Homological properties of cochain Differential Graded algebras

HOMOLOGICAL PR OPER TIES OF COCHAIN DIFFERENTIA L G RADED ALGEBRAS ANDERS J. FRANKILD AND PETE R JØRGENSEN Abstract. Consider a lo cal chain Differential Graded algebra , such as the singular chain complex of a path wise connected topo - logical gr o up. In t w o previous pap ers, a num ber of homolo gical results w ere prov ed for suc h a n alg ebra: An Amplitude Inequality , an Auslan- der-Buchsbaum Equality , and a Gap Theorem. These were inspired by homological ring theory . By the s o-called lo oking gla ss principle, one would expect that analogo us results exist for simply connected co chain Differen tial Graded algebras, such a s the singular co chain complex of a simply connected top olog ical space. Indeed, this pap er establis hes such ana logous r esults. 0. Intro duction This pap er is a sequel o f [6] a nd [7], or, more accurately , their mirr o r image. The pap ers [6] and [7] in v estigated the homological prop erties of lo cal c hain Diff erential Graded algebras, such as the singular c hain complex o f a path wise connected to p ological group. Sev eral r esults mo delled on ring theory we re pro v ed: An Amplitude Inequalit y , an Auslander-Buc hsbaum Eq ualit y , a nd a Gap The orem for Bass n um b ers. In this pap er, w e shall do the same thing for simply connected co c hain Differen tial Graded algebras, such as the singular co c hain complex of a simply connected top ological space. The resulting co c hain Auslan- der-Buc hsbaum Equality and G ap Theorem are new, while a co chain Amplitude Inequalit y was stated already in [10, prop. 3.11]; our pro of w orks b y differen t metho ds. F or in tro ductions to the theory of Differ- en tial G raded (DG ) algebras, we refer the reader to [2], [5], or [9]. 2000 Mathema tics Subje ct Cla ssific ation. P r imary 16E45 ; Secondary 55P 62. Key wor ds and phr ases. Amplitude Inequalit y , Auslander- Buchsbaum Equality , Different ial Graded mo dules, Gap Theore m, ho mological dimensions, homologica l ident ities, singula r co chain complexes, top olo gical spaces. 1 2 ANDERS J. FR A NKILD AND PETER JØRGENSEN One of the motiv ations for [6] was that the Ga p Theorem answ ered affirmativ ely [1, Question 3.10] b y Avramo v and F oxb y on the so-called Bass n um b ers of lo cal chain DG algebras. The presen t G a p Theorem implies that the a nsw er is also affirmat ive for simply connected co c ha in DG algebras. In fact, it sho ws that fo r these algebras, Avramo v and F o xb y’s conjectural b ound on the gap length o f the Bass num b ers can b e sharp ened b y an a moun t o f one, see Corollary 4.11 and R emark 4.12. As indicated, t he mo ve from lo cal c ha in to simply connected co c ha in DG algebras is a general phenomenon: The so-called “lo oking glass principle” of [4] states tha t eac h result on lo cal c hain DG algebras should hav e a “ mirror imag e” for simply connected co chain DG a lg e- bras. Ho w ev er, pro ofs cannot b e tra nslated in a mec hanical wa y . A lo cal c hain D G algebra sits in non-p ositiv e cohomo lo gical degrees and a simply connected co c hain DG algebra sits in non-nega t ive cohomolog- ical degrees. Ac cordingly , the simplest statemen t of the lo oking glass principle is that it in terchanges p ositive and negativ e degrees. F or instance, ov er lo cal c hain DG alg ebras, in [6] w e used the notion of k -pro j ective dimension of a D G mo dule g iv en b y p d M = sup { j | H j ( k L ⊗ R M ) 6 = 0 } . The lo oking glass principle t ells us that ov er simply connected co c hain DG algebras, w e m ust replace this b y p cd M = sup { j | H − j ( k L ⊗ R M ) 6 = 0 } = sup { j | H j ( k L ⊗ R M ) 6 = 0 } whic h will b e called t he pr oje ctive c o dimension of M . Indeed, while the Auslander-Buc hsbaum Eq uality for lo cal c ha in D G algebras is a state- men t relat ing k -pro jectiv e dimension to o t her homo lo gical in v ariants, see [6, thm. 2.3], for simply connected co c hain DG algebras it will b e a statemen t relating the pro jectiv e c o dimension to other homological in v ariants, see The orem 4.5 and Corollary 4 .7. The metho ds o f this pap er are differen t from the ones of [6] and [7]. They build on D G mo dule adaptatio ns of the ideas used b y Serre to pro v e [11, thm. 10 , p. 217] on the connection b etw een homolog y and homotop y groups of top ological spaces. The main p o int is a list of term wise inequalities of p ow er series giv en in Propositions 3.5 and 3.6. An adv an tag e of the presen t approac