On operations and characteristic classes

ON OPERA TIONS AND CHARA CTERISTIC CLASSES HELGE MAAKEST AD Abstract. W e define for an y abel i an cate gory wi th tensor and exte rior pro d- ucts the grothe ndiec k ring functor. F urthermore w e use exterior pro ducts to define gamma operations and virtual Chern classes and virtual Segre classes for arbitrary elemen ts in the grothendiec k ring. W e apply this to defi ne c haracter- istic classes with v alues in algebraic K-theory and and K-theory of connections. Contents 1. Int ro duction 1 2. Exterior pro ducts and characteristic classe s 2 3. Character is tic cla sses of connections 9 4. Different ial forms, curves and O re-extensions 13 References 18 1. Introduction The aim of this pap er is to g ive a direct a nd elementary construction o f virtual characteristic clas s es with v alues in the K-theory of an arbitrar y ab elian categ ory with tensor and e x terior pro ducts using op era tions. W e apply this to construct characteristic classes with v alues in a lg ebraic K- theory and K- theo ry of connections. T o construc t characteristic c la sses of lo cally free sheaves with v a lues in algebra ic and topolo gical K-theory one us e the pr oje ct ive bund le formula . In the more general case of the K-theor y of an ab elian category with tensor and ex terior product there is no s uch form ula av a ilable. T o define c haracter istic classes using exterior pro ducts is an a lternative a pproach to the construc tio n of characteristic class e s no t relying on the pro jective bundle formula. The main results o f the pap er are the follo wing theorems: Let C at ⊗ ∧ be the category of abelian categor ie s with tensor and exterior pro ducts and morphisms. Let λ -Rings denote the categ ory o f finite dimensional augmented λ -rings and mo r- phisms. Theorem 1.1. Ther e is a c ovariant functor K λ 0 : C at ⊗ ∧ → λ -Rings Date : March 2009. 2020 M athematics Subject Classific ation. 13D15, 17B66, 19A49. Key wor ds and phr ases. Grothendiec k group, exterior pr oduct, virtual Chern class, virtual Segre class, connection, descen t, Gauss-Manin connection, Kodaira-Sp encer class, Cartier opera- tor, Or e extension. 1 2 HELGE MAAKE ST AD define d by K λ 0 ( C ) = { K 0 ( C ) , { λ n } n ≥ 0 } . Her e K 0 ( C ) is the gr othendie ck ring of C and λ n is the n ’th exterior pr o duct. Let C ∈ Ob ( C at ⊗ ∧ ). W e define for any l ≥ 0 a nd x ∈ K 0 ( C ) the virtual char ac- teristic class c l ( x ) ∈ K 0 ( C ). Theorem 1.2. L et x, y ∈ K 0 ( C ) . The fol lowing formulas hold: c l ( x + y ) = X i + j = l c i ( x ) c j ( y ) (1.2.1) c l ( x ) = 0 if x is effe ctive and l > e ( x ) (1.2.2) f ∗ ( c l ( x )) = c l ( f ∗ ( x )) (1.2.3) Her e f : D → C is a morphism in C at ⊗ ∧ . 2. Exterior products an d characteristic cla sses In this section w e prov e ex istence of virtual Che r n and Segre classes c i ( x ) , s i ( x ) ∈ K 0 ( C ) for any integer i ≥ 0 and any x ∈ K 0 ( C ) where C is an ar bitrary ab e lian category with tensor a nd exterior pro ducts. Let C b e a small ab elian category . W e sa y that C is an ab elian c ate gory with tensor and exterior pr o duct s (for short ACTEP) if the following holds. There is a tensor pr o duct ⊗ : C × C → C satisfying the fo llowing prop erties : The triple { C , ⊕ , ⊗} is a n ab elian tensor category . There are canonical iso morphisms x ⊗ y ∼ = y ⊗ x and ( x ⊕ y ) ⊗ z ∼ = x ⊗ z ⊕ y ⊗ z for any ob jects x, y , z ∈ O b ( C ). There is a unique o b ject 1 ⊗ ∈ O b ( C ) and canonica l isomorphisms 1 ⊗ ⊗ x ∼ = x ⊗ 1 ⊗ ∼ = x for an y ob ject x ∈ O b ( C ). More over the endofunctor x ⊗ − is ex act for any ob ject x ∈ O b ( C ). There is a function rk : O b ( C ) → N such that rk ( 1 ⊗ ) = 1 and rk is additive on exact sequences. There is for ev ery k ≥ 0 an e ndo functor - exterior pr o du ct λ k : C → C with the following prop erties: λ i ( x ) = 0 for i > r k ( x ), λ 0 ( x ) = x fo r all x ∈ O b ( C ). Moreov er, for any exact sequence 0 → x ′ → x → x ′′ → 0 in C and any p ≥ 2 there is a filtr ation 0 = F p +1 ⊆ F p ⊆ · · · ⊆ F 1 ⊆ F 0 = λ p ( x ) ON OPERA TIONS AND CHARA CTERISTIC CLASSES 3 and canonical isomor phis ms F i /F i +1 ∼ = λ i ( x ′ ) ⊗ λ p − i ( x ′′ ) in C . W e say a functor f : D → C betw een A CTE P’s is a morphism of ACTEP ’s if f is an additiv e tensor functor commuting with the exter ior pro duct. T his means for any l ≥ 0 there are canonical isomorphisms f ( λ l ( x )) ∼ = λ l ( f ( x )) in Ob ( C ). Definition 