. Recall that, ignoring group ac- tions, there alwa ys exist gr ade d R -fr e e r esolutions F of M in whic h all terms are ﬁnitely generated, that is, an exact sequence · · · d i +1 → F i d i → F i − 1 d i − 1 → · · · F 1 d 1 → F 0 d 0 → M → 0 with eac h F i a grad ed free R -mo dule of ﬁnite rank, and grade-preserving diﬀeren tials d i . F rom any su c h resolution one can compute the b igraded k -v ector space T or R ( M , N ) = { T or R i ( M , N ) j } for any graded R -mo d ule N , b y taking the homology of the tensored complex F ⊗ R N . This m eans that T or R i ( M , N ) for i ∈ N is a graded k -vect or space; the index i is called the homological grading , and the gradin g on T or R i ( M , N ), namely the index j , is called the internal gradin g. Dep ending on th e con text we w ill use the notation T or R ( M , N ) f or the bigraded T or-fu n ctor, or its ungraded EXTENDING COINV ARIANT THEOREM S 11 analog obtained b y taking the direct su m of the homogeneous comp onents T or R i ( M , N ) j for all i and j . It is p ossible to c ho ose the resolution F to b e minimal in the sense that the ranks β i of the resolve n ts F i ∼ = R β i are sim ultaneously all minimized; this turns out to b e equiv alen t to eac h diﬀerentia l d i ha ving en tries in R + = ⊕ i> 0 R i . In particular, w hen N = k = R/R + is the trivial R -mo dule, if the complex is minimal then F ⊗ R N b ecomes a complex of k -vec tor spaces with all zero diﬀerenti als, sho wing that β i = d im k T or R i ( M , k ) . The length of a minimal resolution is call ed the homological dimension hd R ( M ), th at is, h d R ( M ) := m in { i : T or R i ( M , k ) 6 = 0 } . Note that hd R ( M ) need not b e ﬁnite. Ho w ev er, Hilb ert’s syzygy the or em asserts that wh en R is a p olynomial algebra on n generators, one alw a ys has hd R ( M ) ≤ n . Giv en an N -graded k -vec tor space U = P d ≥ 0 U d , let start( U ) := min { d : U d 6 = 0 } . The u sual constru ction of a minimal fr ee R -resolution F of M (see e.g., the pro of of Pr op osition 2.3.1 (i) to follo w) shows that it en j o ys the pr op erty start( F i +1 ) > start( F i ). W e will sh o w in S ection 2.3 that a sim ilar p rop erty holds after incorp orating a ﬁnite group action. 2.2. T he Group Action on T or . W e start b y p oin ting out that for R ⋊ G - mo dules M and N (wh ere G is a group wh ic h acts on R b y graded k - algebra automorphism s) there are diagonal actions of R and G on M ⊗ R N and also on T or R i ( M , N ) making them into R ⋊ G -mo du les. Since R is comm u tativ e, we allo w our selv es to take the tensor pro d uct of t w o left R - mo dules. W e claim that for eac h g ∈ G th e map M × N → M ⊗ R N give n b y ( m, n ) 7→ g ( m ) ⊗ g ( n ) is R -balanced. T o establish th is w e m ust s ho w that for eac h r ∈ R w e hav e g ( r m ) ⊗ g ( n ) = g ( m ) ⊗ g ( r n ). This is so b ecause g ( r m ) ⊗ g ( n ) = g ( r ) g ( m ) ⊗ g ( n ) = g ( m ) ⊗ g ( r ) g ( n ) = g ( m ) ⊗ g ( r n ) . F r om this we obtain the diagonal action of R ⋊ G on M ⊗ R N . In fact for eac h g ∈ G we ha v e a natural transformation fr om the fun ctor − ⊗ R N : R ⋊ G -mo d → R -mo d to itself, giving a f unctor − ⊗ R N : R ⋊ G -mo d → R ⋊ G -mo d . W e next sho w that the diagonal action of G extends to an action on T or R i ( M , N ) for i > 0. Regard T or R i ( − , N ) as a functor R ⋊ G -mo d → R -mo d. F or eac h g ∈ G and i ∈ N w e constru ct a natural transformation η g ,i from T or R i ( − , N ) to itself so that th ese maps also commute with the b ound ary maps in the long exact sequences whic h arise f rom an y short exact sequence of R ⋊ G -mo dules 0 → M 1 → M 2 → M 3 → 0. F or i = 0 the n atural tr ansformation will b e the one already constru cted. T o extend this for arb itrary i , w e ma y take an y complex of pro jectiv e R ⋊ G -mo dules 12 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB P = ( · · · → P 2 → P 1 → P 0 → 0) which is acyclic except in degree zero where its h omology is M . Sin ce R ⋊ G is free as an R -mo d u le this is also a pro jectiv e resolution of M as an R -mo du le, so T or R i ( M , N ) = H i ( P ⊗ R N ). As th e action of G is by natural transformations of th e functor − ⊗ R N it passes to an action on the complex P ⊗ R N and hen ce to an action on its homology . T he v eriﬁcation that this action comm u tes with th e b oundary homomorphisms is a standard argum en t in homological algebra. W e ﬁn ally observe that if higher natural transf ormations exist with these prop erties, they m ust b e uniqu e. This f ollo ws fr om a homological degree shifting argum ent (cf [7, Ch ap ter I I I ]), since give n an y R ⋊ G -mo dule M one m ay tak e a s h ort exact sequence 0 → K → P → M → 0 where P is a pro jectiv e R ⋊ G -mo dule. This give s a long exact sequence 0 = T or R i ( P , N ) → T or R i ( M , N ) → T or R i − 1 ( K, N ) → T or R i − 1 ( P , N ) → · · · and the sp eciﬁcation of η g ,i − 1 on T or R i − 1 ( K, N ) and T or R i − 1 ( P , N ) (whic h is only non-zero for i = 1) d etermine the sp eciﬁca tion of η g ,i on T or R i ( M , N ) since th ese maps comm ute with the connecting h omomorphisms. Then the uniqueness implies that T or R i ( M , N ) b ecomes an R ⋊ G -modu le, since the relations on the action of G wh ic h are n eeded f or this hold if i = 0, h ence also for higher v alues of i . 2.3. T he Group Action on Resolutions. Give n these preliminaries we can state and pro v e our ﬁrst rationalit y resu lt. T his p rop osition can b e regarded as an equiv ariant generalization of the theorem of Hilb ert–Serre (see e.g., [31, Theorem 4.2]) on the rationalit y of Poincar ´ e series of graded No etherian mo dules ov er No etherian k -alg ebras. Prop osition 2.3.1. L et R b e a c ommutative N -gr ade d No etherian k algebr a, G is a gr oup which acts on R by gr ade d k - algebr a automorphisms, and M a No etherian R ⋊ G -mo dule. (i) Ther e exists an R -r esolution F = { F i } i ≥ 0 of M in which e ach r e solvent F i is not only a fr e e R - mo dule but also a k ( G ) -mo dule, the maps ar e k ( G ) -mo dule morphisms, and start ( F i +1 ) > start( F i ) . Conse qu ently, the inﬁnite sum X i ≥ 0 ( − 1) i [T or R i ( M , k )]( t ) gives rise to a wel l-deﬁne d element in R ( k ( G ))[[ t ]] . (ii) When k ( G ) is semisimple (i.e., |G | ∈ k × ), the r esolution in (i) c an in addition b e chosen minimal as an R -r esolution. (iii) If h d R ( M ) is ﬁnite, the r esolution in (i) or in (i i ) c an i n addition b e chosen with length hd R ( M ) . EXTENDING COINV ARIANT THEOREM S 13 (iv) In R ( k ( G ))[[ t ]] , the series [ R ]( t ) is invertible, and one has the fol lowing r elation: X i ≥ 0 ( − 1) i [T or R i ( M , k )]( t ) = [ M ]( t ) [ R ]( t ) . (v) A l l thr e e series [ R ]( t ) , [ M ]( t ) , X i ≥ 0 ( − 1) i [T or R i ( M , k )]( t ) in R ( k ( G ))[[ t ]] have the pr op erty that, for e ach simple k ( G ) -mo dule S , the p ower series in t c ounting the multiplicity of [ S ] actual ly lies in Q ( t ) . (vi) F or e ach simple k ( G ) -mo dule S , the c o eﬃcie nt se rie s of [ S ] in [ M ]( t ) and [ R ]( t ) have p oles at t = 1 of or der at most the Krul l dimension of R . Pr o of. (i): Because M is ﬁn itely-generated as an R -mo d ule, the quotien t M /R + M is a ﬁ nite-dimensional, graded k -vect or space. By the graded ve r- sion of Nak a y ama’s Lemma any homogeneous k -b asis { ¯ m j } for M /R + M lifts to a minimal h omogeneous generating set { m j } for M as an R -mo dule. Let U b e the k ( G )-submo d ule of M generated by any choi ce of suc h lifts { m j } . Then U is a graded ﬁnite-dimensional su bspace of M b ecause G is ﬁnite and acts in a degree-preserving fashion on M . Start a resolution F with the surjection F 0 := R ⊗ k U d 0 − → M , where r ⊗ u 7− → r u . Observe that the d iagonal action of k ( G ) on F 0 := R ⊗ k U is requ ir ed here, b oth to mak e d 0 a k ( G )-mod ule morp hism, and to mak e the R -mo dule structure on F 0 compatible with the k ( G )-mo dule s tructure of R . Observ e also that U ∼ = k ⊗ R F 0 , a relationship which w ill b e used in pr oving (iv). Replacing M b y ke r( d 0 ), we can iterate this p ro cess, and pro du ce the desired resolution F , p ro vided we can s ho w that the inequalit y start(k er( d 0 )) > µ := start( M ) = s tart( F 0 ) holds. How ev er, this f ollo ws easily from the observ ation that the restriction of the ab o v e map d 0 to the µ th homogeneous comp onents is the k -v ector space isomorph ism R 0 ⊗ k U µ = k ⊗ k U µ d 0 − → U µ = M µ , where 1 ⊗ u 7− → u , and hence ke r( d 0 ) is n onzero only in degrees strictly larger than µ . (ii): If k ( G ) is semisimple, then in th e construction of F in (i), th e k ( G )- submo d ule U ⊂ M is a direct sum mand so there is a k ( G )-mo d ule direct sum decomp osition M = U ⊕ R + M . Since U ∼ = M /R + M , iterating this construction w ill pro duce a minimal resolution. 14 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB (iii): T his w e pro v e by induction on hd R ( M ). In th e base case, i.e., wh ere hd R ( M ) = 0, M is a free R -mo dule and the assertion is trivial. In the inductiv e step, note that after one step of the constru ction in (i) or (ii), there is a s h ort exact sequence 0 → k er( d 0 ) → F 0 → M → 0 whose long exact sequence in T or R ( − , k ) sh ows that hd R (k er ( d 0 )) = hd R ( M ) − 1 . Applying th e indu ctiv e hyp othesis to k er( d 0 ) giv es the result. (iv): The fact that [ R ]( t ) lies in R ( k ( G ))[[ t ]] × follo ws from the assumption that R is a connected k -alg ebra with G acting trivially on R 0 = k . Th is means that th e ser ies expansion of [ R ]( t ) b egins [ 1 ] + [ R 1 ] t 1 + [ R 2 ] t 2 + · · · , and [ 1 ] is a u nit of R ( k ( G )). F or the remaining assertion, start with a resolution F of M p ro du ced as in (i). Th er e is the follo wing string of equalities in R ( k ( G ))[[ t ]], whic h are justiﬁed b elo w. [ M ]( t ) = X i ≥ 0 ( − 1) i [ F i ]( t ) = X i ≥ 0 ( − 1) i [ R ⊗ k ( k ⊗ R F i )]( t ) = X i ≥ 0 ( − 1) i [ R ]( t ) · [ k ⊗ R F i ]( t ) = [ R ]( t ) X i ≥ 0 ( − 1) i · [ k ⊗ R F i ]( t ) = [ R ]( t ) X i ≥ 0 ( − 1) i · [T or R i ( M , k )]( t ) The ﬁ rst equalit y comes from looking at the Euler c h aracteristic for the (ﬁnite) exact sequen ce in eac h h omogeneous comp onent. The s econd equalit y comes from the fact that F i is constru cted as R ⊗ k U i and U i ∼ = k ⊗ R F i , s o F i ∼ = R ⊗ k ( k ⊗ R F i ). The thir d equalit y comes fr om th e f act that in the isomorph ism F i ∼ = R ⊗ k ( k ⊗ R F i ) the actio n of G on the tensor pro duct on the righ t is diagonal, and this tensor pro d uct deﬁnes the pr o duct in R ( k ( G ))[[ t ]]. The four th equality is trivial. The ﬁ fth equalit y holds b ecause T or R ( M , k ) is the h omology of the com- plex F ⊗ R k . In eac h h omogeneous comp onen t ( F ⊗ R k ) d one has a ﬁ- nite complex of ﬁn ite-dimensional k -ve ctor spaces, an d the alternating su m F i ⊗ R k o v er i r ep resen ts the same element in R ( k ( G )) as the alternating sum of the T or R i ( M , k ) d in that comp onent (i.e., taking homology preserves Euler charact eristics). EXTENDING COINV ARIANT THEOREM S 15 (v): It su ﬃ ces to