h is that, whereas the metho ds of [6] and [7] fail for un b ounded DG algebras, as dem onstrated for example in [6, sec. 4], this pa p er is able to treat b ounded and un b ounded D G algebras on the same fo oting. COCHAIN DG ALGEBRA S 3 The pap er is organized as follo ws: Section 1 giv es notatio n and e- lementary prop erties f o r DG mo dules ov er simply connected co chain DG alg ebras. S ections 2 and 3 set up the D G mo dule adapta tion o f Serre’s ideas from the pro of of [11, thm. 10, p. 217]. Section 4 pro ves the co chain Amplitude Inequalit y , the Auslander-Buch sbaum Equality , and the Gap Theorem for Ba ss n umbers in Corollaries 4.4, 4.7, and 4.11. These results a r ise as sp ecial cases o f the stronger statemen ts Theorems 4.3 , 4.5, and 4.8. Finally , Section 5 applies the Auslander-Buc hsbaum Equalit y and the Gap Theorem to the singular co c hain DG alg ebra of a top o logical space. The contex t will b e a fibratio n of top o logical space s, and w e recov er in Theorem 5.5 the classical fact that homological dimension is additiv e on fibrations. Theorem 5 .6 sho ws that a gap of length g in the Betti n umbers of the fibre space implies that the total space has cohomology in a dimension bigger than or equal to g + 1. This pap er sup ersedes the man uscript “Homological identitie s for Dif- feren tial Graded Algebras, I I” from the spring of 2002. That manu- script suffers from techn ical problems whic h remain unsolv ed, a nd it w as nev er submitted. Anders J. F r a nkild, my coauthor a nd friend o f many years, died in June 2007, b efore the presen t, more successful approach to ok the form of this pap er. He will b e bitterly missed, but his memory will live on. Since Anders has not b een able to c hec k the final vers ion of the pap er, the resp onsibility for an y mistak es rests with me. 1. Back ground This section gives notation and elementary prop erties for DG mo dules o v er simply connected co c hain D G alg ebras. F or intro ductions to the theory of DG alg ebras, see [2], [5], or [9]. The notation will sta y close to [6], [7], and [8]. Setup 1.1. By k is denoted a field and by R a co c hain DG algebra o v er k whic h has the form · · · → 0 → k → 0 → R 2 → R 3 → · · · and satisfies dim k H i ( R ) < ∞ for eac h i . 4 ANDERS J. FR A NKILD AND PETER JØRGENSEN Remark 1.2. In particular, R 0 = k , R 1 = 0, H 0 ( R ) = k and H 1 ( R ) = 0. Notation 1.3. There are derive d categories D ( R ) of DG left- R -mo- dules a nd D ( R o ) of DG righ t- R - mo dules which supp ort the deriv ed functors L ⊗ R and RHom R . W e defin e full sub categor ies of D ( R ), D + ( R ) = { M ∈ D ( R ) | H j ( M ) = 0 for j ≪ 0 } , D − ( R ) = { M ∈ D ( R ) | H j ( M ) = 0 for j ≫ 0 } , and similarly for D ( R o ). A DG left- R -mo dule M is c omp act precise ly if it is finitely built from R in D ( R ) using distinguished triangles, (de)susp ensions, finite direct sums, a nd direct summands, cf. [9, thm. 5.3]. The full sub category of D ( R ) consisting of compact DG mo dules is denoted D c ( R ), and similarly for D ( R o ). The susp ension functor on DG modules is denoted b y Σ . The op eration ( − ) ♮ forgets the differential of a complex; it sends DG algebras and DG mo dules to graded algebras and gra ded mo dules. A DG R -mo dule M is called lo c al ly finite if it satisfies dim k H j ( M ) < ∞ for each j . The infimum and the supr e m um of a DG module are defined b y inf M = inf { j | H j ( M ) 6 = 0 } , sup M = sup { j | H j ( M ) 6 = 0 } , and the amplitude is amp M = sup M − inf M . Note that we use the con ven tion inf ( ∅ ) = ∞ and sup( ∅ ) = −∞ , so inf (0) = ∞ , sup(0) = −∞ , a nd amp(0) = −∞ . In fa ct, these sp ecial v alues o ccur precisely when a DG mo dule has zero cohomolo gy , and this prop ert y c hara cterizes DG mo dules whic h a re zero in the derive d category , so M ∼ = 0 in the deriv ed category ⇔ inf M = ∞ ⇔ sup M = −∞ ⇔ amp M = −∞ . (1.a) Moreo ver, M 6 ∼ = 0 in the deriv ed category ⇒ inf M ≤ sup M . (1.b) W e can view k as a DG bi- R -mo dule concen trated in cohomolog ical degree 0. The Betti numb ers of a DG left- R -mo dule M are β j R ( M ) = dim k H j ( k L ⊗ R M ) . COCHAIN DG ALGEBRA S 5 The pr oje ctive c o dimens ion of M is p cd R M = sup( k L ⊗ R M ) = sup { j | β j R ( M ) 6 = 0 } ; (1.c) it is an inte ger or ∞ or −∞ . If β is a cardinal n umber, then a direct sum of β copies of M will b e denoted by M ( β ) . The following lemma holds by [3]. Lemma 1.4. L et M b e in D + ( R ) . Ther e i s a sem i-fr e e r esolution ϕ : F → M with