2.1. Le t C at ⊗ ∧ denote the categor y with A CTEP’s as ob jects and mor- phisms of ACTEP’s as morphisms. Theorem 2.2. Ther e is a c ovariant functor K 0 : C at ⊗ ∧ → R ing s define d by C → K 0 ( C ) wher e K 0 ( C ) si t he gr othendie ck ring of t he c ate gory C . Pr o of. Let for an y C ∈ O b ( C at ⊗ ∧ ) K 0 ( C ) b e the grothendieck r ing o f the category C . Direct sum induce a n addition opera tion and tensor pro duct induce a m ultiplica tio n with [ 1 ⊗ ] as mult iplicative unit. One chec ks that for a ny mor phism f : D → C there is an induced map of ring s f ∗ : K 0 ( D ) → K 0 ( C ) . Finally for tw o composa ble morphisms f , g it follows ( g ◦ f ) ∗ = g ∗ ◦ f ∗ and the Theorem follows.  Define for any e lemen t x ∈ Ob ( C ) and a ny integer l ≥ 0 λ l [ x ] = [ λ l ( x )] where λ l is the l ′ th exterior p ow er . It follows that for any exa ct sequence 0 → x ′ → x → x ′′ → 0 in C there is for a ll p ≥ 2 an equality λ p [ x ] = X i + j = p λ i [ x ′ ] λ j [ x ′′ ] in K 0 ( C ). Let K 0 ( C )[[ t ]] b e the r ing of forma l p ow er series in t with co efficients in K 0 ( C ). Let 1 + K 0 ( C )[[ t ]] b e the m ultiplicative subg roup of K 0 ( C )[[ t ]] consisting of formal p ow erser ies with constant term equal to one. Let Φ( C ) denote the ab elian monoid on C with direct sum as addition op eration. Define the following ma p λ t : Φ( C ) → 1 + K 0 ( C )[[ t ]] by λ t ( x ) = X l ≥ 0 λ l ( x ) t l . F or any ex a ct sequence 0 → x ′ → x → x ′′ → 0 in C we get an equality of formal p ow e r series λ t ( x ) = λ t ( x ′ ) λ t ( x ′′ ) . 4 HELGE MAAKE ST AD W e get a well defined map of ab elian groups λ t : K 0 ( C ) → 1 + K 0 ( C )[[ t ]] defined by λ t ( n [ x ] − m [ y ]) = λ t ( x ) n λ t ( y ) − m . W e can define for any elemen t ω ∈ K 0 ( C ) a formal p ow erserie s λ t ( ω ) = X l ≥ 0 λ l ( ω ) t l . Let O p (K 0 ) be the set o f natural transformations of the underlying set v a lued functor of K 0 . It follows λ n ∈ O p (K 0 ) Lemma 2 .3. The set Op (K 0 ) is an asso ciative ring. Pr o of. The pro o f is left to the reader as an ex ercise.  Let λ -Rings denote the ca tegory of finite dimensional a ugmented λ -rings . Theorem 2.4. Ther e is a c ovariant functor K λ 0 : C at ⊗ ∧ → λ -Rings define d by K λ 0 ( C ) = { K 0 ( C ) , { λ n } n ≥ 0 } . Her e K 0 ( C ) is the gr othendie ck ring of C and λ n is the op er ation define d ab ove. Pr o of. One c hecks that for an y categ ory C { K 0 ( C ) , { λ n } n ≥ 0 } is a λ -ring. An y morphism f : D → C induce a morphism f ∗ : K 0 λ ( D ) → K 0 λ ( C ) o f λ -ring s bec ause f comm utes with exterio r pro ducts. Moreo ver since any ob ject in C has finite rank it follows that K 0 λ ( C ) is a finite dimensiona l λ -ring . Finally the rank function r k defines a map rk : K 0 ( C ) → Z which is a map of λ -r ings wher e Z has the canonical λ -str uc tur e, a nd the Theorem is prov ed.  Let u = t/ 1 − t and le t γ t ( x ) = λ u ( x ) = P l ≥ 0 γ l ( x ) t l . It follows tha t for any exact sequence in C 0 → x ′ → x → x ′′ → 0 there is an equa lit y γ t ( x ) = γ t ( x ′ ) γ t ( x ′′ ) . It follows that for any l ≥ 0 we hav e γ l ∈ O p (K 0 ). Definition 2 .5. Let ω = P i n i [ x i ] ∈ K 0 ( C ). Define e : K 0 ( C ) → Z by e ( ω ) = X i n i rk ( x i ) . Define d : K 0 ( C ) → K 0 ( C ) by d ( ω ) = e ( ω )[ 1 ⊗ ] ON OPERA TIONS AND CHARA CTERISTIC CLASSES 5 where 1 ⊗ is the unit ob ject for ⊗ . Define for all l ≥ 0 c l ( ω ) = ( − 1) l γ l ( ω − d ( ω )) to be the l ’th c ha racteristic class of ω ∈ K 0 ( C ). W e say an element ω is effective if n i ≥ 1 for all i . Theorem 2.6. L et x, y ∈ K 0 ( C ) . The fol lowing formulas hold: c l ( x + y ) = X i + j = l c i ( x ) c j ( y ) (2.6.1) c l ( x ) = 0 if x is effe ctive and l > e ( x ) (2.6.2) f ∗ ( c l ( x )) = c l ( f ∗ ( x )) (2.6.3) Her e f : D → C is a morphism in C at ⊗ ∧ . Pr o of.  