pr o ve the assertion for [ M ]( t ); one then tak es M = R to deduce it for [ R ]( t ), and uses (iv) to dedu ce it for P i ≥ 0 ( − 1) i [T or R i ( M , k )]( t ). T o pro v e [ M ]( t ) is inv ertible, one can reduce to the case where R is a p oly- nomial algebra A = k [ f 1 , . . . , f n ] w ith trivial G -action as follo w s . Note th at since R is No etherian, by a result of Emmy No ether R G is also No etherian. Hence by the No ether Normalizatio n Lemma, R G con tains a homogeneous system of parameters f 1 , . . . , f n , and we pu t A = k [ f 1 , . . . , f n ]. T he rin g extensions A ֒ → R G ֒ → R are b oth integ ral (i.e. mod ule-ﬁnite), and hen ce M is also a ﬁnitely generated A -mo d ule. In th is case, one can app ly (iv) to giv e (2.3.1) [ M ]( t ) =   X i ≥ 0 ( − 1) i [T or R i ( M , k )]( t )   · [ A ]( t ) . Note th at the su m on the righ t is ﬁnite, i.,e., lies in R ( k ( G ))[ t ], b ecause Hilb ert’s syzygy theorem says T or A ( M , k ) is ﬁnite dimensional. Note also that b ecause G acts trivially on A , one h as [ A ]( t ) = n Y i =1 1 1 − t deg( f i ) ! [ 1 ] , whic h is an elemen t of Q ( t ) times th e class [ 1 ] of the trivial mo d ule. Hence for an y s im p le mo dule S the series counting the m ultiplicit y of [ S ] h as Q ( t ) co eﬃcien ts. (vi): Again it suﬃces to p ro v e it for [ M ]( t ), and then tak e M = R to deduce it for [ R ]( t ). F or [ M ]( t ) it is implied b y equ ation (2.3.1 ) and th e comments af- ter it, as after forgetting the group action the sum P i ≥ 0 ( − 1) i [T or R i ( M , k )]( t ) has Z [ t ] co eﬃcients and the p ole at t = 1 in [ A ]( t ) is the Krull dimension of A . T his is the same as th e Kru ll dimension of R since A ֒ → R is an in tegral extension.  2.4. A Short Review of Brauer theory. At sev eral p oin ts w e will n eed facts ab out the Grothendieck r in g R ( k (Γ)) f or a ﬁnite group Γ which can b e con v eniently d educed from the th eory of Brauer c haracters. W e review this theory here and r efer to [26, Part I I I] for details. Let p d enote the charact eristic of the ground ﬁeld k . W e say th at an elemen t γ in Γ is p -regular if its order lies in k × . Let m b e th e least common m ultiple of the orders of all p -regular elemen ts of Γ, and let ξ b e a p rimitiv e m th ro ot of u nit y in some extension ﬁeld of k . Then every p -regular element γ in Γ acting on a ﬁnite-dimensional k ( G )-mo dule U has all its eigenv alues among the group of m th ro ots of unit y µ m ( k ( ξ ) × ) = h ξ i . Pic k a primitive complex m th ro ot of u nit y ˆ ξ ∈ C wh ic h will lift ξ , and consider the r esulting homomorphism 16 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB whic h lifts m th ro ots of un it y from k ( ξ ) to C : µ m ( k ( ξ ) × ) lift − → µ m ( C × ) ξ j 7− → ˆ ξ j . The Brauer character v a lue χ U ( γ ) for γ is then deﬁned to b e the sum of the lifts of the eigen v alues of g on U . F urthermore the ﬁeld k ( ξ ) is a splitting ﬁeld for Γ b y a theorem of Brauer (see e.g., [26, § 12.3, T heorem 24]). The Br au er charac ter χ U of a k (Γ)-mo d ule U determines the comp osition factors of U , and th is has sev eral imp ortant consequences. T o b egin with, the collection of restriction homomorphisms R ( k (Γ)) − → R ( k h γ i ) , where γ ranges ov er all p -regular elemen ts γ , determines elemen ts of R ( k (Γ)) uniquely; that is, the m ap R ( k (Γ)) → L γ R ( k h γ i ) is injectiv e. It implies also that whenever on e h as a ﬁeld extension k ֒ → K , the map R ( k (Γ)) ψ k,K − → R ( K (Γ)) that is indu ced by extension of scalars U 7→ K ⊗ k U is in j ectiv e, since the Brauer c haracter of a mo dule remains the same after extending scalars. S o to pr o ve an equ alit y in R ( k (Γ)) it suﬃces to prov e the equ alit y in R ( k h γ i ) for the p -regular elemen ts γ in Γ. If γ ∈ Γ is a p -r egular elemen t th en k h γ i is semisimple. O v er a splitting ﬁeld k ( ξ ) the simple k h γ i -mo d ules U j are all 1-dimensional and are ind exed b y j ∈ Z /d Z , where d is the order of γ , with γ acting as the scalar ζ j for some primitiv e d th ro ot of unity ζ in k × . An elemen t in R ( k h γ i ) is d etermined b y the (vir tu al) comp ositi on multiplicit ies of eac h U j . If this elemen t is of th e form [ U ] for some genuine k (Γ)-modu le U , then [ U ] w ill b e determined by the dimensions dim k ( U ⊗ k U j ) h γ i of its U j -isot ypic comp onen ts. Observe also that if one has a comm utin g action of another ﬁnite group Γ ′ , and one w ants to pr o ve an equalit y in R ( k (Γ × Γ ′ )), it suﬃces to pro v e it in R (( k h γ i × Γ ′ )) for eac h p -regular γ ∈ Γ. F urthermore, this can b e d one for gen u ine k (Γ × Γ ′ )-mo dules by pro ving equalit y in R ( k (Γ ′ )) for eac h isotypic comp onent . 2.5. T he Case of Domains wit h T rivial Gro up Ac tion. W e assume the n otations intro d uced in Pr op osition 2.3.1, an d in addition require that the graded k -alg ebra R b e an in tegral domain on wh ic h Γ-acts trivially . W e let K b e the fr action ﬁeld of R and recall from the d iscussion of Section 2.4 that extension of scalars giv es an inclusion of Grothendiec k rings ψ k ,K : R ( k (Γ)) → R ( K (Γ)). W e will mak e v arious assertions ab out isomorphism of K (Γ)-mo dules, but w here these mo du les are in fact deﬁned o ver k we ma y also d educe a corresp onding result for k (Γ) mo d u les(b ecause ψ k ,K is injectiv e) whic h we lea ve to the reader to formulate . EXTENDING COINV ARIANT THEOREM S 17 The main result in this section is an abs tract version of Theorem 1.1.1, from which Theorem 1.1.1 will immediately follo w. Before we state it, we present a lemma whic h will b e needed in the pro of. Lemma 2.5.1. L et M b e a ﬁnitely gener ate d gr ade d R (Γ) -mo dule, wher e R is a c ommutat ive gr ade d No etherian k -algebr a on which Γ acts trivial ly. Assume that R is an inte gr al domain and let R ′ ⊆ R b e a gr ade d subring over which R is inte gr al. L et the ﬁelds of fr actions of R and R ′ b e K and K ′ , r esp e ctively. Then the map ϕ : K ′ × M − → K ⊗ R M sending ( a, m ) ∈ K ′ × M to a ⊗ R m ∈ K ⊗ R M induc es an isomorphism of K ′ (Γ) -mo dules K ′ ⊗ R ′ M → K ⊗ R M . Note that her e K is r e g ar de d as a ( K ′ , R ) -bimo dule, and may even b e r e- gar de d as a ( K ′ (Γ) , R (Γ)) -bimo dule, on which Γ acts trivial ly. Pr o of. Note that the m ap ϕ is R ′ -balanced simp ly b ecause R ′ ⊂ R . Hence it ind uces a w ell-deﬁned map ϕ : K ′ ⊗ R ′ M → K ⊗ R M , whic h one can see is K ′ -linear and eve n a K ′ (Γ)-morphism. It remains to show th at ϕ is a K ′ -v ector space isomorp hism, w hic h is facilitat ed by ﬁrst observing that the K ′ -dimensions of the d omain and range are the same, viz., dim K ′  K ′ ⊗ R ′ M  = r ank R ′ ( M ) = r ank R ′ ( R ) · rank R ( M ) = [ K : K ′ ] · dim K ( K ⊗ R M ) = d im K ′ ( K ⊗ R M ) . Th us it suﬃces to sh o w ϕ is sur jectiv e. F or this one need only c h ec k for an y s, r ∈ R with r 6 = 0, and any m ∈ M , th at the decomp osable tensor s r ⊗ R m is in the image of ϕ . F or this w e us e that R is integral o v er R ′ , s o there is a d ep enden ce r n + a n − 1 r n − 1 + · · · + a 2 r 2 + a 1 r + a 0 = 0 with a i ∈ R ′ . W e ma y assu m e a 0 6 = 0, since one can divide by r in the domain R . Let m ′ := − ( r n − 1 + a n − 1 r n − 2 + · · · a 2 r + a 1 ) m, whic h is an element of M , satisfying r · m ′ = a 0 · m. Hence s r ⊗ R m = s r a 0 ⊗ R a 0 m = s r a 0 ⊗ R r m ′ = 1 a 0 ⊗ R sm ′ = ϕ  1 a 0 ⊗ R ′ sm ′  lies in the image of ϕ .  18 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB Theorem 2.5.2. L et M b e a ﬁnitely gener ate d gr ade d R (Γ) -mo dule, wher e R is a c ommutative gr ade d c onne cte d No etherian k -algebr a on which Γ acts trivial ly. Assume in addition that R is an inte gr al domain. (i) In R ( K (Γ)) we have [ K ⊗ k ( M ⊗ R k )] ≥ [ K ⊗ R M ] . F urthermor e, one has e qu ality if and only if M i s R -fr e e, in which c ase the K (Γ) -mo dule K ⊗ R M has a ﬁltr ation 0 = F − 1 ⊆ F 0 ⊆ F 1 ⊆ · · · ⊆ F d = K ⊗ R M so that for e ach j ther e is an isomorphism of K (Γ) -mo dules K ⊗ k ( M ⊗ R k ) j → F j /F j − 1 wher e ( M ⊗ R k ) j denote the j th homo gene ous c omp onent of M ⊗ R k . (ii) Mor e gener al ly, for any m ≥ 0 , in R ( k (Γ)) , m X i =0 ( − 1) i X j ≥ 0  K ⊗ k T or R i ( M , k ) j   ≥ [ K ⊗ R M ] if m is even, ≤ [ K ⊗ R M ] if m is o dd, and e quality holds if and only if h d R ( M ) ≤ m , that is, if and only if T or R i ( M , k ) v anishes for i > m . (iii) Even if hd R ( M ) is not ﬁnite, the formal p ower series X i ≥ 0 ( − 1) i  K ⊗ k T or R i ( M , k )  ( t ) of R ( K (Γ))[[ t ]] has the pr op erty that, for e ach simple K (Γ) -mo dule S , the p ower series in t giving the c o eﬃcient of [ S ] lies in Q ( t ) . Mor e over, t = 1 is a r e gular value for these r ational func tions, and X i ≥ 0 ( − 1) i  K ⊗ k T or R i ( M , k )  ( t )    t =1 = [ K ⊗ R M ] in R ( K (Γ)) . Observe in the s tatement of this r esu lt that the action of R on K coming from the inclusion R ⊆ K is not th e same as the action coming fr om the comp osite homomorph ism R → k ֒ → K , th us distinguishing K ⊗ R M f rom K ⊗ k ( M ⊗ R k ). Pr o of. W e b egin by proving (i). Let { e α } b e a min imal homogeneous R - spanning subset for M . Then { e α ⊗ 1 } forms a k -basis for M ⊗ R k , b y th e graded version of Nak ay ama’s Lemma. Hence { 1 ⊗ ( e α ⊗ 1) } forms a K -basis for K ⊗ k ( M ⊗ R k ). Also { 1 ⊗ e α } is a K -spannin g set for K ⊗ R M . Filter K ⊗ R M by letting F j b e the K -span of 1 ⊗ e α for whic h e α has degree at most j . The mo d u le K ⊗ k ( M ⊗ R k ) h as a direct su m d ecomp osition coming from its inh erited grading, and there is a comp osite mapping deﬁned by ( K ⊗ k ( M ⊗ R k )) j → F j → F j /F j − 1 1 ⊗ ( e α ⊗ 1) 7→ 1 ⊗ e α 7→ 1 ⊗ e α EXTENDING COINV ARIANT THEOREM S 19 where e α is assumed to ha v e d egree exactly j . Th is comp osite mappin g is a surjection of K (Γ)-mo dules. These su rjections show the in equalit y asserted in (i). O n e has equalit y if and only if all these su r jections are isomorph isms, that is, if and only if th e { 1 ⊗ e α } are a K -basis for K ⊗ R M , whic h hap p ens if and only if they are K -linearly indep end en t. Th is in turn h app ens if and only if the { e α } are R -linearly indep enden t, and hence an R -basis for M , so that M is fr ee ov er R . W e turn next to the p ro of of (ii) and (iii). I n b oth of these p ro ofs, it is con v enient to red uce to the case w here | Γ | lies in k × and hence k (Γ) is semisimple: Recall from Section 2.4 that virtual mo d u les in R ( k (Γ)) are determined by their restrictions to the cyclic su b groups generated by p - regular elemen ts γ ∈ Γ, so one may replace Γ with h γ i with ou t loss of generalit y . F or the pro of of (ii), s emisimplicit y of k (Γ) allo ws us to w rite do wn the ﬁrst m steps in a minimal R -free resolution of M as in Prop osition 2.3.1 (ii), so that all diﬀerentials are k (Γ)-module maps. Let L denote the kernel after the m th stage, so that one h as the exact sequence 0 → L → F m → F m − 1 → · · · → F 1 → F 0 → M → 0 . Applying the fu nctor K ⊗ R ( − ) is th e same as a localization, and hen ce giv es rise to an exact sequen ce, wh ose i th term for i = 0 , 1 , . . . , m lo oks lik e K ⊗ R F i ∼ = K ⊗ R R ⊗ k ( F i /R + F i ) ∼ = K ⊗ k T or R i ( M , k ) due to minimalit y of the r esolution. This sh o ws that ( − 1) m m X i =0 ( − 1) i [ K ⊗ k T or R i ( M , k )] − [ K ⊗ R M ] ! = [ K ⊗ R L ] ≥ 0 in K (Γ), w hic h giv es th e inequalit y in R ( K (Γ)) asserted in (ii), with equalit y if and only if K ⊗ R L = 0. Since L is a s u bmo du le of the free R -mo d u le F m , it is torsion-free as an R -mo dule and h ence K ⊗ R L = 0 if and only if L = 0. Minimalit y of the resolution then shows that v anishing of L (whic h is sometimes called the ( m + 1) st syzygy mo dule for M ) is equiv alen t to hd R ( M ) ≤ m . F or the p ro of of (iii), we n ote b y Hilb ert’s syzygy theorem that it holds when R is a p olynomial algebra as a sp ecial case of (ii). T o deduce th e general case of (iii), extend the ﬁeld k if necessary in order to pic k a graded No ether n orm alizatio n R ′ ⊆ R , that is a graded p olynomial su balgebra R ′ o ver which R is mo du le-ﬁnite, and let K ′ ⊆ K b e the associated extension of fraction ﬁelds with degree [ K : K ′ ]. W e will tak e adv an tage of the injectiv e ring homomorp h isms from Section 2.4 R ( k (Γ)) ψ k,K ′ ֒ → R ( K ′ (Γ)) ψ K ′ ,K ֒ → R ( K ′ (Γ)) R ( k (Γ))[[ t ]] ψ k,K ′ ֒ → R ( K ′ (Γ))[[ t ]] ψ K ′ ,K ֒ → R ( K ′ (Γ))[[ t ]] . 