semi-fr e e filtr ation 0 = F ( − 1) ⊆ F (0) ⊆ F (1) ⊆ · · · ⊆ F wher e the fr e e quotients F ( j ) /F ( j − 1 ) ar e dir e ct sums of (de)susp en- sions Σ ℓ R with ℓ ≤ − inf M . Lemma 1.5. L et P b e in D + ( R o ) and let M b e in D + ( R ) . Then inf ( P L ⊗ R M ) = inf P + inf M . Pr o of. If P or M is zero then the equation reads ∞ = ∞ , so suppose that P and M are non-zero in the deriv ed categor ies. Then j = inf P and i = inf M are integers. By [8, lem. 3.4 (i)], w e can replace P with a quasi-isomorphic DG mo- dule whic h is zero in cohomological degrees < j . Lemma 1.4 say s that M has a semi-free resolution F with a semi-free filtration where the success iv e quotien ts are direc t sums of DG modules Σ ℓ R with ℓ ≤ − i . This implies that F ♮ is a direct sum o f graded mo dules Σ ℓ R ♮ with ℓ ≤ − i , so ( P ⊗ R F ) ♮ = P ♮ ⊗ R ♮ F ♮ is a direct sum of graded mo dules Σ ℓ P ♮ with ℓ ≤ − i . Since P is zero in cohomolog ical degrees < j , this implies that ( P ⊗ R F ) ♮ is zero in cohomological degrees < j + i . In particular we hav e inf ( P ⊗ R F ) ≥ j + i , that is, inf ( P L ⊗ R M ) ≥ j + i = inf P + inf M . (1.d) On the other ha nd, a morphism of DG left- R - mo dules Σ − i R → M is determined b y the imag e z of Σ − i 1 R , and z is a cycle in M i , the i th comp onent of M . Since H i (Σ − i R ) ∼ = H 0 ( R ) ∼ = k , the induced map H i (Σ − i R ) → H i ( M ) is just the map k → H i ( M ) whic h sends 1 k to the cohomology class of z . He nce if w e pic k cycles z α suc h that the corresp onding cohomolo gy classes form a k -basis of H i ( M ) and construct a morphism Σ − i R ( β ) → M b y sending t he elemen ts Σ − i 1 R to 6 ANDERS J. FR A NKILD AND PETER JØRGENSEN the z α , then the induced map H i (Σ − i R ( β ) ) → H i ( M ) is an isomorphism. Complete to a distinguished triangle Σ − i R ( β ) → M → M ′′ → ; (1.e) since w e hav e H i +1 (Σ − i R ( β ) ) ∼ = H 1 ( R ( β ) ) = 0, the long exact cohomo- logy sequence sho ws inf M ′′ ≥ i + 1 . (1.f ) T ensoring the distinguished triangle (1.e) with P giv es Σ − i P ( β ) → P L ⊗ R M → P L ⊗ R M ′′ → whose long exact cohomology sequen ce con ta ins H j + i − 1 ( P L ⊗ R M ′′ ) → H j + i (Σ − i P ( β ) ) → H j + i ( P L ⊗ R M ) . (1.g) The inequalit y (1.d) can b e applied to P and M ′′ ; b ecause of (1.f), this giv es inf ( P L ⊗ R M ′′ ) ≥ j + i + 1 so the first term of the exact sequence (1.g) is zero. The second term is H j + i (Σ − i P ( β ) ) ∼ = H j ( P ( β ) ) whic h is non-zero since j = inf P . So the third term is non-zero whence inf ( P L ⊗ R M ) ≤ j + i = inf P + inf M . Com bining with (1.d) completes the pro of.  Lemma 1.6. L et P b e no n -zer o in D + ( R o ) and let M b e in D + ( R ) . (i) M ∼ = 0 in D + ( R ) ⇔ P L ⊗ R M ∼ = 0 in D ( k ) . (ii) M 6 ∼ = 0 in D + ( R ) ⇒ inf M ≤ p cd M . Pr o of. (i) F ollows from Lemma 1.5 and Equation (1.a). (ii) When M is non- zero in D + ( R ), it follows from (i) that k L ⊗ R M is non-zero in D ( k ). Then inf ( k L ⊗ R M ) ≤ sup( k L ⊗ R M ) b y Equation (1.b). By Lemma 1.5 and Equation (1.c), this r eads inf M ≤ p cd M .  2. A construction This section starts to set up the DG mo dule adaptation o f Serre’s ideas from the pro of of [1 1, thm. 10, p. 217]. The main item is Construction 2.2 whic h approxim ates a DG mo dule b y (de)suspensions o f the DG algebra R . COCHAIN DG ALGEBRA S 7 Lemma 2.1. L et M b e in D + ( R ) and let i b e an inte ger with i ≤ inf M . Ther e is a distinguishe d triangle in D ( R ) , Σ − i R ( β ) → M → M ′′ → , which satisfies the fol lowing . (i) β = β i ( M ) . (ii) β j ( M ) = β j ( M ′′ ) for e ach j ≥ i + 1 . (iii) inf M ′′ ≥ i + 1 ; in p a rticular, M ′′ is in D + ( R ) . If M is lo c al ly fin ite then β = β i ( M ) < ∞ and M ′′ is als o lo c al ly fi n ite. Pr o of. The distinguished triang le is just (1.e) f rom the pro of of Lemma 1.5, constructed by pic king cycles z α in M i suc h that the corresp onding cohomology classes form a k -ba sis for H i ( M ), defining Σ − i R ( β ) → M b y sending the elemen ts Σ − i 1 R to the z α , and completing to a distinguished triangle. Prop ert y (iii) is t he inequality (1 .f) in the pro of of Lemma 1.5. T enso- ring the distinguished triangle with k , prop ert y ( ii) is immediate and prop ert y (i) follo ws b y using Lemma 1.5. If M is lo cally finite, then there ar e only finitely many cycles z α , so β < ∞ . The lo ng exact cohomolog y sequence then shows that M ′′ is also lo cally finite.  