Pr o of. W e prov e 2.6.1: c l ( x + y ) = ( − 1) l γ l ( x + y − d ( x + y )) = ( − 1 ) l γ l ( x − d ( x ) + y − d ( y )) = ( − 1) l X i + j = l γ i ( x − d ( x )) γ j ( y − d ( y )) = X i + j = l ( − 1) i γ i ( x − d ( x ))( − 1) j γ j ( y − d ( y )) = X i + j = l c i ( x ) c j ( y ) , and 2.6.1 is proved. W e next pr ov e 2.6.2: Note: u = t/ 1 − t . W e see that γ t ( 1 ⊗ ) = λ u ( 1 ⊗ ) = 1 + [ 1 ⊗ ] u = 1 + t/ 1 − t = 1 / 1 − t. W e get: γ t ( x − d ( x ) ) = γ t ( x ) γ t ( 1 ⊗ ) − e ( x ) = γ t ( x )(1 − t ) e where e = e ( x ) and x = P i n i [ x i ] = [ ⊕ i x n i i ] = [ ω ] ∈ K 0 ( C ). W e g et: γ t ( x − d ( x )) = X l ≥ 0 γ l ( x − d ( x )) = γ t ( x )(1 − t ) e = λ u ( ω )(1 − t ) e = (1 + λ 1 ( ω ) u + λ 2 ( ω ) u 2 + · · · + λ e ( ω ) u e )(1 − t ) e = (1 − t ) e + λ 1 ( ω ) t (1 − t ) e − 1 + · · · + λ e ( ω ) t e . It follows that γ t ( x − d ( x )) = X l ≥ 0 γ l ( x − d ( x ) ) = p 0 + p 1 t + · · · + p e t e hence γ l ( x − d ( x ) ) = 0 for l ≥ e = e ( x ) and hence c l ( x ) = ( − 1 ) l γ l ( x − d ( x )) = 0 for l > e ( x ), and 2 .6.2 is prov e d. W e next pr ov e 2.6.3: f ∗ ( c l ( x )) = f ∗ (( − 1) l γ l ( x − d ( x ))) = ( − 1) l γ l ( f ∗ ( x ) − f ∗ ( e ( x )[ 1 ⊗ ])) = ( − 1) l γ l ( f ∗ ( x ) − e ( f ∗ ( x ))[ 1 ⊗ ]) = c l ( f ∗ ( x )) , and 2.6.3 is proved.  6 HELGE MAAKE ST AD Definition 2 .7. Let x ∈ K 0 ( C ) be an element. Let c t ( x ) = X k ≥ 0 c k ( x ) t k ∈ 1 + K 0 ( C )[[ t ]] be the Chern p ower series of the element x . It follows fro m Theor em 2.6 that for any functor f : D → C in C at ⊗ ∧ there is a commutativ e diagra m K 0 ( D ) f ∗   c t / / 1 + K 0 ( D )[[ t ]] 1 ⊗ f ∗   K 0 ( C ) c t / / 1 + K 0 ( C )[[ t ]] . Hence the Chern p ow er series define a natural transforma tion c t : K 0 ( − ) → 1 + K 0 ( − )[[ t ]] of functors. Her e w e view K 0 ( − ) and 1 + K 0 ( − )[[ t ]] a s functors K 0 ( − ) , 1 + K 0 ( − )[[ t ]] : C at ⊗ ∧ → Ab where Ab is the categor y of ab elian gr oups. Definition 2 .8. W e say that a natural trans formation u : K 0 ( − ) → 1 + K 0 ( − )[[ t ]] of functors is a the ory of char acteristic cla sses with values in K 0 . Let C l ass (K 0 ) denote the set of the ories of char acteristic classes with values in K 0 . It follo ws the Chern p ow er series define a theo ry o f characteristic classes c t ∈ C l ass (K 0 ). Since the Chern power series c t ( x ) is a unit for the multiplication in K 0 ( C )[[ t ]] there exists a p ow er series s t ( x ) ∈ 1 + K 0 ( C )[[ t ]] - the Se gr e p ower series - defined by s t ( x ) = c t ( x ) − 1 . Let s t ( x ) = X k ≥ 0 s k ( x ) t k . W e get s t ( x + y ) = c t ( x + y ) − 1 = ( c t ( x ) c t ( y )) − 1 = c t ( x ) − 1 c t ( y ) − 1 = s t ( x ) s t ( y ) and f ∗ ( s t ( x )) = f ∗ ( c t ( x ) − 1 ) = ( f ∗ ( c t ( x ))) − 1 = c t ( f ∗ ( x )) − 1 = s t ( f ∗ ( x )) . Definition 2.9. W e let the classes s k ( x ) ∈ K 0 ( C ) for k ≥ 0 b e the virtual Se gr e classes of the elemen t x ∈ K 0 ( C ). It follows the Segre classes satisfy the following formulas: Let x, y ∈ K 0 ( C ) b e arbitrar y elements and let f : D → C be a morphism in C at ⊗ ∧ . s k ( x + y ) = X i + j = k s i ( x ) s j ( y ) (2.9.1) f ∗ ( s k ( x )) = s k ( f ∗ ( x )) (2.9.2) Hence the Segre p ow er series define a na tural transfor mation s t ( − ) : K 0 ( − ) → 1 + K 0 ( − )[[ t ]] ON OPERA TIONS AND CHARA CTERISTIC CLASSES 7 of functors. It follows s t ∈ C l ass (K 0 ) It is not true in g eneral that s i ( x ) = 0 for x effective and i > e ( x ). Example 2.10. Char acteristic classes in algebr aic and top olo gic al K -the ory. W e include a discuss ion of existence of character istic classes with v a lue s in alge- braic and top olog ical grothendieck gro ups using the constructions ab ove. F ollowing [4 ], let X be an arbitra ry s chem e and let v b ( X ) be the catego r y of lo cally free finite r a nk O X -mo dules. It follows that v b ( X ) is an ACTEP . It is a standard fact tha t the gro thendieck ring K 0 ( X ) = K 0 ( v b ( X )) of vb ( X ) is a comm utative ring with unit. Direct sum induce the addition oper ation and tensor pro duct induce the mult iplication. Given an y morphism f : Y → X the pull-back f ∗ defines a map of rings f ∗ : K 0 ( X ) → K 0 ( Y ) . Let 0 → E ′ → E → E ′′ → 0 be an exact sequence of lo cally free sheav es of r anks e ′ , e and e ′′ . F rom [5], Exer cise II.5 .16, there is for a ll 2 ≤ l ≤ e a filtra tion 0 = F l +1 ⊆ F l ⊆ · · · ⊆ F 0 = ∧ l E (2.10.1) where F i /F i +1 ∼ = ∧ i ( E ′ ) ⊗ ∧ l − i ( E ′′ ) for all 0 ≤ i ≤ l . Let O p (K 0 ) b e the set of natural transforma tions of the underlying set-v alued functor o f the functor K 0 . The oper ation λ k [ E ] = [ ∧ k E ] extends to give a natural tr ansformation λ k ∈ O p (K 0 ) for a ll k ≥ 0. Let S ch denote the categor y of schemes. The construction K 0 and λ n defines a contrav ar iant functor K λ 0 : S c h → λ -Rings by K λ 0 ( X ) = { K 0 ( v b ( X )) , { λ n } n ≥ 0 } . Let u = t/ 1 − t and de fine the p ow erseries γ t ( x ) = λ u ( x ) = X l ≥ 0 λ l ( x ) u l = X l ≥ 0 γ l ( x ) t l ∈ 1 + K 0 ( X )[[ t ]] . F ollowing [4 ] (and [6]) we get op eratio ns - gamma op er ations - γ l ∈ Op (K 0 ) for a ll l ≥ 0. Definition 2 .11. Let x = P i n i [ E i ] ∈ K 0 ( X ) be an element. Define the following map e : K 0 ( X ) → Z by e ( x ) = X i n i rk ( E i ) . Define furthermore d : K 0 ( X ) → K 0 ( X ) by d ( x ) = e ( x )[ O X ] . F or any integer l ≥ 0 define the l’th char acteristic class of x to b e c l ( x ) = ( − 1 ) l γ l ( x − d ( x )) ∈ K 0 ( X ) . 