20 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB that arise b y extension of scalars in eac h case. Applying Prop osition 2.3.1 (iv) twice , one has in R ( k (Γ))[[ t ]] that X i ≥ 0 ( − 1) i [T or R i ( M , k )]( t ) = [ M ]( t ) [ R ]( t ) = [ M ]( t ) [ R ′ ]( t ) · [ R ′ ]( t ) [ R ]( t ) =   X i ≥ 0 ( − 1) i [T or R ′ i ( M , k )]( t )   · [ R ′ ]( t ) [ R ]( t ) . Applying th e map ψ k ,K ′ , one concludes that in R ( K ′ (Γ))[[ t ]] one has (2.5.1) X i ≥ 0 ( − 1) i [ K ′ ⊗ k T or R i ( M , k )]( t ) =   X i ≥ 0 ( − 1) i [ K ′ ⊗ k T or R ′ i ( M , k )]( t )   · ψ k ,K ′  [ R ′ ]( t ) [ R ]( t )  . The ﬁ rst factor on the r igh t in (2.5 .1) h as t = 1 as a regular v alue in R ( K ′ (Γ)), b ecause R ′ is a p olynomial algebra, and th e case already p r o ven sho ws that the v alue tak en there is [ K ′ ⊗ R ′ M ]. F or the s econd factor on the right in (2.5.1), n ote that b oth R ′ , R carr y trivial Γ -actions, and hence either [4, Lemma 2.4.1 (iii) on page 20] or [29, Prop. 5.5.2 on page 123] sho ws that this factor also has t = 1 as a regular v alue in R ( K ′ (Γ)), taking the v alue 1 [ K : K ′ ] [ 1 ]. Consequently , th e left side of (2.5.1) has t = 1 as a regular v alue in R ( K ′ (Γ)) with ev aluation X i ≥ 0 ( − 1) i [ K ′ ⊗ k T or R i ( M , k )]( t )    t =1 = 1 [ K : K ′ ] [ K ′ ⊗ R ′ M ] = 1 [ K : K ′ ] [ K ⊗ R M ] , where th e second equalit y uses Lemma 2.5.1. Applying the map ψ K ′ ,K , on e concludes that the elemen t X i ≥ 0 ( − 1) i [ K ⊗ k T or R i ( M , k )]( t ) in R ( K (Γ))[[ t ]] also has t = 1 as a regular v alue in R ( K (Γ)) and ev aluating giv es X i ≥ 0 ( − 1) i [ K ⊗ k T or R i ( M , k )]( t )    t =1 = 1 [ K : K ′ ] ψ K ′ ,K [ K ⊗ R M ] = 1 [ K : K ′ ] [ K ⊗ K ′ K ⊗ R M ] = [ K ⊗ R M ] . Here the last equalit y uses the fact that b oth K ′ , K carry trivial Γ-actions, and K ⊗ K ′ K ∼ = K [ K : K ′ ] .  EXTENDING COINV ARIANT THEOREM S 21 Remark 2.5.3. Simple examples sh ow that the assu m ption th at Γ act triv- ially on R in Theorem 2.5.2 cannot in general b e remov ed. F or example, consider the follo win g inclusions of algebras. C [ x 2 , y 2 ] | {z } R ′ ⊂ C [ x 2 , xy , y 2 ] | {z } R ⊂ C [ x, y ] | {z } M Let Γ = Z / 4 Z act compatibly on R ′ , R, M via the s calar substitution giv en b y x, y 7→ ix, iy (where i 2 = − 1 in C ), s o that their d th homogeneous comp onent s are scaled by i d . Then R ( k (Γ)) ∼ = Z [ α ] / ( α 4 − 1), where α represent s the class of the 1-dimensional k (Γ)-mo dule that is scaled by i . In R ( k (Γ))[[ t ]] there are equalities [T or R ( M , k )]( t ) = [ M ]( t ) [ R ]( t ) ( by Prop ositio n 2.3.1(iv) ab ov e) = [ R ′ ]( t )[T or R ′ ( M , k )]( t ) [ R ′ ]( t )[T or R ′ ( R, k )]( t ) ( again by Pr op osition 2.3.1(iv) ab o v e) = (1 + αt ) 2 1 + α 2 t 2 = 1 + 2 αt 1 + α 2 t 2 = 1 + α · 2 t 1 − t 4 + α 2 · 0 + α 3 · − 2 t 3 1 − t 4 and the last of these do es n ot ha v e t = 1 as a regular v alue. Note that here R is not ﬁxed b y Γ . 2.6. Pro of of Theorem 1.1.1. T o pro v e Th eorem 1.1.1, one applies Th e- orem 2.5.2 in the sp ecial case where R = k [ V ] G . Then K is its fr action ﬁeld k ( V ) G , and one h as the R -mo dule M = ( U ⊗ k k [ V ]) G , with action by Γ coming from the f act that U is a ( k (Γ) , k ( G ))-bimo du le. What remains to sho w in this situation is that [ K ⊗ R M ] = [ K ⊗ k U ] ∈ R ( K (Γ)) . In fact, these t w o K (Γ)-mo d ules are isomorphic as the follo wing string of equalities and isomorphisms pr ov es. K ⊗ R M (1) = k ( V ) G ⊗ k [ V ] G ( U ⊗ k k [ V ]) G (2) ∼ = ( U ⊗ k k ( V )) G (3) ∼ = ( U ⊗ k K G ) G (4) ∼ = K ⊗ k ( U ⊗ k k ( G )) G (5) ∼ = K ⊗ k U. 22 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB These may b e justiﬁed as follo ws. Equalit y (1) = is ju st a deﬁnition. The isomorphism (2) ∼ = is giv en by th e map k ( V ) G ⊗ k [ V ] G ( U ⊗ k k [ V ]) G → ( U ⊗ k k ( V )) G g h ⊗ X i ( u i ⊗ f i ) 7− → X i u i ⊗ f i g h . whose inv erse sends X i u i ⊗ g i h i = X i u i ⊗ ˆ g i h 7− → 1 h ⊗ X i u i ⊗ ˆ g i ! where ˆ g i , h are c hosen so that ˆ g i h = g i h i and h is G -in v ariant (e.g., choose h to b e the pro du ct of the ﬁnitely many images g ( h i ) as i v aries and as g v aries through the ﬁ nite group G ). T he isomorphism (3) ∼ = comes from th e Normal Basis Th eorem [18, Theorem VI I I.13 .1] applied to the Galois extension K = k ( V ) G ⊂ k ( V ) , whic h asserts k ( V ) ∼ = K ( G ) as k ( G )-mo dules. T he isomorp h ism (4) ∼ = comes from th e fact that G acts trivially on K = k ( V ) G . The isomorph ism (5) ∼ = comes fr om the k -linear m ap deﬁn ed by U → ( U ⊗ k k ( G )) G u 7− → X g ∈ G ug − 1 ⊗ t g where t g is the k -basis elemen t in k ( G ) indexed by g . This completes the pro of of T heorem 1.1.1.  2.7. Remarks on Group C ohomology and other relat ed C onstruc- tions. W e rein terpret some of the foregoing r esults and comment on their implications for the higher group cohomology H j ( G, U ⊗ k k [ V ]). Giv en a ﬁnitely generated graded integ ral domain R o v er the ﬁeld k and a ﬁnite group Γ, we let Γ act trivially on R and denote by R (Γ) -mo d the ab elian catego ry of ﬁnitely generated non -n egativ ely graded R (Γ)-mo dules M . T he assignment M 7→ T or R ( M , k ) is a fun ctor from the category R (Γ) - mo d to the category bigraded- k (Γ) -mo d of bigraded- k (Γ)-mo d ules th at are ﬁnite-dimensional in eac h bidegree. One can comp ose th is with the for- getful fu nctor bigraded- k (Γ) -mo d → graded- k (Γ) -mo d that forgets the in ternal grading b y taking the direct sum o v er it, and lea v es the homolog ical grading. Theorem 2.5.2(iii) asserts that the comp osite of these t wo fun ctors follo wed by taking th e alternating su m ov er the homological gradin g yields a well -deﬁned homomorp hism of Grothendiec k rings R ( R (Γ)) → R ( k (Γ)), whic h sends [ M ] 7→ ψ − 1 [ K ⊗ R M ], w here K is the ﬁeld of fractions of R and ψ is in duced from the inclusion k ֒ → K . EXTENDING COINV ARIANT THEOREM S 23 Next we place ourselv es in th e con text of Theorem 1.1.1 (iii) where R = k [ V ] G and we are considering the r elative invariants fun ctor in the form k (Γ × G ) -mo d → R (Γ) -mo d that sends U 7→ ( U ⊗ k k [ V ]) G . Then Th eo- rem 1.1.1 (iii) says that if we follo w this by the comp osite functor d escrib ed ab o v e, we get a we ll-deﬁned h omomorphism R ( k (Γ × G )) → R ( k (Γ)) whic h sends [ U ] 7→ [ U ], that is, it coincides with the restriction h omomorphism that simp ly forgets the k ( G )- structure. This is a b it surpr isin g, as the r elativ e inv ariant s fu n ctor is not exact; it is the case j = 0 of a family of (generally nontrivial) group cohomology functors U 7→ M j ( U ) := H j ( G, U ⊗ k k [ V ]) for j ≥ 0, which measure the inexactness of taking relativ e in v arian ts. In fact, it is not hard to sho w using the same ideas as ab o ve, that for any strictly p ositive j , if one follo ws M j ( − ) by th e comp osite fu nctor and tak es an alternating sum (i.e., Euler c haracteristic) as discussed ab o ve , the r esult induces the zer o homomorphism R ( k (Γ × G )) → R ( k (Γ)). That is, one has the follo wing conclusion. Theorem 2.7.1. L et G b e a ﬁnite sub gr oup of GL ( V ) , let R = k [ V ] G and let U b e any ﬁnite-dimensional k ( G × Γ) -mo dule. F or al l j ≥ 0 , let M j ( U ) := H j ( G, U ⊗ k k [ V ]) b e the j th c ohomolo gy gr oup of G with c o eﬃcients in U ⊗ k k [ V ] , c onsider e d as a R (Γ) -mo dule. Then for strictly p ositive j one has X i ≥ 0 ( − 1) i [T or R i ( M j ( U ) , k )]( t )      t =1 = 0 in R ( k (Γ)) Before p ro ving this, w e qu ote a standard fact ab out the b eha vior of group cohomology un der c hange of ground rings, whic h will also b e of u se fur ther on in the p ro of of Pr op osition 3.3.1. Prop osition 2.7.2. L et G b e a ﬁnite g r oup and let R → S b e a homomo r- phism of c ommutative rings which is ﬂat, so that M → S ⊗ R M i s an exact functor fr om R - mo dules to S -mo dules. Then for any R ( G ) -mo dule M , one has S ⊗ R H j ( G, M ) ∼ = H j ( G, S ⊗ R M ) as S -mo dules. Pr o of. See [13, Sec. 10.3, Prop. 7].  Pr o of of The or em 2.7.1. App lyin g Theorem 2.5.2(iii ), and th e r emarks in Section 2.4 concerning extension of scalars, it suﬃ ces for us to sh ow that K ⊗ R H j ( G, U ⊗ k k [ V ]) = 0 for j > 0 , 24 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB where K is the ﬁeld of fr actions of R . Note that K ⊗ R ( − ) is exact, b eca use it is a lo calization, so Prop osition 2.7.2 implies the ﬁr st isomorph ism in the follo wing strin g of isomorphisms and equalities. K ⊗ R H j ( G, U ⊗ k k [ V ]) ∼ = H j ( G, K ⊗ R ( U ⊗ k k [ V ])) ∼ = H j ( G, U ⊗ k k ( V )) ∼ = H j ( G, U ⊗ k K G ) ∼ = H j ( G, K ⊗ k ( U ⊗ k k ( G ))) = 0 The equalit y on the second line is by deﬁnition, while th e next three isomor- phisms ap p eared as (2) ∼ = , (3) ∼ = , (4) ∼ = in S ection 2.6. Th e last v anishin g assertion comes f r om the fact that U ⊗ k k ( G ) is alwa ys fr ee as a k ( G )-mo dule, and hence K ⊗ k ( U ⊗ k k ( G ))) is free as a K ( G )-mod ule, so its higher cohomology will v anish.  W e close this section by noting that whereas we ha v e b een dealing with the ﬁxed p oints of G and its d eriv ed functors, we could instead ha ve w ork ed with the ﬁxed quotients by th e action of G . Sp eciﬁcally , th e same argumen ts that we ha ve used in the pro of of Theorem 1.1.1 can also b e used if we replace M by M ′ := U ⊗ k ( G ) k [ V ], namely the ﬁ x ed quotien t of G acting on U ⊗ k k [ V ] rather than the ﬁ xed p oin ts wh ich we hav e used b efore. Of course, when | G | is inv ertible in k these constru ctions are isomorph ic, and so they are b oth v alid interpretations of the notion of relativ e inv ariant s in the mo dular case. With th is deﬁnition of M ′ the crucial chain of equations in the pr o of of Theorem 1.1.1 b ecomes K ⊗ R M ′ = k ( V ) G ⊗ k [ V ] G ( U ⊗ k ( G ) k [ V ]) = U ⊗ k ( G ) ( k [ V ] ⊗ k [ V ] G k ( V ) G ) ∼ = ( U ⊗ k ( G ) k ( V )) ∼ = ( U ⊗ k ( G ) K G ) ∼ = K ⊗ k ( U ⊗ k ( G ) k ( G )) ∼ = K ⊗ k U and this sho ws that it is v alid to replace M b y M ′ in the statemen t of Theorem 1.1.1. 