Construction 2.2. Let M b e non- zero in D + ( R ) and write i = inf M . Observ e that i is an integer. Set M h i i = M and let u ≥ i b e an in teger. By iterating Lemma 2.1 , w e can construct a sequence of distinguished triangles in D ( R ), Σ − i R ( β i ) → M h i i → M h i + 1 i → , Σ − i − 1 R ( β i +1 ) → M h i + 1 i → M h i + 2 i → , . . . Σ − u +1 R ( β u − 1 ) → M h u − 1 i → M h u i → , Σ − u R ( β u ) → M h u i → M h u + 1 i → , where (i) β j = β j ( M ) for eac h j . (ii) β j ( M ) = β j ( M h ℓ i ) for eac h j ≥ ℓ . (iii) inf M h ℓ i ≥ ℓ for each ℓ . In particular, each M h ℓ i is in D + ( R ). If M is lo cally finite, then so is eac h M h ℓ i , and then each β j = β j ( M ) = β j ( M h j i ) is finite. 8 ANDERS J. FR A NKILD AND PETER JØRGENSEN Prop osition 2.3. (i) If M is in D c ( R ) , then it is lo c al ly fi n ite and b elongs to D + ( R ) . (ii) L et M b e lo c al ly finite in D + ( R ) . Then M is in D c ( R ) ⇔ p cd M < ∞ . Pr o of. Let M b e in D c ( R ), that is, M is finitely built from R in D ( R ). Since R is lo cally finite and belongs to D + ( R ), the same holds for M . This prov es (i). Moreo ver, w e ha v e sup( k L ⊗ R R ) = sup k = 0 < ∞ so we mus t also hav e sup( k L ⊗ R M ) < ∞ , that is, pcd M < ∞ . This pro v es (ii), implication ⇒ . (ii), implication ⇐ : If M is zero then it is certainly in D c ( R ), so assume that M is non- zero in D + ( R ). Then inf M is an inte ger and inf M ≤ p cd M by Lemma 1 .6(ii). On the ot her hand, pcd M < ∞ , so p = p cd M is an integer . But p cd M = sup ( k L ⊗ R M ) so β j ( M ) = dim k H j ( k L ⊗ R M ) = 0 for j ≥ p + 1 . In Construction 2.2, b y part (ii) this implies β j ( M h p + 1 i ) = 0 for j ≥ p + 1 . Ho wev er, part (iii) of the construction sa ys inf M h p + 1 i ≥ p + 1 , so inf ( k L ⊗ R M h p + 1 i ) = inf M h p + 1 i ≥ p + 1 b y Lemma 1.5, that is β j ( M h p + 1 i ) = dim k H j ( k L ⊗ R M h p + 1 i ) = 0 for j < p + 1 . Altogether, β j ( M h p + 1 i ) = 0 fo r each j . That is, eac h cohomolog y group of k L ⊗ R M h p + 1 i is zero and so k L ⊗ R M h p + 1 i is itself zero. Lemma 1.6(i) hence gives M h p + 1 i ∼ = 0 . But now the distinguished triangles in Construction 2.2, starting with Σ − p R ( β p ) → M h p i → M h p + 1 i → and running bac kw ards to the first one, Σ − i R ( β i ) → M → M h i + 1 i → , sho w that M is finite ly built from R since eac h β j is finite. That is, M is in D c ( R ).  COCHAIN DG ALGEBRA S 9 Remark 2.4. Let M b e non-zero in D c ( R ). Prop osition 2.3 implies that M is lo cally finite in D + ( R ) with p cd M < ∞ . The pro of of the prop osition actually give s a bit more: F irst, p = p cd M is an integer. Secondly , in Cons truction 2.2, we ha ve M h p + 1 i ∼ = 0 (2.a) in D ( R ). Com bining this isomorphis m with the distinguished tr ia ngle Σ − p R ( β p ) → M h p i → M h p + 1 i → sho ws M h p i ∼ = Σ − p R ( β p ( M )) (2.b) in D ( R ). 3. Inequalities This section con tin ues to set up the DG mo dule adaptation of Serre’s ideas from the pro of of [1 1, thm. 10, p. 217]. The main items a re Prop ositions 3.5 and 3.6 whic h use Construction 2.2 to prov e some term wise inequalities of p ow er s eries. Setup 3.1. Let F : D ( R ) → M o d ( k ) b e a k -linear homological functor which resp ects copro ducts. F or M in D ( R ), we set f M ( t ) = X ℓ dim k F (Σ ℓ M ) t ℓ . Remark 3.2. In the g eneralit y of Setup 3.1, the expression f M ( t ) ma y not b elong to a n y reasonable set. So me of the co efficien ts ma y b e infinite, and there ma y b e non- zero co efficien ts in arbit r a rily high p ositiv e and nega tiv e degrees at the same time. Ho wev er, we will see that there are circu mstances in whic h f M ( t ) is a Lauren t series. Notation 3.3. T erm wise inequalities ≤ of co efficien ts betw een expres- sions like f M ( t ) mak e sense; they will b e denoted b y 4 . Lik ewise, it mak es sense to add suc h expres sions, and to m ultiply them b y a n um b er or a p o w er of t . 10 ANDERS J. FR A NKILD AND PETER JØRGENSEN Finally , the de gr e e of an expres sion lik e f M ( t ) is defined b y deg  X ℓ f ℓ t ℓ  = sup { ℓ | f ℓ 6 = 0 } . Lemma 3.4. L et M b e in D ( R ) . (i) f Σ j M ( t ) = t − j f M ( t ) . (ii) f M ( β ) ( t ) = β f M ( t ) . (iii) I f M ′ → M → M ′′ → is a distinguishe d triangl e in D ( R ) , then ther e is a termwise ine quality f M ( t ) 4 f M ′ ( t ) + f M ′′ ( t ) . Pr o of. Parts (i) and ( ii) ar e clear. In part (iii), the distinguished tri- angle give s a long exact sequence consisting of pieces F (Σ ℓ M ′ ) → F (Σ ℓ M ) → F (Σ ℓ M ′′ ) , whence dim k F (Σ ℓ M ) ≤ dim k F (Σ ℓ M ′ ) + dim k F (Σ ℓ M ′′ ) and the lemma fo llows.  