8 HELGE MAAKE ST AD Prop ositio n 2.12. L et f : Y → X b e a map of schemes and let x, y ∈ K 0 ( X ) b e arbitr ary elements. The fol lowing formulas hold: c l ( x + y ) = X i + j = l c i ( x ) c j ( y ) (2.12.1) c l ( x ) = 0 if l > e ( x ) (2.12.2) f ∗ ( c l ( x )) = c l ( f ∗ ( x )) (2.12.3) Pr o of. The pro of is similar to the pro of of Theorem 2.6 and is left to the r eader as an exercise.  Let c t ( x ) = P k ≥ 0 c k ( x ) t k ∈ 1 + K 0 ( X )[[ t ]] b e the C he r n p ow er series of x ∈ K 0 ( X ). Define s t ( x ) = c t ( x ) − 1 . It follo ws we get Segr e clas s es s k ( x ) ∈ K 0 ( X ) with k ≥ 0 for any elemen t x = P i n i [ E i ]. If one co nsiders the gr othendieck r ing K 0 ( B ) of finite rank contin uous vector bundles on a top ologica l space B a similar co nstruction using tenso r op eratio ns defines characteristic cla sses c i ( E ) ∈ K 0 ( B ) wher e E is a co mplex contin uous vector bundle on a top olo g ical spa ce B . If E is a r ank n vector bundle on B it follows we get a total char acteristic class c ( E ) = n X i =0 c i ( E ) ∈ K 0 ( B ) . Let π : P ( E ∗ ) → B be the asso ciated pro jective bundle of E . Its fiber o ver a p oint x ∈ B is the pro jectivizatio n P ( E ∗ x ) of the dual vector space E ∗ x where E x is the fiber of E at x . By [6] Pro p os ition IV.7.4 the following holds: The map π ∗ : K 0 ( B ) → K 0 ( P ( E ∗ )) is injective and the ring K 0 ( P ( E ∗ )) is free o f rank n on the e lemen t h = 1 − [ O ( − 1 )] where O ( − 1) is the tautological bundle on P ( E ∗ ). It follows we g et an equation h n − c 1 ( E ) h n − 2 + c 2 ( E ) h n − 2 + · · · + ( − 1 ) n c n ( E ) = 0 (2.12.4) where the c lasses c i ( E ) in Equation 2.12.4 a re the ones defined b y ope rations on K 0 . The cla sses defined by E quation 2.1 2.4 are the Chern-classes of the vector bundle E with v alues in K 0 ( B ). Defining the Chern-c la sses of a vector bundle using the pro jective bundle P ( E ∗ ) is usually referred to a s the pr oje ct ive bund le formula . The c har acteristic cla s s c i ( E ) ∈ K 0 ( B ) is related to the exterior pr o duct o f bundles in the following wa y (see [6] Section IV.2.18 ) c i ( E ) =  n i  [ ∧ 0 E ] −  n − 1 i − 1  [ ∧ 1 E ] + · · · + ( − 1 ) i  n − i 0  [ ∧ i E ] , hence we may use the ex terior pro duct to define well b ehav ed characteristic classe s with v alues in K 0 -theory . Defining the characteristic classe s c ( x ) directly using oper ations is an alter na- tive approach to the theor y of characteris tic cla sses going “ar ound” the pro jective bundle form ula. As w e ha ve seen: In the case of the catego ry C a t ⊗ ∧ there is no replacement for the pro jective bundle formula. Hence o per ations give a direc t, int rinsic and elementary approa ch to the construction of characteristic classes. ON OPERA TIONS AND CHARA CTERISTIC CLASSES 9 3. Charac teristic classes of connections In this section we define characteristic classes with v alues in K 0 ( conn - g ) where g is a res tr icted Lie-Rineha r t algebra using exterior pro ducts. W e introduce λ - op erations on the gr othendieck ring K 0 ( conn - g ) using techniques s imila r to the ones found in [6] and [12]. Let k ⊆ K be fields of characteristic p > 0. Definition 3.1. A sub k -Lie-alge br a and K -vector space g ⊆ Der k ( K ) is a p - ( k , K ) -Lie algebr a if for any ∂ ∈ g it follows tha t ∂ [ p ] = ∂ ◦ · · · ◦ ∂ ∈ g . Given tw o p -( k , K )-Lie a lgebras g a nd h a morphism of p -( k , K )-Lie algebra s is given by an an inclusion I : g → h . Let Lie K/k denote the c a tegory of p -( k , K )-Lie algebra s and morphisms. A c onne ction ρ is a map o f K -vector spaces ρ : g → E nd k ( V ) where V is a finite dimensiona l K -vector space satisfying the following formula: ρ ( ∂ )( ax ) = aρ ( ∂ )( x ) + ∂ ( a ) x for all a ∈ K and x ∈ V . The curvatur e R ρ of ρ is the map R ρ ( ∂ , ∂ ′ ) = ρ ([ ∂ , ∂ ′ ]) − [ ρ ( ∂ ) , ρ ( ∂ ′ )] . The map ψ ρ ( ∂ ) = ρ ( ∂ [ p ] ) − ρ ( ∂ ) [ p ] is the p-curvatur e o f ρ . The co nnection is flat if R ρ = 0. It is p-flat if R ρ = ψ ρ = 0. The