3. Proof of Theorem s 1.2.1 and 1.3.1 Let u s recall the setting and statemen t of Theorem 1.3.1. W e sup p ose k is an arbitrary ﬁeld, G, Γ, and C ﬁ n ite groups, and V a ﬁnite dimensional ( k ( G ) , k ( C ))-bimo du le on whic h G acts faithfully , so G ⊂ GL ( V ). W e regard V , k [ V ] and k [ V ] G as trivial k (Γ)-mo dules. Supp ose there is a No ether normalization R ⊂ k [ V ] G that is s table u n der the action of C on k [ V ]. F urther su pp ose th at one has a vec tor v in V suc h EXTENDING COINV ARIANT THEOREM S 25 that the ﬁb er Φ v := φ − 1 ( φ ( v )) con taining v for the map φ : V → S p ec( R ) carries b oth a free (but not n ecessarily transitiv e) G -action and that this ﬁb er Φ v is stable und er the action of C . Denote b y m φ ( v ) the maximal ideal in R corresp onding to φ ( v ) in S p ec( R ) and in tro duce the co ord inate ring of Φ v namely , A (Φ v ) = k [ V ] / m φ ( v ) k [ V ], whic h is a k ( C )-mo dule. Let U b e a ﬁnite-dimensional ( k (Γ) , k ( G ))-bimo dule which we regard as a trivial k ( C )-mo dule. In this situation Th eorem 1.3.1 asserts that the relativ e inv ariants M := ( U ⊗ k k [ V ]) G satisfy th e equ ation X i ≥ 0 ( − 1) i X j ≥ 0  T or R i ( M , k ) j  =  ( U ⊗ k A (Φ v )) G  . in R ( k (Γ × C )). Note th at Hilb ert’s syzygy theorem tells us that the sum is ﬁn ite since R is a p olynomial algebra. In the subs ections that follo w we make v arious redu ctions leading up to the pro of of Theorem 1.3.1, with the goal of separating out the d iﬀeren t ideas inv olv ed . 3.1. Reduction 1: Remo ving the Γ -action. T h e inte n tion here is to pro v e the follo win g lemma. Lemma 3.1.1. The or em 1.3.1 fol lows fr om its sp e ci al c ase wher e Γ acts trivial ly on U ; that is, the c ase with only a C -action and no Γ -action. This will essential ly b e a consequence of the Brauer theory review ed in Section 2.4, app lied to the k (Γ)-mo d ules ( U ⊗ k k [ V ]) G and ( U ⊗ k A (Φ v )) G . Giv en a p -regular elemen t γ in Γ, let ˜ G := h γ i × G . F or an y ( k (Γ) , k ( G ))- bimo du le W , whic h we regard as a left k (Γ × G )- mo dule, its G -ﬁxed subspace W G is a semisimple k h γ i -mod ule, and one can express its k h γ i -isot ypic direct sum decomp ositi on in terms of ˜ G -ﬁxed sub spaces: (3.1.1) W G ∼ = M j ∈ Z /d Z ( W ⊗ k U j ) ˜ G If W happ ens also to ha v e an action of C commuting with the actions of Γ and G this b ecomes a direct sum of k ( C )-mo du les. Pr o of of L emma 3.1.1. Apply the p receding discuss ion to the ( k ( G ) , k (Γ))- bimo du les W = U ⊗ k A (Φ v ) and W = U ⊗ k k [ V ]. Using the fact that functors lik e tensor p ro du ct and T or commute with direct sums , along with the Brauer th eory from Section 2.4, one sees that Theorem 1.3.1 is equiv alen t to sh owing for eac h p -regular elemen t γ ∈ Γ (3.1.2) X i ≥ 0 ( − 1) i X j ≥ 0 h T or R (( ˜ U s ⊗ k k [ V ]) ˜ G , k ) j i = h ( ˜ U s ⊗ k A (Φ v )) ˜ G i in R ( k ( C )), where U s ranges o ver the simple k ( h γ i )-modu les and ˜ U s := U ⊗ k U s . Ho wev er, the equality (3.1.2) is an instance of Theorem 1.3.1 with trivial Γ-action: Simply rep lace G by ˜ G , and U by ˜ U s .  26 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB 3.2. Reduction 2: Replacing Sp ecial Fib er w ith Genera l Fib er. A k ey idea in the pro of of T h eorem 1.3.1 go es bac k to Borho and K raft [5], b e- fore that to Kostant [17], and p erhaps earlier: Given th e ﬁnite ramiﬁed co v er φ : V → Sp ec( R ), one should compare the group actions on the sp ecial ﬁ b er Φ 0 and its co ordinate ring k [ V ] /R + k [ V ] with the (often b ette r und er s to o d) actions on a m ore general ﬁb er Φ v and its co ordinate rin g k [ V ] / m φ ( v ) k [ V ]. T o set this up, w e return to the situation of Section 2 with an R -mo dule M and compatible ﬁnite group G acting as ab ov e. Then k = R /R + carries the trivial R -mo du le and k ( G )-action. Let k ′ b e a diﬀerent , p ossibly non- graded, R -mo d ule stru cture on the ﬁeld k . In other wo rds, k ′ = R / m ′ where m ′ is some (generally inhomogeneous) maximal id eal of R wh ic h happ ens to b e G -stable. Note that the action of G on k ′ remains trivial, sin ce k ′ is spanned by the image of 1, on whic h G acts trivially . Prop osition 3.2.1. L e t G b e a ﬁnite gr oup, R an N -gr ade d No etherian algebr a over the ﬁeld k and M a ﬁnitely gener ate d R -mo dule. Assume that b oth R and M have c omp atible G -actions and that h d R ( M ) is ﬁnite. Denote by m the (tautolo gic al) maxima l ide al R + of R (so R/ m ∼ = k is the tautolo gic al R -mo dule structur e on k ), and by m ′ some (p ossibly inhomo ge nous) maximal ide al of R which is G -stable. Set k ′ = R / m ′ . Then one has the fol lowing (u ng r ade d) e quality i n R ( k ( G )) of two (ﬁnite) sums: X i,j ≥ 0 ( − 1) i [T or R i ( M , k ) j ] = X i,j ≥ 0 ( − 1) i [T or R i ( M , k ′ ) j ] . Pr o of. Compute either of the tw o T or’s by starting with a ﬁn ite (but not necessarily minimal) free R -resolution F of M pro duced as in Prop osi- tion 2.3.1(iii), tensorin g ov er R w ith k or k ′ , and then taking the h omology of either F ⊗ R k or F ⊗ R k ′ . Eac h term F i ⊗ R k or F i ⊗ R k ′ is a ﬁnite- dimensional k -ve ctor space, and taking alternating sums in R ( k ( G )) giv es the follo wing equalities X i,j ≥ 0 ( − 1) i  T or R i ( M , k ) j  = X i ≥ 0 ( − 1) i [ F i ⊗ R k ] X i,j ≥ 0 ( − 1) i  T or R i ( M , k ′ ) j  = X i ≥ 0 ( − 1) i [ F i ⊗ R k ′ ] . Note the su ms are ﬁnite b ecause M is ﬁ nitely generated and hd R ( M ) is ﬁnite. It therefore suﬃces to sho w that as k ( G )-mod ules only (disregarding their R -mo du le str u cture), one has [ F i ⊗ R k ] = [ F i ⊗ R k ′ ] in R ( k ( G )), which we pro v e by a ﬁ ltration argumen t. Giv en a homogeneous R -basis { e α } for F i , one has ﬁltrations A , A ′ on the t wo k -v ector spaces F i ⊗ R k , F i ⊗ R k ′ deﬁned as follo ws: let A j , A ′ j b e the k -sp an of those k -basis elemen ts e α ⊗ R 1 in whic h deg( e α ) ≤ j . Since G acts in a grade-preserving fashion, it preserve s these ﬁltrations. W e claim that there is also a k ( G )-mo dule isomorphism A j / A j − 1 → A ′ j / A ′ j − 1 sending the k -basis elemen t e α ⊗ 1 to the k -basis EXTENDING COINV ARIANT THEOREM S 27 elemen t e α ⊗ 1. T o chec k that this isomorphism is G -equiv arian t, giv en g ∈ G , let g ( e α ) = P β r β ,α ( g ) e β for some homogeneous elemen ts r β ,α ( g ) in R . One then has, the same computation in either of A j / A j − 1 or A ′ j / A ′ j − 1 , g ( e α ⊗ 1) = X β :deg( e β )=deg( e α ) r β ,α ( g ) ( e β ⊗ 1) . Note that in this last sum, the coeﬃcient r β ,α ( g ) in R represents the same elemen t in the quotient ﬁelds k or k ′ , s in ce it m u st b e of degree zero by homogeneit y considerations.  W e comment that in th e last p art of the pro of of Pr op osition 3.2.1 we do not necessarily get an isomorphism of k ( G )-mo dules, as can b e seen by considering an example where R = F 2 [ x ] acted on trivially b y a cyclic group G of ord er 2. W e m a y tak e F i = Re 0 ⊕ R e 1 to b e free of rank 2, where e 0 lies in degree 0 and e 1 lies in d egree 1. L et G act on F i via the matrix  1 x 0 1  and let k ′ = R / ( x − 1). Then G acts on F ⊗ R k via  1 0 0 1  and on F ⊗ R k ′ via  1 1 0 1  . These tw o actions are non-isomorphic. The particular case of Prop osition 3.2.1 that will interest us most is where one has a ﬁnite group G ⊆ GL ( V ), with R ⊆ k [ V ] G an integral extension of graded algebras. Then giv en any ﬁnite-dimensional k ( G )-mo dule U , as in th e introdu ction, one can form the R -mo d ule U ⊗ k k [ V ], w ith d iagonal k ( G )-actio n, having G -ﬁxed su bspace M := ( U ⊗ k k [ V ]) G whic h retains the structure of an R -mo du le (b eca use G acts trivially on R ). Prop osition 3.2.2. L e t G b e a ﬁnite gr oup that acts on the ﬁnite dimen- sional k - v e ctor sp ac es V and R ⊂ k [ V ] G a No ether norma lization. Then for any ﬁnite dimensional k ( G ) -mo dule U the mo dule of r elative invariants M := ( U ⊗ k k [ V ]) G is ﬁnitely-gener ate d as an R - mo dule. Pr o of. Recall the to w er of integ ral extensions R ⊆ k [ V ] G ⊆ k [ V ] . Note that U ⊗ k k [ V ] is ﬁnitely-generated as a k [ V ]-mo du le, hence also ﬁnitely-generated as a k [ V ] G -mo dule. Hence it is a No etherian k [ V ] G - mo dule, and its k [ V ] G -submo d ule M = ( U ⊗ k k [ V ]) G will b e No etherian as we ll, so is ﬁ nitely-generated ov er k [ V ] G . But then M is also ﬁnitely- generated ov er R .  28 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB In the situation of Theorem 1.3.1, with R, M , Γ , C as deﬁned th ere, c h o ose k ′ = k v := R / m φ ( v ) . Note hd R ( M ) is ﬁnite and b ounded ab o v e by d im k ( V ) via Hilb ert’s syzygy theorem since R is a p olynomial algebra. Thus Prop osition 3.2.1 together with Lemma 3.1.1 s ho w that to prov e Theorem 1.3.1 redu ces to sho wing (3.2.1) X m ≥ 0 ( − 1) i X j ≥ 0  T or R i ( M , k v ) j  =  ( U ⊗ k A (Φ v )) G  in R ( k ( C )). The plan for proving (3.2.1) is pr ett y clear: T ry to pr o ve the v anishing result T or R i ( M , k v ) = 0 for i > 0 , and then try to iden tify T or R 0 ( M , k v ) ∼ = ( U ⊗ k A (Φ v )) G as k ( C )-mo dules. 3.3. Reduction 3: W orking Lo cally via Completions. Both parts in the aforemen tioned plan for pro ving (3.2.1) in Section 3.2 ha ve the adv anta ge that one can work with local and semi-local r ings, by taking completions with resp ect to m φ ( v ) -adic top ologies. Comp letion will b e s h o w n to commute with T or, reducing the v anishin g state men t T or R i ( M , k v ) = 0 for i > 0 to sh o w ing that th e completion ˆ M of M is free as a mo du le o ver the completion ˆ R of R . W orkin g lo cally also allo w s us to tak e adv an tage of the Chinese Remainder Theorem for semi-lo cal rings. W e b egin with a r eview of some prop erties of completion in a general setting. Muc h of this material can b e found in [1, Ch ap. 10], [10, C hap. 7], [19, § 8]. Let R b e a comm utativ e No etherian ring, M an R -mo dule, and l an y ideal of R . Denote by ˆ R the completion of R with resp ect to the l -adic top ology , an d ˆ M ∼ = ˆ R ⊗ R M the corr esp onding completion of M . Prop osition 3.3.1. L et R b e a c ommutative N o etherian ring, M an R - mo dule, l an ide al of R , and G a ﬁnite gr oup acting on M by R -mo dule homomo rphisms. Then taking ﬁxe d p oints c ommutes with l -adic c ompletion:  ˆ M  G ∼ = d M G that is, ( ˆ R ⊗ R M ) G ∼ = ˆ R ⊗ R M G as ˆ R -mo dules. Pr o of. Completion is an exact f unctor from R -mo dules to ˆ R -mo du les, so comm u tes w ith the cohomology functors H j ( G , − ) b y Pr op osition 2.7.2. In particular, th is holds for j = 0, the ﬁxed-p oin t f unctor ( − ) G .  