Prop osition 3.5. L et M b e no n -zer o in D + ( R ) . Write i = inf M , l e t u ≥ i b e an inte ger, a n d c onsider C o nstruction 2 .2. Ther e ar e termwise ine qualities (i) f M ( t ) 4 ( β i ( M ) t i + · · · + β u ( M ) t u ) f R ( t ) + f M h u +1 i ( t ) , (ii) f M h u +1 i ( t ) 4 f M ( t ) + t − 1 ( β i ( M ) t i + · · · + β u ( M ) t u ) f R ( t ) . Pr o of. This fo llo ws b y a pplying Lemma 3.4 succes siv ely to the distin- guished triangles of Construction 2.2. F or instance, (i) can b e prov ed as follows , f M ( t ) = f M h i i ( t ) 4 f Σ − i R ( β i ) ( t ) + f M h i +1 i ( t ) = β i ( M ) t i f R ( t ) + f M h i +1 i ( t ) 4 β i ( M ) t i f R ( t ) + f Σ − i − 1 R ( β i +1 ) ( t ) + f M h i +2 i ( t ) = β i ( M ) t i f R ( t ) + β i +1 ( M ) t i +1 f R ( t ) + f M h i +2 i ( t ) 4 · · · = ( β i ( M ) t i + · · · + β u ( M ) t u ) f R ( t ) + f M h u +1 i ( t ) , and (ii) is prov ed b y s imilar manipulations.  COCHAIN DG ALGEBRA S 11 Prop osition 3.6. L et M b e non-zer o in D c ( R ) and write i = inf M and p = p cd M . Then i and p ar e inte gers with i ≤ p , we have β p ( M ) 6 = 0 , and ther e ar e termwise ine qualities (i) f M ( t ) 4 ( β i ( M ) t i + · · · + β p ( M ) t p ) f R ( t ) , (ii) β p ( M ) t p f R ( t ) 4 f M ( t ) + t − 1 ( β i ( M ) t i + · · · + β p − 1 ( M ) t p − 1 ) f R ( t ) . Pr o of. Prop osition 2.3(i) says that M is in D + ( R ), and since M is non-zero it follo ws that i = inf M is an in teger. Remark 2.4 sa ys that p = p cd M is an in teger. Lemma 1.6(ii) sa ys i ≤ p . Since p = sup( k L ⊗ R M ), it is c lear that β p ( M ) = dim k H p ( k L ⊗ R M ) 6 = 0. Consider Construction 2 .2 fo r M . By Remark 2.4, Equations (2.a) and (2.b), we ha ve M h p i ∼ = Σ − p R ( β p ( M )) and M h p + 1 i ∼ = 0. Inserting this into the ineq ualities of Proposition 3.5 give s the inequalities of the presen t prop osition.  As an immediate application, consider the follo wing le mma. Lemma 3.7. L et M b e in D c ( R ) . If f R ( t ) is a L aur ent se rie s in t − 1 then so is f M ( t ) , and deg f M ( t ) = deg f R ( t ) + p cd M . Pr o of. If M is zero then f M ( t ) = 0 is trivially a Laurent series in t − 1 , and the equation o f the lemma reads −∞ = −∞ so the lemma holds. Supp ose that M is non-zero in D c ( R ). Since f R ( t ) is a Lauren t series in t − 1 , Prop o sition 3.6(i) implies that so is f M ( t ) since eac h β j ( M ) is finite, cf. Prop osition 2.3(i) and Construction 2.2. If f R ( t ) has all co efficien ts equal to zero then Proposition 3.6(i) forces f M ( t ) to hav e all co efficien ts equal to zero, and the equation of the lemma reads −∞ = −∞ so the lemma holds. Supp ose that not all co efficien ts o f f R ( t ) are equal to zero. Then Prop o- sition 3 .6(i) implies deg f M ( t ) ≤ deg f R ( t ) + p = deg f R ( t ) + p cd M . On the other hand, consider the inequalit y of Prop osition 3.6(ii). The left hand side con ta ins a non-zero monomial of degree deg f R ( t ) + p . The righ t hand side consists of t wo terms, and the second one, t − 1 ( β i ( M ) t i + · · · + β p − 1 ( M ) t p − 1 ) f R ( t ), consists of monomials of degree < deg f R ( t ) + 12 ANDERS J. FR A NKILD AND PETER JØRGENSEN p . Hence the first term, f M ( t ), mus t contain a non-zero monomial of degree deg f R ( t ) + p whence deg f M ( t ) ≥ deg f R ( t ) + p = deg f R ( t ) + p cd M . Com bining the display ed inequalities giv es the desired equation.  4. M ain resul ts This section sho ws the co chain Amplitude Inequality , Auslander-Buc hs- baum Equality , and Gap Theorem for Bass n umbers in Corollaries 4 .4, 4.7, a nd 4.11. These results are special cases of Theorems 4.3, 4.5, and 4.8. Setup 4.1. F rom no w on, w e will only consider a sp ecial form of F and f from Setup 3.1. Namely , let P b e in D ( R o ) and set F ( − ) = H 0 ( P L ⊗ R − ) . This means that f M ( t ) = X ℓ dim k H ℓ ( P L ⊗ R M ) t ℓ (4.a) and in particular f R ( t ) = X ℓ dim k H ℓ ( P ) t ℓ . (4.b) Lemma 4.2. L et M b e in D c ( R ) , and let P b e lo c al ly finite in D − ( R o ) . Then sup( P L ⊗ R M ) = sup P + p cd M . Pr o of. The express ion f R ( t ) is giv en by Equation (4.b) and it is a La u- ren t series in t − 1 b ecause P is lo cally finite in D − ( R o ). Lemma 3.7 giv es deg f M ( t ) = deg f R ( t ) + p cd M . (4.c) Ho wev er, Equations (4.a) and (4.b) imply deg f M ( t ) = sup( P L ⊗ R M ) and deg f R ( t ) = sup P , so Equation (4.c) reads sup( P L ⊗ R M ) = sup P + p cd M as claimed.  