connection ( V , ρ ) is nilp otent of exp onent ≤ n if there is a filtra tion of g - connections 0 = F n ⊆ F n − 1 ⊆ · · · ⊆ F 1 ⊆ F 0 = V where the induced c o nnection F i /F i +1 has p -c ur v ature zero for all i . Let ( V , ρ ) and ( W , η ) b e g -connectio ns. A K -linear map φ : V → W is a map of g -co nnections if for all x ∈ g there is a commutativ e dia gram V φ / / ρ ( x )   W η ( x )   V φ / / W . Let conn - g (resp. flat - g , mo d - g ) denote the categor y of g -connections (resp flat g -connections, p- flat g -connections) of finite dimens io n ov er K and mo r phisms of g -connections. Theorem 3.2. Ther e is a c ontr avariant functor K 0 : L ie K/k → R ing s wher e for al l g ∈ Lie K/k K 0 ( g ) is the gr othendie ck ring of t he c ate gory conn - g ) . Pr o of. The pro o f is left to the reader as an ex ercise.  10 HELGE MAAKE ST AD Note: A p -( k , K )-Lie algebr a is also r eferred to as a r estricte d Lie-Ri nehart alge- br a . A p -flat g -connection is also r eferred to as a r estricte d g -mo dule . A co nnection ρ : g → End k ( V ) is flat if and only if ρ is a morphism of k - Lie alg ebras. There exists a restricted env eloping algebra U [ p ] ( g ) of the res tricted Lie-Rinehart alg ebra g with the following prop erty: There is an equiv alence of categor ies mo d - g ∼ = mo d - U [ p ] ( g ) , where mo d - U [ p ] ( g ) is the categor y of finite dimensio nal left U [ p ] ( g )-mo dules. Lemma 3 .3. L et ( V , ρ ) b e a g -c onne ction, wher e g ⊆ Der k ( K ) is a r estricte d Lie- Rineh art algebr a. It fol lows V ⊗ l , ∧ l V and Sym l ( V ) ar e g -c onne ctions. Pr o of. The pro o f is left to the reader as an ex ercise.  Let 0 → U → V → W → 0 be an exa ct sequence in conn - g , where g ⊆ Der k ( K ) is a restricted Lie- Rinehart algebra. There is the following Lemma: Lemma 3 .4. Ther e is for al l l ≥ 2 a g -s table fi ltr ation 0 = F l +1 ⊆ F l ⊆ · · · ⊆ F 1 ⊆ F 0 = ∧ l V wher e F i is a g -c onn e ction, with the fol lowing pr op erty: F i /F i +1 ∼ = ∧ i U ⊗ ∧ ( l − i ) W is an isomorphism of g -c onne ctions. Pr o of. Let π : V ⊗ l → ∧ l V b e the canonical pro jection map. It follows π is a map of g -connections. Ther e is a filtra tion of g -connections 0 ⊆ U ⊗ l ⊆ U ⊗ ( l − 1) ⊗ V ⊆ · · · ⊆ U ⊗ ( l − i ) ⊗ V ⊗ i ⊆ · · · ⊆ V ⊗ l . Make the following definition: F i = π ( U ⊗ i ⊗ V ⊗ ( l − i ) ) ⊆ ∧ l V . Since π is a map of g -co nnections, it follows that the filtration 0 = F l +1 ⊆ F l ⊆ · · · ⊆ F 1 ⊆ F 0 = ∧ l V is a filtration of g -connections. Ther e is a commutativ e diag ram of g -connectio ns U ⊗ i ⊗ V ⊗ ( l − i ) 1 ⊗ p   π / / F i   U ⊗ i ⊗ W ⊗ ( l − i ) ˜ π / /   F i /F i +1   ∧ i U ⊗ ∧ ( l − i ) W g / / F i /F i +1 . The claim is that the b ottom hor iz o nt al map g : ∧ i U ⊗ ∧ ( l − i ) W → F i /F i +1 is an is omorphism of g -c o nnections. It is eno ugh to prove it is an is omorphism o f K - vector spaces. ON OPERA TIONS AND CHARA CTERISTIC CLASSES 11 The sequence 0 → U → V → W → 0 is split as se q uence of K -vector spac es, hence V = U ⊕ W as K -vector space. Consider the diagr am U ⊗ i ⊗ V ⊗ ( l − i ) / /   V ⊗ l   F i / / ∧ l V . Since V = U ⊕ W as K -vector s pace the following holds: F i = f ( U ⊗ i ⊗ V ⊗ ( l − i ) ) = ∧ i U ∧ ( ∧ ( l − i ) ( U ⊕ W )) = ∧ i U ∧ ⊕ l − i j =0 ∧ j U ⊗ ∧ l − i − j W = ⊕ l − i j =0 ∧ i + j U ⊗ ∧ l − ( i + j ) W = ⊕ l s = i ∧ s U ⊗ ∧ l − s W . F rom this it follows that F i /F i +1 = ∧ i U ⊗ ∧ l − i W as K -vector s pace, hence g is an isomorphism o f g -connectio ns , a nd the claim of the Lemma follows.  Let Op (K 0 ) denote the set of all natural transforma tions o f the underlying set v alued functor of K 0 . It follows that O p (K 0 ) is an asso c iative ring . The exterior pro duct ∧ l from Lemma 3.4 defines op erations λ l ∈ O p (K 0 ). Theorem 3.5. F or any fi eld ex t ension k ⊆ K of fields of char acteristic p ther e is a c ontr avariant fun ctor K λ 0 : L ie K/k → λ -Rings define d by K λ 0 ( g ) = (K 0 ( conn - g ) , λ l l ≥ 0 ) . Her e K 0 ( conn - g ) is the gr othendie ck ring of the c ate gory conn - g and λ l [ V , ρ ] = [ ∧ l V , ∧ l ρ ] ∈ K 0 ( conn - g ) . Pr o of. The pro o f uses Lemma 3.4 a nd is left to the r eader as an exe r cise.  