EXTENDING COINV ARIANT THEOREM S 29 If m is a maximal ideal of R , then the ideal ˆ m := m ˆ R is also m aximal in ˆ R , and the t w o residue ﬁelds are the same, viz., k m = R / m = ˆ R/ ˆ m . Th us k m -v ector s paces can b e regarded as b oth R -mo dules and ˆ R -mo du les. F u rthermore, ˆ R is a lo c al ring. The n ext pr op osition exploits this to translate the v anishin g of T or i for all i > 0 into freeness of the completed mo d ule. Prop osition 3.3.2. L et m b e a maximal ide al of a c ommutative N o etherian ring R , and M an R -mo dule. Then as k m -ve ctor sp ac es one has T or ˆ R i ( ˆ M , k m ) ∼ = T or R i ( M , k m ) . Conse qu ently T or R i ( M , k m ) vanishes for al l i > 0 if and only if ˆ M is a fr e e ˆ R -mo dule. Pr o of. The asserted isomorphism is a consequence of the string of isomor- phisms that follo ws. T or ˆ R i ( ˆ M , k m ) ∼ = T or ˆ R i ( ˆ R ⊗ R ˆ M , ˆ R ⊗ R k m ) ∼ = ˆ R ⊗ R T or R i ( M , k m ) ∼ = T or R i ( M , k m ) The second of these u s es th e f act that ˆ R ⊗ R ( − ) is exact, and th e last that T or R ( M , N ) is an R/ Ann R ( N )-mo d ule, so a ve ctor space o ver k m = R / m = ˆ R/ ˆ m , and ˆ R ⊗ R k m = k m . Since ˆ R is a lo ca l rin g w ith residue ﬁeld k m , the ˆ R -mo du le ˆ M is free if and only if T or ˆ R i ( ˆ M , k m ) = 0 for i > 0. S o the previous isomorphism implies T or R i ( M , k m ) = 0 for all i > 0.  3.4. ( Semi-)lo cal Analysis of the Fib ers. Let k b e a ﬁeld and V a ﬁnite- dimensional k -vect or sp ace. Supp ose that G is a ﬁn ite subgroup of GL ( V ) and R ⊂ k [ V ] G a No ether normalization, so w e hav e inte gral extensions of graded algebras R ⊆ k [ V ] G ⊆ k [ V ] . Let U b e a ﬁn ite-dimensional ( k (Γ) , k ( G ))-bimo dule for some ﬁn ite group Γ with mo du le of relativ e inv ariant s M := ( U ⊗ k k [ V ]) G . S upp ose that there exists a ve ctor v ∈ V whose ﬁb er Φ v for the comp osite φ in the tow er of ramiﬁed co v erings V → V /G → Sp ec( R ) is p ermuted freely (bu t not n ecessarily transitive ly) by G . In this section w e let Γ act trivially on V and ignore all C -actions. The C -actions will b e put bac k in Section 3.5. F or an y w ∈ V , d enote by k w the k [ V ]-mo d u le s tructure on k deﬁned by f · α := f ( w ) α for all f ∈ k [ V ] and α ∈ k . Th e corresp ond ing maximal ideal of p olynomials v anish ing at w is denoted b y m w ⊂ k [ V ]. By restriction 30 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB w e also consider k w as m o dule o v er k [ V ] G , R , etc. T he n otations for the corresp ondin g m aximal ideals of k [ V ] G and R are m Gw = m w ∩ k [ V ] G m φ ( w ) = m w ∩ R. The ideal m Gw can also b e charact erized as those G -in v ariant p olynomials that v anish at the G -orbit Gw regarded as a p oin t of V /G . L ikewise, m φ ( w ) can b e charac terized as those p olynomials in R that v anish on th e w hole ﬁb er Φ w := φ − 1 ( φ ( w )) = { u ∈ V : φ ( u ) = φ ( w ) } , and the generators are easy to describ e: If R = k [ f 1 , . . . , f m ] then m φ ( w ) = ( f 1 − f 1 ( w ) , f 2 − f 2 ( w ) , . . . , f m − f m ( w )) R . Let ˆ R denote the complete lo cal ring obtained by completing R at the max- imal ideal m φ ( w ) . An imp ortan t p rop erty of completion, the C hinese remaind er theorem (see e.g., [10, C orollary 7.6], [19, Theorem 8.15]), gives cartesian p ro du ct decomp ositions of complete semi-lo cal rin gs. Thus in the con text ju st d e- scrib ed one has the follo w ing table in which the ﬁrst column lists v arious R -mo du les M , the second column lists th eir asso ciat ed semi-lo cal com- pleted ˆ R -mo du les ˆ M := ˆ R ⊗ R M along with Chinese remainder theorem isomorphisms, and the third column lists th eir quotien t k w -mo dules , viz., M / m φ ( w ) M ∼ = ˆ M / ˆ m φ ( w ) ˆ M . R -mo du le M ˆ R -mo dule ˆ M k w -mo dule M / m φ ( w ) M A := k [ V ] ˆ A ∼ = Q w ∈ Φ w ˆ A w A (Φ w ) B := k [ V ] G ˆ B (= ˆ A G ) ∼ = Q Gw ∈ Φ w /G ˆ B Gw B (Φ w /G ) R ˆ R k w T able 3.4.1: Ch inese Remainder T able The heart of the matter n o w lies in using this to prov e the follo wing lemma. Lemma 3.4.1. L et k b e a ﬁeld and V a ﬁnite-dimensional k -ve ctor sp ac e. Supp ose that G is a ﬁnite sub gr oup of GL ( V ) and R ⊂ k [ V ] G a No ether normaliza tion. A ssume that ther e is a v ∈ V whose ﬁbr e Φ v c arries a fr e e G - action. L et U b e a ﬁnite- dimensional ( k (Γ) , k ( G )) -bimo dule for some ﬁnite gr oup Γ with mo dule of r elative invariants M := ( U ⊗ k k [ V ]) G c onsider e d as a B (Γ) -mo dule in the notations of T able 3.4.1. Then with the notations of that table the fol lowing hold. (i) As ˆ B (Γ) - mo dules, one has isomorphisms ˆ M ∼ = ( U ⊗ k ˆ B ( G )) G ∼ = ˆ B ⊗ k ( U ⊗ k k ( G )) G . EXTENDING COINV ARIANT THEOREM S 31 In p articular, ˆ A ∼ = ˆ B ( G ) as ˆ B ( G ) -mo dules; this is the sp e cial c ase U = k ( G ) . (ii) ˆ M is a fr e e ˆ R -mo dule, and henc e T or ˆ R i ( ˆ M , k v ) = 0 for i > 0 . (iii) In R ( k (Γ)) ther e is an e quality X i ≥ 0 ( − 1) i X j ≥ 0 [T or R i ( M , k ) j ] = [ ˆ M ⊗ ˆ R k v ] . (iv) As k (Γ) - mo dules ther e ar e isomorphisms ˆ M ⊗ ˆ R k v ∼ = ( U ⊗ k A (Φ v )) G ∼ = ( U ⊗ k B (Φ v /G )( G ) ) G . In p articular, A (Φ v ) ∼ = B (Φ v /G )( G ) as B (Φ v /G )( G ) -mo dules; this is the sp e cial c ase U = k ( G ) . Pr o of. W e b egin with some preparations making use of the isomorphism ˆ A ∼ = Q w ∈ Φ v ˆ A w from T able 3.4.1. Here the r igh t side has comp onent wise m ultiplicatio n, and has a k ( G )-mo du le structure giv en via th e isomorphisms ˆ A w g → ˆ A g w whic h p ermute the factors. As a consequence of the assumption that G acts freely on Φ v , th ere is a decomp osition of th e ﬁ b er Φ v = Gw 1 ⊔ · · · ⊔ Gw r in to free G -orbits. This give s tw o wa ys to r egroup the factors, viz., ˆ A ∼ = Y g ∈ G r Y i =1 ˆ A g w i ! | {z } := ˆ A g and ˆ A ∼ = r Y i =1   Y g ∈ G ˆ A g w i   | {z } := ˆ A Gw i . Note that here ˆ A g = g ( ˆ A e ) w here ˆ A e = Q r i =1 ˆ A w i . View ˆ A as an ˆ A e -algebra and ˆ A Gw i as a ˆ A w i -algebra via the diagonal em b eddings ˆ A e ∆ − → Q g ∈ G ˆ A g ∼ = ˆ A ˆ A w i ∆ − → Q g ∈ G ˆ A g w i ∼ = ˆ A Gw i deﬁned by mapping a to the p ro du ct ( g ( a )) g ∈ G . These emb eddings give ring isomorphisms of ˆ A e and ˆ A Gw i on to their diagonal images, wh ic h are exactly the G -in v arian t subrin gs of the relev ant completions, or the completions of the relev ant G -in v ariant su brings by Prop osition 3.3.1. T o wit, (3.4.1) ˆ A e ∼ = ∆( ˆ A e ) ∼ = ˆ A G ∼ = ˆ B ˆ A w i ∼ = ∆( ˆ A w i ) ∼ = ( ˆ A Gw i ) G ∼ = ˆ B Gw i Next consider the group algebras ˆ A e ( G ) and ˆ A w i ( G ) for the group G , with either ˆ A e or ˆ A w i as co eﬃcien ts; in either case, denote b y t g the basis elemen t 32 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB corresp ondin g to the elemen t g in G . W e claim th at there are isomorphisms of ˆ A e ( G )-mod u les and ˆ A w i ( G )-mod u les (3.4.2) ˆ A e ( G ) α − → Q g ∈ G ˆ A g ∼ = ˆ A ˆ A w i ( G ) α − → Q g ∈ G ˆ A g w i ∼ = ˆ A Gw i deﬁned in b oth cases by at g 7− → g ( a ) e g where e g is the stand ard basis vec tor/idemp oten t corresp ond ing to the factor in the pro duct indexed by g , and w here a lies either in ˆ A e or ˆ A w i (so that g ( a ) lies either in g ( ˆ A e ) = ˆ A g or g ( ˆ A w i ) = ˆ A g w i ). The in v erse isomorphism α − 1 in either case is d eﬁned b y ( a ( g ) ) g ∈ G = X g ∈ G a ( g ) e g 7− → X g ∈ G g − 1 a ( g ) t g . With this preparation, w e can start to pr o ve the assertions of the Lemma, b eginning w ith assertion (i). In light of the ﬁrs t isomorphism ˆ A e ∼ = ˆ B in (3.4.1) the ﬁrst isomorphism in (3.4.2 ) shows ˆ A ∼ = ˆ B ( G ) as a ˆ B ( G )-mo du les. Since G acts trivially on R , us ing Prop ositio n 3.3.1 again, one has ˆ M := ˆ R ⊗ R (( U ⊗ k A ) G ) ∼ = ( ˆ R ⊗ R ( U ⊗ k A )) G ∼ = ( U ⊗ k ˆ A ) G ∼ = ( U ⊗ k ˆ B ( G )) G ∼ = ˆ B ⊗ k ( U ⊗ k k ( G )) G as ˆ B -mo dules. How ev er, these are also ˆ B (Γ)-mo d ule isomorp h isms since the Γ-action o ccurs enti rely in the U f actor and acts trivially on R, B , A and their completions. F or (ii), it suﬃces to sh o w that ˆ M is a f ree ˆ R -mo dule, and then to app ly Prop osition 3.3.2(ii). S in ce (i) implies ˆ M is a free ˆ B -mod ule it suﬃces to sho w that ˆ B is a free ˆ R -mo du le. F rom T able 3.4.1, one has ˆ B ∼ = Q r i =1 ˆ B Gw i , and hence it is enough to verify that eac h ˆ B Gw i is a free ˆ R -mo dule. Note that the ring ˆ B Gw i is a ﬁn ite extension of the ring ˆ R . It turns out that b oth of these are regular lo cal r ings, b ecause they are isomorphic to completions of p olynomial algebras at m aximal ideals ([10, Corollary 19.14 ]): in the case of ˆ R this is due to the assumption that R is p olynomial, and in the case of ˆ B Gw i this is d ue to the second isomorphism ˆ B Gw i ∼ = ˆ A w i of (3.4.1 ). Hence according to th e Au s lander-Buc hsbaum Theorem [10 , Th eorem 19.9], using dim( − − ) to indicate Kru ll d imension of − − , we obtain hd ˆ R ( ˆ B Gw i ) = dim( ˆ R ) − depth ˆ R ( ˆ B Gw i ) = d im( ˆ R ) − dim( ˆ B Gw i ) = d im( ˆ R ) − dim( ˆ R ) = 0 . EXTENDING COINV ARIANT THEOREM S 33 The second equalit y here is due to the fact that r egular lo cal r ings are Cohen- Macaula y , so their depth and Krull dimension are the same, while the third follo ws fr om the fact th at ˆ B Gw i is a ﬁnite extension of ˆ R . Thus ˆ B Gw i is ˆ R -free, and hen ce so is ˆ M . F or (iii), note that the sum is ﬁn ite by Hilb ert’s Syzygy Theorem. In R ( k Γ), one then has the follo win g equalities, justiﬁed b elo w: X i ≥ 0 ( − 1) i X j ≥ 0 [T or R i ( M , k ) j ] = X i ≥ 0 ( − 1) i X j ≥ 0 [T or R i ( M , k v ) j ] = X i ≥ 0 ( − 1) i X j ≥ 0 [T or ˆ R i ( ˆ M , k v ) j ] = [T or ˆ R 0 ( ˆ M , k v )] = [ ˆ M ⊗ ˆ R k v ] . The ﬁ rst equalit y comes from applying Prop osition 3.2.1 with k ′ = k v , the second fr om Prop osition 3.3.2 with k m = k v , the third fr om assertion (ii) ab o v e, and the last from th e deﬁn ition of T or 0 . F or (iv), n ote that ˆ M ⊗ ˆ R k v ∼ = ˆ M / ˆ m φ ( v ) ˆ M ∼ =  U ⊗ k ˆ A ) G  / ˆ m φ ( v )  U ⊗ k ˆ A ) G  ∼ =  U ⊗ k  ˆ A/ ˆ m φ ( v ) ˆ A  G ∼ =  U ⊗ k  A/ ˆ m φ ( v ) A  G =: ( U ⊗ k A (Φ v )) G . This giv es the ﬁ r st isomorph ism in (iv). F or the second, note th at ˆ A/ ˆ m φ ( v ) ˆ A ∼ = ˆ B ( G ) / ˆ m φ ( v ) ˆ B ( G ) ∼ = ( ˆ B / ˆ m φ ( v ) ˆ B )( G ) ∼ = ( B / ˆ m φ ( v ) B )( G ) = B (Φ v /G )( G )  3.5. F inishing the pro ofs: Incorp orat ing the C -Action. Assertions (ii) and (iii) of Lemma 3.4.1 complete the pro of of Theorem 1.3.1, except that we ha v e n ot y et accoun ted f or the C -action. W e rectify th is here, and explain how Theorem 1.3.1 imp lies T heorem 1.2.1. So we assu me there is also a ﬁ nite subgroup C ⊂ GL ( V ), comm u ting with G , that preserves R and the maximal ideal m φ ( v ) , making C act on the ﬁb er Φ v . One then has compatible C -actions on R ⊆ B = k [ V ] G ⊆ A = k [ V ] ˆ R ⊆ ˆ B ⊆ ˆ A k v ⊆ B (Φ v /G ) ⊆ A (Φ v ) . 