COCHAIN DG ALGEBRA S 13 Theorem 4.3. L et M b e in D c ( R ) . L et P b e lo c al ly finite in D ( R o ) and supp ose amp P < ∞ . The n amp( P L ⊗ R M ) = amp P + p cd M − inf M . Pr o of. Lemma 4.2 sa ys sup( P L ⊗ R M ) = sup P + p cd M . Subtracting the equation o f Lemma 1.5 pro duces sup( P L ⊗ R M ) − inf ( P L ⊗ R M ) = sup P + p cd M − inf P − inf M , and this is the equation of the presen t theorem.  Corollary 4.4 (Amplitude Inequalit y) . L et M b e non-z er o in D c ( R ) . L et P b e lo c al ly finite in D ( R o ) and supp ose amp P < ∞ . T h en amp( P L ⊗ R M ) ≥ amp P . Pr o of. Combine Theorem 4.3 with Lemm a 1.6(ii).  Theorem 4.5. Assume that ther e is a P which is non-zer o in D c ( R o ) and satisfies sup P < ∞ . Set d = p cd P − sup P . If M is in D c ( R ) and s a tisfies sup M < ∞ , then p cd M = sup M + d. Pr o of. Prop osition 2.3( i) giv es that M and P are lo cally finite. They are also in D − b ecause they ha ve finite suprem um. Hence Lemma 4.2 sa ys sup( P L ⊗ R M ) = sup P + p cd M , and Lemma 4.2 with M and P in terc hang ed sa ys sup( P L ⊗ R M ) = sup M + p cd P . The tw o righ t hand sides m ust be equal, sup P + p cd M = sup M + p cd P , and rearranging terms prov es the prop osition.  Question 4.6. F or which DG alg ebras R do es there exist a DG mo dule lik e P ? F or suc h D G a lgebras, the in v ariant d = pcd P − sup P app ears to b e in teresting, and it w ould b e useful to find a form ula express ing it directly in terms of R . 14 ANDERS J. FR A NKILD AND PETER JØRGENSEN The following corollary considers tw o easy , sp ecial cases of Theorem 4.5 whic h can reasonably b e termed Auslander-Buc hsbaum Eq ualities. Corollary 4.7 (Auslander-Buchs baum Equalities) . L et M b e in D c ( R ) . (i) I f R has sup R < ∞ , then p cd M = sup M − sup R. (ii) I f sup M < ∞ a n d k is in D c ( R ) , then p cd M = sup M + p cd k . Pr o of. Both pa r t s follow from Theorem 4.5, b y us ing R and k in place of P . In part (i), note that when M is in D c ( R ), it is finitely built from R , so sup R < ∞ implies sup M < ∞ .  Theorem 4.8. L et M b e lo c al ly finite in D + ( R ) . L et P b e lo c al ly finite and non-zer o in D ( R o ) and supp ose amp P < ∞ . L et g ≥ amp P . If the Be tti numb ers of M h a ve a gap of length g in the sense that ther e is a j such that β ℓ ( M )      6 = 0 for ℓ = j , = 0 for j + 1 ≤ ℓ ≤ j + g , 6 = 0 for ℓ = j + g + 1 , then amp( P L ⊗ R M ) ≥ g + 1 . Pr o of. Since β j ( M ) 6 = 0, it is clear t ha t M is non-zero in D + ( R ). By (de)suspending, we can supp ose inf M = inf P = 0. W rite s = sup P ; then s = a mp P and we hav e the assumption g ≥ s. Lemma 1.5 giv es inf ( P L ⊗ R M ) = inf P + inf M = 0, so H 0 ( P L ⊗ R M ) 6 = 0. T o sho w the lemma, we need to prov e H ℓ ( P L ⊗ R M ) 6 = 0 for some ℓ ≥ g + 1, so let us assume H ≥ g +1 ( P L ⊗ R M ) = 0 and sho w a con tr a diction. Lemma 1.5 implies β ℓ ( M ) = dim k H ℓ ( k L ⊗ R M ) = 0 fo r ℓ < 0, so the in teger j from the prop osition satisfies j ≥ 0 . COCHAIN DG ALGEBRA S 15 Hence in particular H ≥ j + g +1 ( P L ⊗ R M ) = 0 . (4.d) Inserting (4.a) and (4 .b) in to the inequalit y of Pro p osition 3.5 (i) gives X ℓ dim k H ℓ ( P L ⊗ R M ) t ℓ 4 ( β 0 ( M ) t 0 + · · · β u ( M ) t u ) X ℓ dim k H ℓ ( P ) t ℓ + X ℓ dim k H ℓ ( P L ⊗ R M h u + 1 i ) t ℓ where u ≥ 0 is an in teger and M h u + 1 i is defined b y Construction 2.2. W e hav e β ℓ ( M ) = 0 for j + 1 ≤ ℓ ≤ j + g while P ℓ dim k H ℓ ( P ) t ℓ has terms only of degree 0 , . . . , s , so the first term on the righ t hand side is zero in degree ℓ for j + s + 1 ≤ ℓ ≤ j + g . And inf M h u + 1 i ≥ u + 1 b y Construction 2 .2 (iii) so Lemma 1.5 implies inf ( P L ⊗ R M h u + 1 i ) ≥ u + 1, so b y pick ing u larg e w e can mov e the second term on the right hand side into large degrees and thereb y ignor e it . It f o llo ws that the left hand side is also zero in degree ℓ for j + s + 1 ≤ ℓ ≤ j + g ; that is, H ℓ ( P L ⊗ R M ) = 0 for j + s + 1 ≤ ℓ ≤ j + g . Com bining with Eq uation (4.d) sho ws H ≥ j + s +1 ( P L ⊗ R M ) = 0 . (4.e) No w insert (4.a) and (4.b) in to Prop osition 3.5(ii) with u = j , X ℓ dim k H ℓ ( P L ⊗ R M h j + 1 i ) t ℓ 4 X ℓ dim k H ℓ ( P L ⊗ R M ) t ℓ + t − 1 ( β 0 ( M ) t 0 + · · · + β j ( M ) t j ) X ℓ dim k H ℓ ( P ) t ℓ . Again, P ℓ dim k H ℓ ( P ) t ℓ only has terms of degree 0 , . . . , s , so on the righ t hand side, the second term is zero in degrees ≥ j + s . In particular, it is zero in degrees ≥ j + s + 1, a nd since Equation (4 .e) implies t ha t 16 ANDERS J. FR A NKILD AND PETER JØRGENSEN the same holds for t he first term, it mus t also hold for the left