Example 3.6. R adicial desc en t . There is a n is o morphism of λ -rings K 0 ( mo d - g ) ∼ = Z . There is by Theorem 2.3 in [1] an equiv alence between the category m o d - g o f p -fla t g -connections o f finite dimension and the category V ect K g of K g -vectorspaces of finite dimension, hence ther e is an isomo rphism of grothendieck g roups e : K 0 ( mo d - g ) ∼ = K 0 (V ect K g ) ∼ = Z . In the following we use the opera tions defined above to define characteristic classes o f arbitrar y virtual connections in the grothendieck ring K 0 ( conn - g ) a s done in Section 2. Consider the functor K 0 : L ie K/k → R ing s where fo r any g ∈ Lie K/k K 0 ( conn - g ) is the gr othendieck ring of the catego ry conn - g . 12 HELGE MAAKE ST AD W e aim to show that K 0 has a theory of c haracter is tic clas ses c using oper ations defined in the previo us section. Let u = t/ 1 − t and c onsider the formal p ow erseries γ t ( x ) = λ u ( x ) = X l ≥ 0 λ l ( x ) u l = X l ≥ 0 γ l ( x ) t l . It has the pr op e rty that for a n y exact sequence 0 → U → V → W → 0 in conn - g there is an equality of formal powerseries in 1 + K 0 ( conn - g )[[ t ]]: γ t ( V ) = γ t ( U ) γ t ( W ) . F or all l ≥ 0 it follows γ l ∈ O p (K 0 ). W e get well defined op erations - gamma op er ations - γ l : K 0 ( conn - g ) → K 0 ( conn - g ) . satisfying γ l ( V ) = X i + j = l γ i ( U ) γ j ( W ) . Definition 3. 7. Let x = P i n i [ V i , ρ i ] = P i n i [ V i ] ∈ K 0 ( conn - g ) b e a n element. Define the following map e : K 0 ( conn - g ) → Z by e ( x ) = X i n i dim K ( V i ) . Define furthermore d : K 0 ( conn - g ) → K 0 ( conn - g ) by d ( x ) = e ( x )[ θ K ] where θ K is the trivial r a nk one g -connection. F or a ny integer l ≥ 0 define the l’th char acteristic class of x to b e c l ( x ) = ( − 1 ) l γ l ( x − d ( x )) ∈ K 0 ( conn - g ) . One chec ks immedia tely that the notions ab ov e are w ell defined. Let F : h → g be a morphism in Lie K/k . There is for every g -connection ( W , ρ ) a canonica l h -connection F ∗ W = K ⊗ W . This cons truction commutes with exterior pro duct: F ∗ ( ∧ l W ) = ∧ l ( F ∗ W ). W e say that an elemen t x = P n i [ V i ] ∈ K 0 ( conn - g ) is effe ct ive if n i > 0 for all i . W e g et the following result for the group K 0 ( conn - g ): Theorem 3.8. L et x, y ∈ K 0 ( conn - g ) b e arbitr ary elements with x effe ctive. The fol lowing formulas hold: c l ( x + y ) = X i + j = l c i ( x ) c j ( y ) (3.8.1) c l ( x ) =0 if l > e ( x ) (3.8.2) F ∗ c l ( x ) = c l ( F ∗ x ) (3.8.3) Pr o of. The pro of is similar to the pro of of Theorem 2.6 and is left to the r eader as an exercise.  ON OPERA TIONS AND CHARA CTERISTIC CLASSES 13 Definition 3 .9. Let for any effectiv e x ∈ K 0 ( conn - g ) c ( x ) = X l ≥ 0 c l ( x ) ∈ K 0 ( conn - g ) be it’s virtual char acteristic class . By Lemma 3.8 , F orm ula 3.8.2 it follows c ( x ) is w ell de fined for any x ∈ K 0 ( conn - g ). 4. Differential f orms, cur ves and Ore-extensions In this section we make the constructions in Sectio n 2 explic it and cons truct characteristic c lasses for connections on curves a nd connections de fined in terms of differential for ms on fields. First w e in tro duce the Cartier op er ator (following the presen tation in [1] a nd [8]) and rela te it to the p -curv ature of a c o nnection defined on the field K . Let K p ⊆ L ⊆ K b e fields o f characteristic p > 0 and let g = Der L ( K ) b e the p - Lie alg ebra of deriv ations of K ov er L . Let K 1 /p be the field of p ’th roo ts of elements of K ie K 1 /p is the splitting field of all p olynomials T p − a with a ∈ K . It has the prop erty that for all elemen ts a ∈ K there is a unique element x = a 1 /p ∈ K 1 /p with x p = a . F urthermor e o ne ha s that ( a + b ) 1 /p = a 1 /p + b 1 /p . Let ω = xdy ∈ Ω 1 K/L be a differential form and define the following map: C ω : g → K 1 /p by C ω ( ∂ ) = ( ω ( ∂ p ) − ∂ p − 1 ( ω ( ∂ ))) 1 /p . Definition 4.1. The map C is the Cartier op era to r for the field extension L ⊆ K . Prop ositio n 4.2. The fol lowing holds: C ( ω + ω ′ ) = C ω + C ω ′ (4.2.1) C ( xω ) = x 1 /p C ω , x ∈ L (4.2.2) C ( dx ) = 0 (4.2.3) Pr o of. W e prov e 4.2.1: C ( ω + ω ′ )( ∂ ) = (( ω + ω ′ )( ∂ ) − ∂ p − 1 (( ω + ω ′ )( ∂ ))) 1 /p = ( ω ( ∂ p ) − ∂ p − 1 ( ω ( ∂ )) + ω ′ ( ∂ ) − ∂ p − 1 ( ω ′ ( ∂ ))) 1 /p = ( ω ( ∂ ) − ∂ p − 1 ( ω ( ∂ )) 1 /p + ( ω ′ ( ∂ ) − ∂ p − 1 ( ω ′ ( ∂ ))) 1 /p = C ω ( ∂ ) + C ω ′ ( ∂ ) . W e prove 4.2.2: Let x ∈ L . W e get C ( xω )( ∂ ) = ( xω ( ∂ ) − ∂ p − 1 ( xω ( ∂ ))) 1 /p = ( x ( ω ( ∂ ) − ∂ p − 1 ( ω ( ∂ )))) 1 /p = x 1 /p C ω ( ∂ ) . 4.2.3 is obvious.  Define the following connection r : Der L ( K ) → End L ( K ) by r ( ∂ )( x ) = ∂ ( x ) + ω ( ∂ ) x. Let R r ( ∂ , ∂ ′ ) = [ r ( ∂ ) , r ( ∂ ′ )] − r ([ ∂ , ∂ ′ ]) b e the curv ature of r and ψ r ( ∂ ) = r ( ∂ p ) − r ( ∂ ) p the p -curv ature. 