34 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB W e wish to describ e these actio ns more explicitly under the assu mption that the G -orbits { Gw i } r i =1 in Φ v /G are all regular. Note that C p erm utes these G -orbits, sin ce it comm utes with G , and if c ∈ C stabilizes some G -orbit Gw i , th en there will b e a unique element g c,w i ∈ G for which (3.5.1) cw i = g c,w i w i . One chec ks that this element g c,w i dep end s only on the choice of the r ep re- sen tativ e w i for the orbit Gw i up to conju gacy as f ollo ws. First g c,hw i = hg c,w i h − 1 . Ho wev er, once a c hoice of r epresen tativ e w i is made, one has cg w i = g cw i = g g c,w i w i for all g ∈ G . Also recall that for eac h i = 1 , 2 , . . . , r , there is an isomorp h ism (see (3.4.1) of § 3.4) of the completed lo cal rings ˆ A w i ∼ = ˆ B Gw i , which are ﬁ n ite extensions of the lo cal ring ˆ R m φ ( v ) . Let B ( Gw i ) := B Gw i / m φ ( v ) B Gw i ∼ = ˆ B Gw i / ˆ m φ ( v ) ˆ B Gw i ∼ = ˆ A w i / ˆ m φ ( v ) ˆ A w i ∼ = A w i / m φ ( v ) A ( w i ) . This ring is a ﬁn ite-dimensional k v -v ector space; it ma y b e viewed either as the co ordinate ring for the (p ossibly non-redu ced) structure on the ﬁb er Φ v lo cal to the p oin t w i , as a subscheme of V , or for th e s tr ucture on the orbit space Φ v /G lo cal to the orbit Gw i , as a subscheme of V /G . Lemma 3.5.1. Assume the notation and hyp otheses of L emma 3.4.1. F ur- ther assume, as in this se ction, that ther e is a ﬁnite sub g r oup C ⊂ GL ( V ) which pr eserves the No ether normalization R and the ﬁb er Φ v , Cho ose for e ach G -orbit Gw i within Φ v a r epr esentative w i , and deﬁne g c,w i in G via (3.5.1) whenever cGw i = Gw i . Then the isomorph ism of B (Φ v /G )( G ) -mo dule asserte d in p art (iv) of that lemma, viz., ˆ M ⊗ ˆ R k v ∼ = ( U ⊗ k A (Φ v )) G ∼ = ( U ⊗ k B (Φ v /G )( G ) ) G . is also a k ( C ) -mo dule isomorp hism, obtaine d by using the k ( C ) -mo dule structur e induc e d fr om the fol low ing isomorphisms: A (Φ v ) ∼ = B (Φ v /G )( G ) ∼ = r Y i =1 B Gw i ( G ) . If c ∈ C has cGw i = Gw j for j 6 = i , then ther e is a ring isomorphism B Gw i c → B Gw j that extends to a ring isomorphism B Gw i ( G ) c → B Gw j ( G ) . EXTENDING COINV ARIANT THEOREM S 35 F or c ∈ C with cGw i = Gw i , the ring automorph ism B Gw i c → B Gw i extends to a ring automorphism B Gw i ( G ) c − → B Gw i ( G ) at g 7− → c ( a ) t g g c,w i . Conse qu ently, ther e is the fol lowing identity r elating Br auer char acter val- ues: (3.5.2) χ ˆ M ⊗ ˆ R k v ( c ) = X i : cGw i = Gw i χ B Gw i ( c ) · χ U ( g − 1 c,w i ) . Pr o of. The assertions will b e derived by passin g to the qu otien t b y m φ ( v ) from the analogous s tatement f or the k ( C )-mo dule str uctures on the com- pleted rings ˆ A, ˆ B , etc. Note th at the Chinese Remainder T heorem isomorphism ˆ A ∼ = Y w ∈ Φ v ˆ A w translates the C -action on ˆ A to a C -action b y isomorph ism s ˆ A w c → ˆ A c ( w ) p ermuting the factors on th e r igh t. F rom th is, an d the isomorphisms ˆ B ∼ = r Y i =1 ˆ B Gw i ˆ A ∼ = ˆ B ( G ) ∼ = r Y i =1 ˆ B Gw i ( G ) , it is straigh tforward to c hec k the assertions for the case when c ∈ C has cGw i = Gw j for j 6 = i . If c ∈ C has cGw i = Gw i , then there is an automorphism c of ˆ A Gw i = Q g ∈ G ˆ A g w i , whic h acts by isomorphisms ˆ A g w i c → ˆ A g g c,w i w i b et w een the com- p onents. This action of c translates ov er to ˆ A w i ( G ) usin g the isomorph ism α fr om equation (3.4.2 ) in th e pro of Lemma 3.4.1 whic h send s at g α 7→ g ( a ) e g . So c sends g ( a ) e g to cg ( a ) e g g c,w i , and this maps u nder α − 1 to ( g g c,w i ) − 1 cg ( a ) t g g c,w i = g − 1 c,w i g − 1 cg ( a ) t g g c,w i = g − 1 c,w i c ( a ) t g g c,w i , where the last equalit y uses th e fact that C and G comm u te w ithin GL ( V ). Th us c ( at g ) = g − 1 c,w i c ( a ) t g g c,w i . F r om here one can translate the C -action to ˆ B Gw i ∼ = Q g ∈ G ˆ A g w i using the diagonal embed d ing ˆ A w i ∆ → ˆ B Gw i that maps a 7→ ∆( a ) = ( g ( a )) g ∈ G . One then ﬁn d s that g − 1 c,w i c ( a ) ∆ 7→ c (∆( a )). In other words, a t ypical elemen t bt g in ˆ B Gw i ( G ) is sent by c to c ( b ) t g g c,w i , as claimed. The assertion ab out Brauer c haracters is then a consequence of Lemma 3.5.2 wh ic h follo ws.  36 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB Lemma 3.5.2. Given an element g 0 in the ﬁnite gr oup G and a ﬁnite- dimensional k ( G ) -mo dule U , let a cyclic gr oup C = h c i act on U via c ( u ) := g − 1 0 ( u ) , and let C act on ( k ( G ) ⊗ k U ) G via c   X g ∈ G t g ⊗ u g   := X g ∈ G t g g 0 ⊗ u. Then ( k ( G ) ⊗ k U ) G ∼ = U as k ( C ) -mo dules, and c onse quently, if g 0 is p -r e gular, the Br auer char acter value for c acting on ( k ( G ) ⊗ k U ) G is χ ( k ( G ) ⊗ U ) G ( c ) = χ U ( g − 1 0 ) . Pr o of. The map U − → ( k ( G ) ⊗ k U ) G u 7→ X g ∈ G t g ⊗ g ( u ) is easily chec k ed to give the desired k ( C )-mo dule isomorphism .  Pr o of of The or em 1.2.1. Assum e that k [ V ] G is p olynomial, and c in G is a regular elemen t, so that c ( v ) = ζ v for s ome vecto r v w h ose G -orbit Gv is free. W e wish to apply Th eorem 1.3.1 with R = k [ V ] G , so that Φ v consists of only the regular G -orbit Gv (that is, r = 1 and w 1 = v is the r epresen tativ e of the uniqu e G -orbit on Φ v ). In th is case, the lo cal rings ˆ R φ ( v ) = ˆ B Gv are the same, and their quotien t B ( Gv ) by the maximal ideal m φ ( v ) = m Gv is the ﬁeld k v ∼ = k . Thus as k ( G )-mo dules, one h as A (Φ v ) ∼ = k ( G ), and Theorem 1.3.1 implies that X i ≥ 0 ( − 1) i X j ≥ 0  T or R i ( M , k ) j  = [( k ( G ) ⊗ k U ) G ] = [ U ] as k (Γ)-mo d ules, where M = ( U ⊗ k k [ V ]) G , for an y ﬁn ite-dimensional k ( G )- mo dule U and ﬁnite subgrou p Γ of Aut k ( G ) U . W e wish to also tak e into accoun t the action of a cyclic group C = h τ i ⊂ Aut k ( G ) V whose generator τ = c − 1 acts as the scalar ζ − 1 on V . Then τ scales V ∗ b y ζ , and acts on the graded rings and mo d ules R = k [ V ] G , k [ V ] , M := ( U ⊗ k [ V ]) G b y the scalar ζ j in their j th homogeneous comp onent, exactly as in the C -action considered in Theorem 1.2.1. Note that τ acts on Φ v = Gv b y τ ( v ) = c − 1 ( v ) and more generally τ ( g ( v )) = ζ · g ( v ) = g c − 1 ( v ). Thus Lemma 3.5.1 shows that the k ( C )-structure on A (Φ v ) ∼ = k ( G ) has τ ( t g ) = t g c − 1 , in agreemen t with the k ( G × C )-structure on k ( G ) that app eared in Springer’s Theorem, and L emma 3.5.2 then shows that the k (Γ × C ) structure on U agrees with the on e that app ears in Theorem 1.2.1.  EXTENDING COINV ARIANT THEOREM S 37 3.6. I nduced Mo dules and the Pro of of C orollary 1.2.2. W e would lik e to app ly Th eorems 1.2.1 and 1.3.1 to m o dules M whic h are (relativ e) in v ariants f or a su bgroup H of G , r ather than f or G itself; in p articular, we w ould lik e to dr a w conclusions ab out th e H -inv arian t subring k [ V ] H , as in Corollary 1.2.2. This is made p ossible by a suitable notion of in duction of k ( H )-mo du les to k ( G )-mo du les. Regard k ( G ) as a k ( G × H )-mo du le via the action ( g , h ) · t g ′ := g g ′ h − 1 . Then giv en any ﬁnite-dimensional k ( H )-mo dule W , deﬁne its induced k ( G )-mo d ule to b e Ind G H ( W ) := Hom k ( H ) ( k ( G ) , W ) = { f ∈ Hom k ( k ( G ) , W ) : f ( t g h − 1 ) = h ( f ( t g )) } . That this is a k ( G )-mo dule follo ws from the equalit y ( g · f )( t g ′ ) = f ( t g − 1 g ′ ). W e next explain h ow this construction conv erts relati v e in v ariants for G into relativ e inv ariants for H . Prop osition 3.6.1 (F rob enius Recipro cit y) . F or ﬁnite-dimensional k ( G ) - mo dules V and k ( H ) -mo dules W , ( V ⊗ k Ind G H ( W )) G ∼ = (Res G H ( V ) ⊗ k W ) H . Pr o of. This is a standard isomorphism which follo ws f rom [26, § 3.3, Lemma 1] or [3, 3.3.1, 3.3.2 ], giving ( V ⊗ k Ind G H ( W )) G ∼ = (Ind G H (Res G H ( V ) ⊗ k W )) G ∼ = (Res G H ( V ) ⊗ k W ) H . as required.  W e will b e particularly intereste d in the situation w here the k ( H )-mo du le W is the trivial mo dule, so that In d G H ( W ) = Ind G H ( k ) is the p erm utation mo dule for G on the cosets H \ G , and the H -relativ e in v ariants are simply the H -inv arian ts, namely , ( V ⊗ k W ) H = ( V ⊗ k k ) H = V H . In this situation, the isomorp hism in P rop osition 3.6.1 b ecomes a k (Γ)-mo du le isomorphism for the action of the group Γ := N G ( H ) /H . This is a sp ecial case of the follo wing resu lt. Prop osition 3.6.2. Consider neste d sub gr oups H ⊂ L ⊂ G of a ﬁnite gr oup G , and assume that W = Res L H ( f W ) for some ﬁnite-dimensional k ( L ) -mo dule f W . Then (i) The gr oup Γ := N L ( H ) /H acts on U := Ind G H ( W ) v ia ( γ · f )( t g ) := γ ( f ( t g γ )) for γ ∈ N L ( H ) . (ii) F or f W 6 = 0 this action is faithful, that is, it inje cts Γ into Aut k ( G ) ( U ) . 38 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB (iii) The isomorphism ( V ⊗ k Ind G H ( W )) G ∼ = (Res G H ( V ) ⊗ k W ) H . of Pr op osition 3.6.1 is also a k (Γ) -i somorph ism, assuming one lets k (Γ) act as fol lows: • on the left, solely in the f actor I n d G H ( W ) , while • on the right, diagonal ly in b oth factors of Res G H ( V ) ⊗ k W . Pr o of. It is s traigh tforward to c hec k that the deﬁ nition in (i) describ es an action of N L ( H ) on Ind G H ( W ) w ith H acting trivially , and that th is actio n comm u tes w ith the k ( G )-mo du le str u cture. T o see that f W 6 = 0 imp lies the action is faithful, p ic k an y ˜ w 6 = 0 in f W and consider the elemen t f of In d G H ( W ) deﬁned by f ( t g ) = ( g − 1 ( w ) if g ∈ H, 0 otherwise. An element γ in N L ( H ) send s f to the function ( γ · f )( t g ) = ( γ (( gγ ) − 1 ( w )) = g − 1 ( w ) if g γ ∈ H , 0 otherwise. Hence γ · f = f if and only if γ lies in H . It is also not hard to c h ec k that the isomorphism giv en in the p ro of of Prop osition 3.6.1 is k (Γ)-equiv ariant for the actions d escrib ed.  W e n ext provide th e pro of of our m ain application, Corollary 1.2.2, r e- solving in the aﬃrmativ e b oth Conjecture 3 and Qu estion 4 in [23]. Pr o of of Cor ol lary 1.2.2. R ecall that th e ground ﬁ eld k is arbitrary and H ⊂ G are t w o n ested ﬁnite subgroup s of GL ( V ) where V is a ﬁ nite dimens ional k -v ector space. W e ha v e assu med that the ring of in v ariants k [ V ] G is a p olynomial algebra, so it is its o wn No ether normalization and may b e tak en as R in the results established in this s ection. T he group C = h c i is a cyclic subgroup generated by a regular elemen t c in G , with eigen v alue ζ on some regular eigen v ector in V . W e set Γ := N G ( H ) /H and ha v e give n k [ V ] H the k (Γ × C )-structure in w hic h c s cales the v ariables in V ∗ b y ζ , and Γ acts b y linear sub stitutions. What we must p ro v e is that in R ( k (Γ × C )) (3.6.1) X i ≥ 0 ( − 1) i X j ≥ 0 h T or k [ V ] G i ( k [ V ] H , k ) j i = [ k ( H \ G )] . So ignoring the action of Γ this will imply that X ( t ) =  k [ V ] H  ( t ) [ k [ V ] G ] ( t ) ∈ Z [[ t ]] , is a p olynomial in t , and the action of C on the set X = H \ G then giv es a triple ( X , X ( t ) , C ) th at exhib its the cyclic sieving phenomenon of [24]. EXTENDING COINV ARIANT THEOREM S 39 Sp eciﬁcally , f or an y elemen t c j in C , the cardinalit y of its ﬁxed p oint subset X c j ⊂ X is giv en by ev aluating X ( t ) at a complex ro ot-of-unity ˆ ζ j of the same multiplic ativ e order as c j , viz., | X c j | = [ X ( t )] t = ˆ ζ j . In this con text, w e apply Th eorem 1.2.1, together with the r esu lts of this section, taking f W = k to b e the trivial k ( G ) -mo dule. Th en W := Res G H ( f W ) = k is the trivial k ( H )-mo du le, and U := Ind G H ( W ) = Ind G H k = k ( H \ G ) where k ( H \ G ) carries th e k (Γ × C )-mo dule str u cture as describ ed in the statemen t of Corollary 1.2.2. S o we h a ve M = ( U ⊗ k k [ V ]) G ∼ = ( k ⊗ k k [ V ]) H = k [ V ] H , and unrav eling th e notations, Th eorem 1.2.1 tells u s th at equalit y (3.6.1) holds in R ( k (Γ × C )) as required. If we ignore the Γ-action and compare Brauer c haracter v alues for eac h elemen t c − j in C on the t wo sides of equalit y (3.6.1) we ﬁnd, on th e left side X i ≥ 0 ( − 1) i h T or k [ V ] G ( k [ V ] H , k ) i ( t )      t = ˆ ζ j =  k [ V ] H  ( t ) [ k [ V ] G ] ( t )      t = ˆ ζ j = X ( ˆ ζ j ) , where ˆ ζ is the Brauer lift to C × of ζ ∈ k × , wh ile on the righ t it is th e n um b er of ﬁxed p oin ts f or c j p ermuting the elemen ts of the set X = H \ G .  