hand side, that is, H ≥ j + s +1 ( P L ⊗ R M h j + 1 i ) = 0 . (4.f ) No w, inf M h j + 1 i ≥ j + 1 b y Construction 2.2(iii), so Lemma 1.5 implies inf ( k L ⊗ R M h j + 1 i ) ≥ j + 1 . And Construction 2.2(ii) sa ys β ℓ ( M h j + 1 i ) = β ℓ ( M ) for ℓ ≥ j + 1, so β ℓ ( M ) = 0 for j + 1 ≤ ℓ ≤ j + g g iv es β ℓ ( M h j + 1 i ) = 0 for j + 1 ≤ ℓ ≤ j + g , that is, H ℓ ( k L ⊗ R M h j + 1 i ) = 0 for j + 1 ≤ ℓ ≤ j + g , so w e ev en ha v e inf ( k L ⊗ R M h j + 1 i ) ≥ j + g + 1 , that is, inf M h j + 1 i ≥ j + g + 1 b y Lemma 1.5 again. Hence inf ( P L ⊗ R M h j + 1 i ) ≥ j + g + 1 b y Lemma 1.5. How ev er, g ≥ s , so the only w ay this can b e compatible with Equation (4.f) is if we ha ve P L ⊗ R M h j + 1 i ∼ = 0 . By Lemma 1.6(i) this means that M h j + 1 i ∼ = 0. And this g iv es β j + g +1 ( M ) = β j + g +1 ( M h j + 1 i ) = 0 , whic h is the desired con tradiction since w e had assumed β j + g +1 ( M ) 6 = 0.  Corollary 4.9 (Ga p Theorem fo r Betti num b ers) . Supp ose sup R < ∞ . L et M b e lo c al ly finite in D + ( R ) . L et g ≥ sup R . If the Betti numb ers of M have a g a p of leng th g in the sense that ther e is a j such that β ℓ ( M )      6 = 0 for ℓ = j , = 0 for j + 1 ≤ ℓ ≤ j + g , 6 = 0 for ℓ = j + g + 1 , then amp M ≥ g + 1 . Pr o of. This follo ws from Theorem 4.8 by using R in place of P .  COCHAIN DG ALGEBRA S 17 Remark 4. 10. Con v ersely , if sup R < ∞ and M is lo cally finite in D + ( R ) with amp M ≤ sup R , then t he Betti n um b ers of M can ha v e no gaps of length bigger than or equal to sup R . By ev aluating the prev ious theorem on the k -linear dual Hom k ( M , k ), w e immediately get the follow ing result in whic h the Ba s s numb ers of a DG module are µ j ( M ) = dim k H j (RHom R ( k , M )) . Corollary 4.11 (Gap Theorem for Bass n umbers) . Supp ose sup R < ∞ . L et M b e lo c al ly finite in D − ( R ) . L et g ≥ sup R . I f the Bass numb ers of M have a gap of length g in the sense that ther e is a j such that µ ℓ ( M )      6 = 0 for ℓ = j , = 0 for j + 1 ≤ ℓ ≤ j + g , 6 = 0 for ℓ = j + g + 1 , then amp M ≥ g + 1 . Remark 4. 12. Con v ersely , if sup R < ∞ and M is lo cally finite in D − ( R ) with amp M ≤ sup R , then the Bass n um b ers of M can ha ve no gaps of length bigger than or equal to sup R . In par t icular, the Bass num b ers of R itself can ha v e no gaps of length bigger than or equal to sup R . This shows f o r the presen t class of D G al- gebras that the a nsw er is a ffirmativ e to the question ask ed by Avramo v and F oxb y in [1, Question 3.10] for lo cal c hain DG algebras. In fact, it sho ws that for simply connected co chain DG alg ebras, Avramo v and F o xb y’s conjectural b ound on the gap length o f the Bass num b ers can b e sharp ened b y an a moun t of one. 5. Topology This section applies the Auslander-Buc hsbaum Equalit y and t he Gap Theorem to the singular co c hain DG algebra of a t o p ological space. The contex t will b e a fibration o f top ological spaces, and w e reco v er in Theorem 5.5 that homolog ical dimension is additive on fibrations. Theorem 5.6 sho ws that a gap of length g in the Betti num b ers of the fibre space implies that the total space has non- zero cohomology in a dimension ≥ g + 1. A reference for the algebraic top olo gy of this section is [5]. 18 ANDERS J. FR A NKILD AND PETER JØRGENSEN Setup 5.1. Let F → X → Y b e a fibration of top ological spaces where dim k H j ( X ; k ) < ∞ and dim k H j ( Y ; k ) < ∞ for eac h j and where Y is simply connected. Remark 5.2. Recall that the singular c o h omolo gy H j ( Z ; k ) of a top o- logical space Z is defined in t erms of t he sing ular c o chain c omplex C ∗ ( Z ; k ) b y H j ( Z ; k ) = H j (C ∗ ( Z ; k )) . The singular co c hain complex is a D G algebra, and by [5, exa. 6, p. 146] the assumptions on the space Y mean that C ∗ ( Y ; k ) is quasi-isomorphic to a DG algebra whic h falls under Setup 1.1, so the results pro v ed so far apply to it. Moreo ver, the contin uous map X → Y induces a morphism C ∗ ( Y ; k ) → C ∗ ( X ; k ) whereb y C ∗ ( X ; k ) b ecomes a D G bi-C ∗ ( Y ; k )- mo dule whic h is lo cally finite and b elongs to D + b y the assumptions on X . Notation 5.3. The dimensions dim k H j ( Z ; k ) are called the Betti num- b ers of the top o logical space Z . By hd Z = sup { j | H j ( Z ; k ) 6 = 0 } = sup C ∗ ( Z ; k ) (5.a) is denoted the homolo gic al dim ension of Z ; it is a non-negativ e in teger or ∞ . By Ω Z is denoted the Mo or e lo o p sp ac e of Z . Lemma 5.4. (i) We have dim k H j ( F ; k ) = β j C ∗ ( Y ; k ) (C ∗ ( X ; k )) and hd F = p cd C ∗ ( Y ; k ) (C ∗ ( X ; k )) . (ii) We hav e dim k H j (Ω Y ; k ) = β j C ∗ ( Y ; k ) ( k ) and hd Ω Y = p cd C ∗ ( Y ; k ) ( k ) . Pr o of. (i) W e k no w that C ∗ ( F ; k ) ∼ = k L ⊗ C ∗ ( Y ; k ) C ∗ ( X ; k ) in D ( k ) b y [5, thm. 7.5]. T a king the dimension of the j ’th cohomology prov es the dis- pla yed equation, and the other equation is a n immediate conseque nce, cf. Equations (1.c) and (5.a). (ii) This follow s b y using (i) on the fibration Ω Y → P Y → Y where P Y is the Mo or e p ath sp ac e of Y : T he space P Y is con tractible so C ∗ ( P Y ; k ) is isomorphic to the DG mo dule k in D (C ∗ ( Y ; k )).  