14 HELGE MAAKE ST AD Theorem 4.3. The fol lowing holds: R r ( ∂ , ∂ ′ ) = dω ( ∂ , ∂ ′ ) (4.3.1) ψ r ( ∂ ) = ( − 1) p ( C ω ( ∂ ) − ω ( ∂ )) p . (4.3.2) Pr o of. Assume ω = xdy . W e first prove 4.3.1: It follows dω ( ∂ , ∂ ′ ) = ∂ ( x ) ∂ ′ ( y ) − ∂ ′ ( x ) ∂ ( y ) . It follows that ∂ ( ω ( ∂ ′ )) − ∂ ′ ( ω ( ∂ )) − ω ([ ∂ , ∂ ′ ]) = ∂ ( x∂ ′ ( y )) − ∂ ′ ( x∂ ( y )) − x [ ∂ , ∂ ′ ]( y ) = ∂ ( x ) ∂ ′ ( y ) + x∂ ∂ ′ ( y ) − ∂ ′ ( x ) ∂ ( y ) − x∂ ∂ ′ ( y ) − x [ ∂ , ∂ ′ ]( y ) = ∂ ( x ) ∂ ′ ( y ) − ∂ ′ ( x ) ∂ ( y ) = dω ( ∂ , ∂ ′ ) . W e get thus the formula ∂ ( ω ( ∂ ′ )) − ∂ ′ ( ω ( ∂ )) = dω ( ∂ , ∂ ′ ) + ω ([ ∂ , ∂ ′ ]) . W e get [ r∂ , r∂ ′ ]( u ) = ( ∂ + ω ( α ))( ∂ ′ + ω ( ∂ ′ )( u ) − ( ∂ ′ + ω ( ∂ ′ ))( ∂ + ω ( ∂ ))( u ) = ∂ ∂ ( u ) + ∂ ( ω ( ∂ ′ ) u ) + ω ( ∂ ) ∂ ′ ( u ) + ω ( ∂ ) ω ( ∂ ′ ) u − ∂ ′ ∂ ( u ) − ∂ ′ ( ω ( ∂ ) u ) − ω ( ∂ ′ ) ∂ ( u ) − ω ( ∂ ′ ) ω ( ∂ ) u = [ ∂ , ∂ ′ ]( u ) + ∂ ( ω ( ∂ ′ ) u ) + ω ( ∂ ) ∂ ′ ( u ) − ∂ ′ ( ω ( ∂ ) u ) − ω ( ∂ ′ ) ∂ ( u ) = [ ∂ , ∂ ′ ]( u ) + ∂ ( ω ( ∂ ′ )) u − ∂ ′ ( ω ( ∂ )) u = [ ∂ , ∂ ′ ]( u ) + dω ( ∂ , ∂ ′ )( u ) + ω ([ ∂ , ∂ ′ ])( u ) . It follows that R r ( ∂ , ∂ ′ ) = [ r ∂ , r ∂ ′ ] − r ([ ∂ , ∂ ′ ]) = dω ( ∂ , ∂ ′ ) hence 4.3.1 is proved. In the ring E nd L ( K ) the following holds: ( a + ∂ ) p = a p + ∂ p + ∂ p − 1 ( a ) . This is proved using induction. W e prove 4.3.2: ψ r ( ∂ ) = r ( ∂ p ) − r ( ∂ ) p = ∂ p + ω ( ∂ p ) − ( ∂ + ω ( ∂ )) p = ∂ p + ω ( ∂ p ) − ω ( ∂ ) p − ∂ p − ∂ p − 1 ( ω ( ∂ )) = ω ( ∂ p ) − ∂ p − 1 ( ω ( ∂ )) − ω ( ∂ ) p = C ω ( ∂ ) p − ω ( ∂ ) p = ( C ω ( ∂ ) − ω ( ∂ )) p , and the claim follows.  Example 4.4. Differ ential forms. ON OPERA TIONS AND CHARA CTERISTIC CLASSES 15 Let ω ∈ Ω = Ω 1 K/L and let g = Der L ( K ) Define the following map ρ ω : g → End L ( K ) by ρ ω ( ∂ )( x ) = ∂ ( x ) + ω ( ∂ )( x ) . It follows ρ ω is a connection and from Prop os ition 4 .3 one gets R ρ ω ( ∂ , ∂ ′ ) = dω ( ∂ , ∂ ′ ) and ψ ρ ω ( ∂ ) = ( − 1) p ( C ω ( ∂ ) − ω ( ∂ )) p , where C ω is the Ca rtier op er a tor (see [1] and [8]). Note: By Pr op osition 7 in [1] it follows that R ρ ω = ψ ρ ω = 0 if a nd only if ω = dl og ( x ) = x − 1 dx . W e get th us a map φ : Ω → K 0 ( conn - g ) defined by φ ( ω ) = c ( K, ρ ω ) detecting if a differe n tial form ω is a loga rithmic deriv ative. Example 4.5. A djoint r epr esentation. Let g ⊆ Der k ( K ) be a restricted Lie -Rinehart algebr a. There is a representation ad : g → E nd k (Der k ( K )) defined by ad ( x )( y ) = [ x, y ] . One sees ad ( x )( αy ) = [ x, αy ] = x ( αy ) − αy x = x ( α ) y + αxy − αy x = α [ x, y ] + x ( α ) y = αad ( x )( y ) + x ( α ) y, hence ad is a co nnection. The Jaco bi- iden tity sho ws R ad = 0 hence the map a d is a representation of Lie algebra s. In End k (Der k ( K )) the following for m ula holds: [ ∂ [ ∂ [ · · · [ ∂ , η ] · · · ] = n X i =0  n i  ∂ n − i η ∂ i for all n ≥ 1. W e get the formula ad ( ∂ p )( η ) = [ ∂ p , η ] = ∂ p η − η ∂ p = p X i =0  p i  ∂ p − i η ∂ i = [ ∂ [ ∂ [ · · · [ ∂ , η ] · · · ] = ad ( ∂ ) p ( η ) hence it follows ψ ρ ( ∂ ) = ad ( ∂ p ) − ad ( ∂ ) p = 0 and it follows ad is a flat g - connection on Der k ( K ) with zero p - curv ature. One may also check that for any ideal I ⊆ Der k ( K ) it follows I is c losed under p -p ow er s, hence I is a restricted Lie-Rinehart alg ebra. The map ad : Der k ( K ) → End k ( I ) and ad : Der k ( K ) → End k (Der k ( K ) /I ) 16 HELGE MAAKE ST AD makes I and Der k ( K ) /I in to p -repr e s ent ations and the sequence 0 → I → Der k ( K ) → Der k ( K ) /I → 0 is an exact sequence of p -flat g -co nnec tio ns. W e get a characteris tic cla ss c (Der k ( K ) /I , ad ) ∈ K 0 ( conn - g ) for each ideal I ⊆ Der k ( K ). There is a p -flat connection ad : g → E nd k ( U [ p ] ( g )) where U [ p ] ( g ) is the r estricted en veloping algebra of g , hence we get a characteris tic class c ( U [ p ] ( g ) , ad ) ∈ K 0 ( conn - g ) . Example 4.6. Curves and Or e ext ensions. If π : C → C ′ is a finite mor phism of pr o jective cur ves ov er an a rbitrary field k and K ′ ⊆ K is the co r resp onding finite ex tens ion o f function fields, we get an exact sequence of Lie-Rinehar t algebras 0 → h → Der k ( K ) → dπ K ⊗ K ′ Der k ( K ′ ) → 0 . The Lie-Rinehart algebr a h = Der K ′ ( K ) is a finite dimensional K -vector space. Let g = Der k ( K ) and l = Der k ( K ′ ). The Lie -Rinehart algebras g and l