4. Character V alues and Hilber t s eries W e provi de here fur ther con text and applications of Th eorem 1.3 .1, mo d- elled on certain v ariations of Molien’s th eorem, wh ic h we ﬁrst r ecall. 4.1. Molien’s theorem and B rauer character v alues. Molien’s theo- rem is m ost u sually s tated in c haracteristic zero, bu t it is w ell known that it works equally in p ositiv e characte ristic, using Brauer c haracters instead of ordinary c haracters of repr esen tatio ns. W e r ecall a v ersion of this result that works in all charac teristics. In our statemen t we supp ose that a certain k ( G )-mo d ule U is p r o jectiv e. Observe that in charac teristic zero this is no restriction b eca use all mo d ules are p ro jectiv e. In p ositiv e c h aracteristic p the Brauer charac ter of a k ( G )-mo dule is only deﬁ ned on the p -regular el- emen ts of G , that is, th e elemen ts whose order is prime to p . So th at our statemen t w orks in all c haracteristics, w e describ e these elemen ts as the ones whose ord er is inv ertible in k . T h e theorem has to d o with the Hilb ert series for the mo d u le of U -relativ e inv ariants ( U ⊗ k k [ V ]) G , wher e V is a k ( G )- mo dule, and b y a standard isomorph ism this mo dule of relativ e in v ariants is isomorphic to Hom k ( G ) ( U ∗ , k [ V ]). When U = P S is the p r o jectiv e co ver of a simp le k ( G )-mo dule S (in characte ristic 0 this means U ∼ = S ) the Hilb ert series of this mo dule is thus the comp osition factor m ultiplicit y series ∞ X n =0 [ k [ V ] n : S ∗ ] dim End k ( G ) ( S ) t n . 40 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB Prop osition 4.1.1. (Molien; se e [20] § 1 Pr op osition 1.2 and formula (1.5)) L et V b e a ﬁnite dimensional k ( G ) -mo dule and U a ﬁnite dimensional pr o- je ctive k ( G ) -mo dule with Br auer char acter values χ U ( g ) . Deﬁne χ U ( g ) = 0 when the or der of g is not invertible in k . Then one has Hilb(( U ⊗ k k [ V ]) G , t ) = 1 | G | X g ∈ G χ U ( g ) det(1 − tg − 1 ) . The next resu lt is an imm ed iate corollary of Molien’s th eorem, app earing implicitly in the discussion of S pringer [33, (4.5)]. It describ es a situation where information ﬂ o w s in th e other direction, that is, one can compute Brauer charac ter v alues for k ( G )-mo du les from kn o w ledge of the Hilb ert series of their relativ e in v ariants. T o state it, giv en a graded algebra R and graded R -mo dule M , as in Section 2, deﬁ ne X M ,R ( t ) = Hilb( M , t ) Hilb( R, t ) = X i ≥ 0 ( − 1) i Hilb(T or R i ( M , k ) , t ) . where the second equ alit y comes from Pr op osition 2.3.1(iv). Corollary 4.1.2. L e t V b e a ﬁnite dimensional k ( G ) -mo dule and U a ﬁnite dimensional pr oje ctive k ( G ) -mo dule with B r auer c har acter χ U . L et g ∈ G b e such that the or der of g is inv e rtible in k and let ζ b e an element of k × such that the multiplicity of ζ as an eige nvalue of e lements of G is achieve d uniquely by the c onjug acy class of g . L et ˆ ζ denote a Br auer lift of ζ . Then the value of the Br auer char acter χ U at g is giv en by (4.1.1) χ U ( g ) = X M ,R ( ˆ ζ ) , wher e M := ( U ⊗ k k [ V ]) G is c onsider e d as a mo dule over the invariant ring R := k [ V ] G . Pr o of. Let n := d im k V , and let g ha ve eigen v alues ζ 1 , . . . , ζ n on V , with ζ 1 = · · · = ζ m = ζ . If c d enotes the cardinalit y of the conjugacy class of g in G , and if d := Q n j = m +1 (1 − ˆ ζ − 1 ˆ ζ j ), then Molien’s th eorem implies that the Lauren t exp ansions ab out t = ˆ ζ for the Hilb er t series of M = ( U ⊗ k k [ V ]) G and R = k [ V ] G b egin s imilarly: Hilb( M , t ) = c · χ U ( g ) | G | · d · (1 − t ˆ ζ − 1 ) m + O 1 (1 − t ˆ ζ − 1 ) m − 1 ! Hilb( R, t ) = c | G | · d · (1 − t ˆ ζ − 1 ) m + O 1 (1 − t ˆ ζ − 1 ) m − 1 ! . Consequent ly , their qu otien t X M ,R ( t ) has no p ole at t = ˆ ζ , and its v alue at t = ˆ ζ is χ U ( g ).  W e p oin t out t wo old and one new application of Corollary 4.1.2. EXTENDING COINV ARIANT THEOREM S 41 Example 4.1.3 Sprin ger’s original theorem [33, (4.5 )] asserts that th e com- plex c haracter v alues of a regular elemen t g in a ﬁnite complex reﬂ ection group alw a ys ob ey equation (4.1.1). In his pr o of he veriﬁed that the h y- p otheses of Corollary 4.1.2 hold in this situation, that is, the conju gacy class of a regular elemen t u niquely ac hiev es the maxim um eigen v alue multiplici t y for any of its eigen v alues corresp onding to regular eigen v ectors. This should b e compared with the sp ecia l case of T heorem 1.2.1 in w h ic h one f orgets the Γ-action, and lo oks only at the Brau er c haracter v alues for the C -action: it says exactly that, for arbitr ary mo dules U , w hen g is a regular element of a group G that has k [ V ] G p olynomial, equation (4.1.1) still holds for the Brauer characte r v alue χ U ( g ). Example 4.1.4 Sprin ger [33 , Assertion 2.2(iii)] observ ed that the conclu- sion of Corollary 4.1.2 applies to ev ery elemen t g in the binary ic osahe dr al gr oup , the largest of th e three sp oradic ﬁ n ite subgroups of S L 2 ( C ). Ev ery elemen t g in this group has t w o eigen v alues ζ , ζ − 1 , and these eigen v alues happ en to determine the conjugacy class of g un iquely . Example 4.1.5 W e show Corollary 4.1.2 applies to the simple group G of order 168, via its realization as the ind ex tw o sub group G 24 ∩ S L 3 ( C ) inside the complex reﬂection group G 24 from the list of Sheph ard and T o dd [28]; see Sp ringer [33, § 4.6] for details on this realization. Letting α, i, β denote primitive complex ro ots-of-unit y of orders 3 , 4 , 7, resp ecti v ely , the six conju gacy classes within G h a ve these eigen v alues, orders of elemen ts, and sizes: eigen v alues order size of conju gacy class (1 , 1 , 1) 1 1 (1 , − 1 , − 1) 2 21 (1 , i, − i ) 4 42 (1 , α, α 2 ) 3 56 ( β , β 2 , β 4 ) 7 24 ( β 3 , β 5 , β 6 ) 7 24 T able 4.1.1: Conju gacy classes in the simple group of order 168 Scrutiny of the ﬁr st column of this table shows th at every g in G has at least one eigen v alue ζ to whic h Corollary 4.1.2 app lies. Hence ev ery g in G h as its complex c haracter v alues χ U ( g ) satisfying equation (4.1.1). 4.2. A mo dular version. As w ith Example 4.1.3 and Th eorem 1.2.1, one w ould sometimes lik e to d ed uce resu lts lik e Corollary 4.1.2 f or the Brauer c h aracter v alues χ U ( g ) of all mo dules U in p ositi v e charact eristic, not just 42 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB the p ro jectiv e mo du les. W e derive here on e su c h general result from Th eo- rem 1.3.1, and then app ly it to the 3-mo dular redu ction of the group G of order 168 considered in Examp le 4.1.5. Prop osition 4.2.1. L et G b e a ﬁnite sub gr oup of GL ( V ) , and g a r e gular element of G with r e gular eigenve ctor v and asso ciate d eigenvalue ζ . L et U b e a k ( G ) -mo dule and denote its Br auer char acter by χ U : G → C . L et ˆ ζ ∈ C × b e a Br auer lift of ζ . Assume ther e exists a gr ade d No ether normalization R ⊂ k [ V ] G having the fol lowing pr op erties. 2 (i) G acts fr e ely on the ﬁb er Φ v = φ − 1 ( φ ( v )) c ontaining v for the ﬁnite map V φ → S p ec( R ) . (ii) F or every G -orbit Gw i in Φ v that has ζ Gw i = Gw i , the unique element g ζ ,w i ∈ G for which ζ w i = g ζ ,w i w i has the same Br auer char acter value χ U ( g ζ ,w i ) = χ U ( g ) . (iii) X k [ V ] G ,R ( ˆ ζ ) 6 = 0 . Then χ U ( g ) = X M ,k [ V ] G ( ˆ ζ ) . wher e M = ( U ⊗ k k [ V ]) G is c onsider e d as a mo dule over k [ V ] G and over R . Observe th at the h yp otheses imply g ζ − 1 ,w i = g − 1 ζ ,w i , hence χ U ( g − 1 ζ − 1 ,w i ) = χ U ( g ). Pr o of. First note that Prop osition 2.3.1(iv) implies X M ,R ( t ) = X M ,k [ V ] G ( t ) · X k [ V ] G ,R ( t ) and hence (4.2.1) X M ,R ( ˆ ζ ) = X M ,k [ V ] G ( ˆ ζ ) · X k [ V ] G ,R ( ˆ ζ ) . Consider the cyclic group C ⊂ Aut k ( G ) V whose generator g acts as the scalar ζ on V , and let τ = g − 1 as in the pro of of Th eorem 1.2.1. Th en τ scales V ∗ b y ζ , and acts on the graded r ings and mo dules R ⊂ k [ V ] G , k [ V ] , M := ( U ⊗ k [ V ]) G b y the scalar ζ j on their j th homogeneous comp onen t. Hyp othesis (i) allo w s us to apply L emm a 3.5.1. 2 These hypoth eses wo uld hold if k [ V ] G w ere p olyn omial and one to ok R = k [ V ] G , as in Theorem 1.2.1, but w e do not assume this here. EXTENDING COINV ARIANT THEOREM S 43 Therefore (4.2.2) X M ,R ( ˆ ζ ) = X i ≥ 0 ( − 1) i χ T or R i ( M ,k ) ( τ ) = χ ˆ M ⊗ ˆ R k v ( τ ) = X i : ζ Gw i = Gw i χ B Gw i ( τ ) · χ U ( g − 1 τ ,w i ) = χ U ( g ) · X i : ζ Gw i = Gw i χ B Gw i ( τ ) where the ﬁrst equ ality uses the deﬁn ition of the scalar action on R, M ; the second equalit y is a consequence of Lemma 3.4.1 (iii); sub stituting c − 1 for τ (to conform to th e notatio ns of th e pro of of Lemma 3.5.1) and remem b ering τ scales by ζ on V ∗ the third equalit y b ecomes equ ation (3.5.2) of that lemma; and the last equ alit y uses hyp othesis (ii) ab ov e and the observ ation made b efore th e p ro of, b earing in m ind that g and τ act in v ers ely on v . In p articular, taking U = k th e trivial k ( G )-mo du le in (4.2.2) giv es (4.2.3) X k [ V ] G ,R ( ˆ ζ ) = X i : ζ Gw i = Gw i χ B Gw i ( τ ) . Putting together equations (4.2.1), (4.2.2), and (4.2.3) , one obtains (4.2.4) X M ,k [ V ] G ( ˆ ζ ) · X k [ V ] G ,R ( ˆ ζ ) = X M ,R ( ˆ ζ ) = χ U ( g ) · X k [ V ] G ,R ( ˆ ζ ) Since b y h yp othesis (iii) X k [ V ] G ,R ( ˆ ζ ) is a nonzer o factor of the extreme left and r igh t terms of this string of equalities, we may divid e by it yielding the desired equalit y .  