COCHAIN DG ALGEBRA S 19 Theorem 5.5 (Additivit y of homological dimension) . (i) If hd F < ∞ and hd Y < ∞ , then hd X = hd F + hd Y . (ii) I f hd X < ∞ and hd Ω Y < ∞ , then hd X = hd F − hd Ω Y . Pr o of. Let us apply Corollary 4.7 to the data R = C ∗ ( Y ; k ) a nd M = C ∗ ( X ; k ). (i) Using Lemma 5.4(i) w e ha v e p cd R ( M ) = p cd C ∗ ( Y ; k ) (C ∗ ( X ; k )) = hd F < ∞ . By Prop osition 2.3(ii), this sa ys that M is in D c ( R ). Moreo v er, Equa- tion (5.a ) giv es sup R = sup C ∗ ( Y ; k ) = hd Y < ∞ . This show s that Coro llary 4.7( i) do es apply , and ev aluating its equation giv es hd F = hd X − hd Y , pr oving (i). (ii) Using Lemma 5 .4(ii) w e ha v e p cd R ( k ) = p cd C ∗ ( Y ; k ) ( k ) = hd Ω Y < ∞ . By Prop osition 2.3(ii), this s a ys that k is in D c ( R ). But w e also ha ve sup M = sup C ∗ ( X ; k ) = hd X < ∞ , and since M is lo cally finite, it follo ws tha t dim k H( M ) < ∞ whence M is finitely built from k in D ( R ). Hence M is a lso in D c ( R ). This sho ws that Corollary 4.7(ii) do es apply , and ev aluating its equa- tion gives hd F = hd X + hd Ω Y , proving (ii).  Theorem 5.6 (Gap) . (i) If hd Y < ∞ and the Betti numb ers of F have a gap of length g ≥ hd Y in the sense that ther e is a j such that H ℓ ( F ; k )      6 = 0 for ℓ = j , = 0 for j + 1 ≤ ℓ ≤ j + g , 6 = 0 for ℓ = j + g + 1 , then hd X ≥ g + 1 . 20 ANDERS J. FR A NKILD AND PETER JØRGENSEN (ii) I f hd X < ∞ and the Be tti n umb e rs of Ω Y have a gap of leng th g ≥ hd X in the sen se that ther e is a j such that H ℓ (Ω Y ; k )      6 = 0 for ℓ = j , = 0 for j + 1 ≤ ℓ ≤ j + g , 6 = 0 for ℓ = j + g + 1 , then hd F ≥ g + 1 , and hd F < ∞ for c es hd Y = ∞ . Pr o of. (i) Let us apply Corollary 4.9(i) to the data R = C ∗ ( Y ; k ) a nd M = C ∗ ( X ; k ). Then sup R = hd Y is clear, β j ( M ) = dim k H j ( F ; k ) holds b y Lemma 5.4(i), a nd a mp M = hd X is clear, so (i) follow s. (ii) Let us apply Theorem 4.8 to the data R = C ∗ ( Y ; k ), M = k , a nd P = C ∗ ( X ; k ). Then amp P = hd X is clear, β j ( M ) = β j ( k ) = dim k H j (Ω Y ; k ) holds b y Lemma 5.4(ii) , and amp( P L ⊗ R M ) = amp (C ∗ ( X ; k ) L ⊗ C ∗ ( Y ; k ) k ) = amp(C ∗ ( F ; k )) = hd F since C ∗ ( X ; k ) L ⊗ C ∗ ( Y ; k ) k ∼ = C ∗ ( F ; k ) b y [5, thm. 7.5], so the ineq ualit y of (ii) follo ws. Moreo ver, if w e had hd F < ∞ and hd Y < ∞ , then Theorem 5.5(i) w ould apply , and we w ould get the con tradiction g ≥ hd X = hd F + hd Y ≥ g + 1 + hd Y ≥ g + 1 .  Example 5.7. Set Y = S n ∨ S n +1 ∨ S n +2 ∨ · · · for an n ≥ 2. Then Y is simply connected with dim Q H j ( Y ; Q ) < ∞ for eac h j , and X j dim Q H j (Ω Y ; Q ) t j = 1 1 − ( t n + t n +1 + t n +2 + · · · ) b y [5, exa. 1, p. 460]. It follows tha t H 1 (Ω Y ; Q ) = · · · = H n − 1 (Ω Y ; Q ) = 0, so the Betti n um b ers of Ω Y hav e a gap of length n − 1. Hence, if F → X → Y is a fibration with dim Q H ∗ ( X ; Q ) < ∞ and hd X ≤ n − 1, then Theorem 5.6(ii) say s hd F ≥ n . COCHAIN DG ALGEBRA S 21 Reference s [1] L. 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Jørg ensen, Amplitude inequalities for Differential Gr aded mo dule s , prepr int (2006). mat h.RA/ 060141 6 . [8] P . Jørgens en, Auslander-R eiten the ory over top olo gic al sp ac es , Co mmen t. Math. Helv. 79 (20 0 4), 1 60–18 2. [9] B. Keller, Deriving DG c ate gories , Ann. Sci. ´ Ecole Norm. Sup. (4) 27 (1994), 63–10 2. [10] K. Sc hmidt, Ausla nder-Reiten theo r y for simply connected differential graded alg ebras, Ph.D. thesis, University of Paderbo r n, Paderb o rn, 2 007. math.R T/080 1.065 1 . [11] J.-P . Serr e , Coho molo gie mo dulo 2 des c omplexes d’Eilenb er g-MacL ane , Com- men t. Math. Helv. 27 (19 53), 1 98–2 32. School of Ma thema tics a nd St a tistics, N ew castle U niversity, New- castle upon Tyne NE 1 7R U , United Kingdom E-mail addr ess : peter .jorg ensen@ ncl.ac.uk URL : http: //www .staff.ncl.ac.uk/peter.jorgensen