are infinite dimensional k -Lie a lgebras. W e get for any flat g -connection ( V , ρ ) and i ≥ 0 canonical flat connec tio ns - Gauss-Manin c onne ctions (see [14]) - ∇ i,ρ GM : l → End k (H i ( h , V )) where H i ( h , V ) is the Chev alley -Ho chsc hild coho mology o f V a s h -mo dule. Let K 0 ( flat - l ) deno te the grothendieck ring of the category flat - l . W e get a well defined cohomolog y clas s dπ ! ( V , ρ ) = X i ≥ 0 ( − 1) i [H i ( h , V )] ∈ K 0 ( flat - l ) . The connec tio n dπ i ( ρ ) is not p -flat in g eneral. (See [7] for examples where the p -curv atur e o f dπ i ( ρ ) is related to the Ko dair a-Sp encer c lass). Let ρ = ad from Example 4.5 with ρ : Der k ( K ) → End k (Der k ( K )) and make the following definition: Definition 4 .7. Let the cohomo logy class ∆( π ) = dπ ! (Der k ( K ) , ρ ) = X i ≥ 0 ( − 1) i [H i ( h , Der k ( K ))] ∈ K 0 ( flat - l ) be the r amific ation class of the morphism π . If π is a separ able morphism, it follows h = 0. In this case ∆( π ) = [Der k ( K )] ∈ K 0 ( flat - l ) . Hence the cohomolo gy class ∆( π ) ∈ K 0 ( flat - l ) ON OPERA TIONS AND CHARA CTERISTIC CLASSES 17 is related to the ramificatio n of the morphism π . W e get a characteristic class c ( π ) = X i ≥ 0 c i (∆( π )) ∈ K 0 ( flat - l ) defined for an ar bitrary finite morphism π : C → C ′ of pro jective curves. Assume there is a separa ble mo rphism of curves C ′ → P 1 k where k is a field of characteristic zer o and let ( V , ρ ) b e a flat g -c o nnection. The function field K ( P 1 k ) equals k ( t ) where t is a transcenden tal v ariable ov er k . W e get an isomorphis m l ∼ = K ′ ∂ where ∂ is partial deriv ative with r esp e ct to t . It follows that the categ ory of flat finite dimensional l -connections is equiv a len t to the catego ry of K ′ { T } -mo dules of finite dimension over K ′ . Here K ′ { T } is the Or e extension of K ′ by T . It is the t wisted p olynomial ring o n the v aria ble T with multiplication defined as follows: T a = aT + ∂ ( a ) for any a ∈ K ′ . In this case the c ohomology gro up H i ( h , V ) is a finite dimensio nal K - vector space. It follows H i ( h , V ) is a finite dimensional K ′ -vector space. The Gauss-Manin connection ∇ i,ρ GM : l → End k (H i ( h , V )) makes H i ( h , V ) into a finite dimensional left K ′ { T } -mo dule. By the cyclic vector theorem (see [9 ] Theorem 5.4.2) any finite dimensional left K ′ { T } -mo dule is on the form K ′ { T } /P ( T ) K ′ { T } where P ( T ) ∈ K ′ { T } is a non-commutative polynomia l. It follows there is a no n- commutativ e p olynomial P i ( T ) ∈ K ′ { T } such that H i ( h , V ) ∼ = K ′ { T } /P i ( T ) K ′ { T } as K ′ { T } -mo dule. In this c a se we ge t a cohomology class ∆( V ) = X i ≥ 0 ( − 1) i [H i ( h , V )] = X i ≥ 0 ( − 1) i [ K ′ { T } /P i ( T ) K ′ { T } ] ∈ K 0 ( flat - l ) . W e get a characteristic class c ( V ) = X i ≥ 0 c i (∆( V )) ∈ K 0 ( flat - l ) . By a Theorem of Quillen (see Theorem 1 in [15]) it follows K n ( flat - l ) ∼ = K n ( K ′ { T } ) ∼ = K n ( K ′ ) ∼ = Z for all n ≥ 0. It follows the class [H i ( h , V ) , ∇ i,ρ GM ] ∈ K 0 ( flat - l ) is indep endent of choice of co nnection whenever C ′ is a separa ble cover of P 1 k . It follows that the g rothendieck group K 0 ( flat - l ) do es not contain enough information to detect inv ar iants of the c o nnection ∇ i,ρ GM . 18 HELGE MAAKE ST AD Ac knowledgemen ts . Thanks to Ken Goo dear l, Max Karoubi a nd Charles W eib e l for discussio ns and comments. References [1] P . Cartier, Questions de rationalit´ e des diviseurs en g ´ eom´ etrie alg´ ebrique, Bul l. So c. Math. F r anc e no. 86 (1958) [2] W. F ulton, S. Lang, Ri emann-Roch algebra, Grund lehr en Math. Wiss. no. 277 (1985) [3] D. Grayson, Exterior p ow er operations in hi gher K-theory , K-t he ory no. 3 (1989) [4] A. Grothendiec k, SGA 6, Lecture Notes in Mathematics no. 225, Springer V erlag (1971) [5] R. Hartshorne, Algebraic geometry , Springe r V erlag GTM no. 52 (1977) [6] M. Karoubi, K-theory - an int ro duction, Grund lehr e n Math. 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Mcconnell, The K-theory of fil tered rings and skew Lauren t extensions, S´ eminair e d’alg ` eb r e Paul Dubr eil et M atri e-Paule Mal liavin, 36` eme an´ ee , Lecture Notes in Mathematics no. 1146 (1985) [16] D. Quillen, Hi gher algebraic K-theory , Lectu re Notes in Mathematics no. 341, Springer V erlag (1973) NTNU, Trondheim Email add r ess : Helge. Maakestad@ math.ntnu.no