Example 4.2.2 W e apply Prop ositio n 4.2.1 to the 3-mo d ular reduction of the s imple group of order 168 that was considered in Ex amp le 3 of sec- tion 4.1 This application also highligh ts the ﬂexibilit y of using v arious d if - feren t No ether normalizations R inside k [ V ] G . As preparation, we consider the 3-mo d ular reduction of th e complex re- ﬂection group G 24 considered b y Shephard and T o d d [28]. K emp er and Malle [16, § 6, p. 74] describ e a realization of it as a s ubgroup (whic h w e also denote G 24 ) of GL 3 ( F 9 ) h a v in g ord er 336. Th eir realization is generated b y three reﬂections, eac h of order t w o, an d eac h unitary with r esp ect to the nondegenerate sesquilinear form on V := ( F 9 ) 3 deﬁned b y ( x, y ) = P 3 i =1 x i ¯ y i where ¯ y := y 3 denotes the F rob eniu s automorph ism of F 9 . Th ey also ex- hibit explicitly three G 24 -in v ariant p olynomials f 4 , f 6 , f 14 of degrees 4 , 6 , 14 resp ectiv ely , with the prop ert y that their Jacobian determinan t f 21 is non- v anishing. Since 4 · 6 · 14 = 336 = | G 24 | a criterion of Kemp er [9, Theorem 3.7.5] implies that k [ V ] G 24 = k [ f 4 , f 6 , f 14 ] 44 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB is a p olynomial algebra for an y extension k of F 9 . The sub group G := G 24 ∩ S L 3 ( F 9 ) has index tw o in G , and is isomorphic to the simple group of ord er 168. Since G 24 con tains th e scalar transformation τ := − 1 V , one has G 24 = G × h τ i . T o analyze the rin g of G -in v ariants, we ﬁ rst analyze the m o dule of det- relativ e inv ariants for G 24 , that is, k [ V ] G 24 , det := (det ⊗ k [ V ]) G 24 where det : G 24 → {± 1 } d enotes th e determinan t c h aracter of G 24 . Because G 24 is generated by inv olutiv e r eﬂections, none of whic h are transv ectio ns, the Jacobian d eterminan t f 21 is the pr o duct of the linear form s deﬁning the reﬂecting h yp erplanes for the 21 r eﬂections in G 24 ; see Bro er [6] or Hartmann and Shepler [14]. One also kno ws th at f 21 is a det-relativ e inv ariant for G . Con v ersely , an y det-relativ e in v ariant is divisible by eac h linear form d eﬁning one of the reﬂecting hyp erplanes, and so m ust also b e d ivisib le by f 24 . Thus k [ V ] G 24 , det = f 21 k [ V ] G 24 is a free k [ V ] G 24 -mo dule of r ank one. W e claim that this implies th e follo w ing decomp ositio n of th e G -in v ariant ring (4.2.5) k [ V ] G eve n = k [ V ] G 24 k [ V ] G odd = k [ V ] G 24 , det . where k [ V ] G eve n , k [ V ] G odd denote the su bspaces sp anned by h omogeneous G - in v ariants of even, o d d d egree. T o see th is, n ote that an y of the generating in v olutive r eﬂections σ for G 24 has σ τ lying in the sub group G . Hence any homogeneous elemen t of f in k [ V ] G , sa y of of degree d , will b e ﬁxed by σ τ and thus s atisfy σ ( f ) = τ ( f ) = ( − 1) d f = ( f if d is ev en , det( σ ) f if d is o d d . Since these reﬂections σ generate G 24 , the d ecomp osition in (4.2.5) follo ws. Consequent ly , k [ V ] G = k [ f 4 , f 6 , f 14 ] ⊕ f 21 · k [ f 4 , f 6 , f 14 ] , and Hilb( k [ V ] G , t ) = 1 + t 21 (1 − t 4 )(1 − t 6 )(1 − t 14 ) . W e wish to use this information to d educe the analogue of Example 1 in sec- tion 4.1 for all 3-mod ular Brauer c h aracters of G . Assume that the extension k of F 9 is alge braically closed. The conjugacy classes of G are describ ed just as in T able 4.1.1, except that in this case one must • r e-interpret i, β as ro ots-of-unit y of orders 4 , 7 in k × , EXTENDING COINV ARIANT THEOREM S 45 • r e-interpret the cub e r o ot-of-unit y α as 1, since the conjugacy class of elements of order 3 are n ot 3-regular and act u nip otent ly . Prop osition 4.2.3. L et G ⊂ S L 3 ( F 9 ) b e the 3 -mo dular r e duction of the simple gr oup of or der 168 . L et k b e an algebr aic al ly close d extension of F 9 , and for any 3 -r e gular element g of G , let ζ in k × b e any eig e nvalue for g having the same multiplic ative or der as g . Then for any k ( G ) -mo dule U one has the Br auer char acter value χ U ( g ) = X M ,k [ V ] G ( ˆ ζ ) wher e M := ( U ⊗ k [ V ]) G , and ˆ ζ is the Br auer lif t of ζ . Pr o of. W e wish to app ly Prop osition 4.2.1 with v c hosen generically from the ζ -eigenspace for g . W e will u se t w o diﬀeren t No ether norm alizatio ns R inside k [ V ] G , dep ending u p on th e ord er of the 3-regular element g . F or elements g of ord er 2, so that ζ = − 1, we u se the sub ring R = k [ f 4 , f 6 , f 21 ] whic h we claim is a No ether normalization. T o c hec k this claim, one must sho w that the only solution to the system (4.2.6) f 4 ( v ) = f 6 ( v ) = f 21 ( v ) = 0 is the v ector v = 0. F or this, w e b orr o w the idea of S pringer [33 , 3.2(i)]. Giv en any solution v to (4.2.6) , scaling v b y a 14 th ro ot-of-unit y ξ in k × giv es a vec tor ξ v w ith the p rop erty that f ( ξ v ) = f ( v ) for f = f 4 , f 6 , f 21 , as w ell as for f = f 14 . Hence f ( ξ v ) = f ( v ) for eve ry f in k [ V ] G , so that ξ v and v must represent th e same p oint in the quotien t space V /G . Thus g ( v ) = ξ v for some g in G . But if v 6 = 0, then suc h an elemen t g would ha v e order divisible by 14, an d there are no su c h elements of G . Hence v = 0. F or this c hoice of R , one has X k [ V ] G ,R ( t ) = 1 + t 14 + t 28 so that X k [ V ] G ,R ( − 1) = 3 6 = 0. Hence hyp othesis (iii) of Prop osition 4.2.1 will b e s atisﬁed. W e claim that hypothesis (i) of Prop ositio n 4.2.1 will b e satisﬁed as long as v is c hosen generically from the 2-dimensional ( − 1)- eigenspace for g inside V . T o see this, note that Φ( v ) = φ − 1 ( φ ( v )) will carry a free G -action as long as v a v oids the union Y := [ h ∈ G : h 6 =1 φ − 1 ( φ ( V h )) . F or eac h h 6 = 1 in G , the ﬁxed sub space V h is of dimen s ion at most 1. Since φ : V → Sp ec( R ) is a ﬁnite morph ism, eac h set φ − 1 ( φ ( V h )) is conta ined in a 1-dimensional algebraic su bset of V = k 3 . Since the union Y is ﬁn ite, Y is also con tained in a 1-dimensional algebraic s et. Th us the v ector v chosen generically ins ide the 2-dimensional ( − 1)-eigenspace will indeed a void Y , since k is algebraically closed (and hence inﬁn ite). Lastly , hyp othesis (ii) of 46 ABRAHAM BROER, VICTOR RE INER, LA R R Y SMITH, AND PETER WEBB Prop osition 4.2.1 w ill b e satisﬁed: all elemen ts g ζ ,w i for th e v arious orbits Gw i in Φ v ha v e the same order 2 as ζ , and all elements of G of ord er 2 are conjugate, so they are all conju gate to g . F or elements g of ord er 1 , 4 , or 7, we u se th e No ether normalization R = k [ V ] G 24 = k [ f 4 , f 6 , f 14 ] X k [ V ] G ,R ( t ) = 1 + t 21 . Note th at X k [ V ] G ,R ( ˆ ζ ) 6 = 0 for such elements g , s o hyp othesis (iii) of Prop o- sition 4.2.1 will b e satisﬁed. Since G 24 = G × h τ i , the ﬁb er Φ v decomp oses int o the tw o G -orbits Φ v = G 24 v = Gv ⊔ − Gv . T o sho w that G acts freely on Φ v , and thereby satisﬁes hypothesis (i) of Prop osition 4.2.1, we consider three cases dep ending up on the order of g . If g has order 1, sin ce v is c hosen generically within all of V , ther e is no problem. If g h as ord er 7, n ote that f 4 ( v ) = f 6 ( v ) = 0 since any f in k [ V ] G whic h is homogeneous of degree d will hav e f ( v ) = f ( g ( v )) = f ( ζ v ) = ζ d f ( v ) . If in add ition, Φ v = G 24 v do es not carry a fr ee G -action, then it carries a non-free G 24 -action, and a theorem of S erre [27] implies th at v lies on some reﬂecting h yp erplane for G 24 , that is, f 21 ( v ) = 0. Hence v is a solution to equation (4.2.6), so v = 0, a con trad iction. If g h as order 4, w e argue dir ectly why the isotropy subgroup G v := { h ∈ G : h ( v ) = v } m ust b e trivial. Firstly , there can b e n o elements h in G v order 7, as suc h elemen ts in G ha ve n o eigenv alues equal to 1. Secondly , if G v con tained an elemen t of eve n order, then withou t loss of generalit y it wo uld con tain some h of order 2. Unitarit y of h would imp ly that h acts on the 2-dimens ional p erp endicular space v ⊥ as the scalar − 1, while unitarit y of g implies that g acts on v ⊥ with its other eigen v alues 1 , − ζ . Thus g and h would comm ute, and g h would hav e eigen v alues ζ (on v ) and − 1 , ζ (on v ⊥ ). As th er e are no elemen ts of G with these eigen v alues, G v can con tain no element s of ev en order. Lastly , this forces G v to b e a 3-torsion group, and since | G | is not divisible by 9, either G v is trivial or G v = { 1 , h, h − 1 } for some h of order 3. But th e latter cannot o ccur : wh en h g i = { 1 , g , g 2 , g 3 } acts on G v b y conjugation, one ﬁnds th at g hg − 1 = h ± 1 , so that g 2 comm u tes with h an d g 2 h wo uld b e an element of order 6 in G . Finally , n ote that sin ce Φ v con tains only the tw o G -orbits Gv , − Gv , one has g ζ , − v = g ζ ,v = g , and hence hyp othesis (ii) of Prop ositio n 4.2.1 is alwa ys satisﬁed.  EXTENDING COINV ARIANT THEOREM S 47 Referen ces [1] M.F. Atiyah and I .G. Macdonald, Intro duction to commutative algebra. Addison- W esley Publishing Co., R eading, Mass.-London-D on Mills, On t. 1969 [2] M. Au slander, I . Reiten and S. S malø, Representation Theory of Artin Algebras. Cambridge studies i n advanc e d mathematics 36 . Cam b rid ge U niv. Press, 1995. [3] D.J. Benson, Representations and cohomology I. Cambridge Studies in A dvanc e d Mathematics 30 , Cambridge Univ. Press, 1991. [4] D.J. Benson, Pol ynomial inv ariants of ﬁnite groups. L ondon M ath. So ciety L e ctur e Notes 190 , Cambridge Univ. Press, 1993. [5] W. Borho and H. Kraft. ¨ Ub er Bahnen un d deren Deformationen bei linearen Aktionen reduktiver Grupp en. Comment. Math. Helvetici 54 (1979), 61-104 . [6] A. 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Gruen b erg, Cohomological topics in group th eory , L e ctur e Notes i n Math. 143 , Springer-V erlag, Berlin-Heidelb erg, 1970. [14] J. Hartmann and A. V . Shepler, Jaco bians of reﬂection groups. T r ans. Amer. Math. So c. 360 (2008), 123– 133 [15] R. Kane, Reﬂection groups and inv arian t theory . C MS Bo oks i n Mathematics/ O u- vr ages de Math´ ematiques de la SMC 5 . Springer-V erlag, N ew Y ork, 2001. [16] G. Kemp er and G. Malle, The ﬁ nite irreducible linear groups with p olynomial ring of in v ariants. T r ansform. Gr oups 2 (1997), 57–89. [17] B. Kostant, Lie group represen tations on p olynomial rings. Amer. J. Math 85 (1963), 327–404 . [18] S. Lang, Algebra, Gr aduate T exts in Mathematics 211 . S pringer-V erlag, N ew Y ork, 2002. [19] H. Matsum ura, Comm utative ring theory , Cambridge Studies in A dvanc e d Mathemat- ics 8 , Cambridge Univ ersit y Press, Cam bridge, 1986 . [20] S.A. Mitchell, Finite complexes with A ( n )-free cohomology . 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Springer, Some remarks on characters of binary p olyhedral groups. J. Algebr a 131 (1990), 641–647. [36] J. T ate, Homology of Lo cal R in gs, I l l. J. of Math. 1 (1957) 14–27. D ´ ep ar temen t de ma th ´ ema thiques e t de st a tistique, Un iversit ´ e de Montr ´ eal, C.P. 6128, succursale Centre -v i lle, Montr ´ eal (Qu ´ ebec), Canada H3C 3J7 E-mail addr ess : broera@DMS.UMontrea l.CA School of Ma thema tics, Universi ty of Minne sot a, Minneapolis, MN 55455, USA E-mail addr ess : reiner@math.umn.edu AG-Inv ari antentheorie, Mittel weg 3, D 37133 Friedland, Federal R e public of Germa n y E-mail addr ess : larry smith@GMX. net School of Ma thema tics, Universi ty of Minne sot a, Minneapolis, MN 55455, USA E-mail addr ess : webb@math.umn.edu

Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer

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