Extension Groups for DG Modules

Transcription

1 Georgia Southern University Digital Southern Mathematical Sciences Faculty Publications Department of Mathematical Sciences Extension Groups for DG Modules Saeed Nasseh Georgia Southern University, Sean Sather-Wagstaff Clemson University Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Nasseh, Saeed, Sean Sather-Wagstaff 26 "Extension Groups for DG Modules" Communications in Algebra, 45 (): doi: 8/ source: This article is brought to you for free and open access by the Department of Mathematical Sciences at Digital Southern It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Southern For more information, please contact

2 EXTENSION GROUPS FOR DG MODULES arxiv:6456v [mathac 2 Apr 26 SAEED NASSEH AND SEAN SATHER-WAGSTAFF Abstract Let M and N be differential graded (DG) modules over a positively graded commutative DG algebra A We show that the Ext-groups Ext i A (M,N) defined in terms of semi-projective resolutions are not in general isomorphic to the Yoneda Ext-groups YExt i A (M,N) given in terms of equivalence classes of extensions On the other hand, we show that these groups are isomorphic when the first DG module is semi-projective Introduction Convention In this paper, R is a commutative ring with identity Given two R-modules M and N, a classical result originating with work of Baer [4 states that Ext R(M,N), defined via projective/injective resolutions, is isomorphic to the abelian group YExt R(M,N) of equivalence classes of exact sequences of the form N X M The purpose of this note is to discuss possible extensions of this result to the abelian category of differential graded (DG) modules over a positively graded commutative DG algebra A See Section 2 for background information on this category Specifically, we show that Baer s isomorphism fails in general in this context: Examples3and32exhibitDGA-modulesM,N withext A (M,N) YExt A (M,N) (See 24 and 26 below for definitions) On the other hand, the following result shows that a reasonable hypothesis on the first module does yield such an isomorphism Theorem A Let A be a DG R-algebra, and let N, Q be DG A-modules such that Q is semi-projective Then there is an isomorphism YExt i A(Q,N) Ext i A(Q,N) of abelian groups for all i This is the main result of Section 3; see Proof 38 In the subsequent Section 4, we discuss some properties of YExt with respect to truncations It is worth noting here that we apply results from this paper in our answer to a question of Vasconcelos in [2 Specifically, in that paper, we investigate DG A- modulesc withext A (C,C) usinggeometrictechniques Thesetechniquesyield an isomorphism between YExt A(C,C) and a certain quotient of tangent spaces; it is then important for us to know when the vanishing of Ext A(C,C) implies the vanishing of related YExt -modules; see Proposition 44 below 2 Mathematics Subject Classification 3D2, 3D7, 3D9 Key words and phrases Differential graded algebras, differential graded modules, Yoneda Ext Sather-Wagstaff was supported in part by North Dakota EPSCoR, National Science Foundation Grant EPS-84442, and NSA grant H

3 2 SAEED NASSEH AND SEAN SATHER-WAGSTAFF 2 DG Modules We assume that the reader is familiar with the category of R-complexes and the derived category D(R) Standard references for these topics are [6, 7, 9,, 3, 4 For clarity, we include some definitions and notation Definition 2 In this paper, complexes of R-modules ( R-complexes for short) M n+2 n+ M are indexed homologically: M M n+ M M n n M n M n The degree of an element m M is denoted m The infimum and supremum of M are the infimum and supremum, respectively, of the set {n Z H n (M) } The tensor product of two R-complexes M,N is denoted M R N, and the Hom complex is denoted Hom R (M,N) A chain map M N is a cycle in Hom R (M,N) Next we discuss DG algebras and DG modules, which are treated in, eg, [, 2, 3, 5, 8, We follow the notation and terminology from [2, 5; given the slight differences in the literature, though, we include a summary next Definition 22 A positively graded commutative differential graded R-algebra (DG R-algebra forshort)isanr-complexaequipped withachainmapµ A : A R A A with ab : µ A (a b) that is associative, unital, and graded commutative such that A i for i < The map µ A is the product on A Given a DG R-algebra A, the underlying algebra is the graded commutative R-algebra A i A i A differential graded module over a DG R-algebra A (DG A-module for short) is an R-complex M with a chain map µ M : A R M M such that the rule am : µ M (a m) is associativeand unital The map µ M is the scalar multiplication on M The underlying A -module associated to M is the A -module M i Z M i The DG A-module Hom A (M,N) is the subcomplex of Hom R (M,N) of the A- linear homomorphisms A morphism M N of DG A-modules is a cycle in Hom A (M,N) Projective objects in the category of DG A-modules are called categorically projective Quasiisomorphisms of DG A-modules are identified by the symbol, also used for the quasiisomorphic equivalence relation Two important DG R-algebras to keep in mind are R itself and, more generally, the Koszul complex over R (on a finite sequence of elements of R) with the exterior product A DG R-module is just an R-complex, and a morphism of DG R-modules is simply a chain map Remark 23 Let A be a DG R-algebra The category of DG A-modules is an abelian category with enough projectives Definition 24 Let A be a DG R-algebra, and let M, N be DG A-modules For each i we have a well-defined Yoneda Ext group YExt i A (M,N), defined in terms of a resolution of M by categorically projective DG A-modules: Q Q M A standard result shows that YExt A (M,N) is isomorphic to the set of equivalence classes of exact sequences N X M of DG A-modules under the Baer sum; see, eg, [5, (346) and the proof of Theorem 35 We now turn to the derived category D(A), and related notions Definition 25 Let A be a DG R-algebra A DG A-module Q is graded-projective if Hom A (Q, ) preserves surjective morphisms, that is, if Q is a projective graded

4 EXTENSION GROUPS FOR DG MODULES 3 R -module The DG module Q is semi-projective if Hom A (Q, ) respects surjective quasiisomorphisms, that is, if Q is graded-projective and respects quasiisomorphisms A semi-projective resolution of M is a quasiisomorphism L M of DG A-modules such that L is semi-projective Fact 26 Let A be a DG R-algebra Then every DG A-module has a semiprojective resolution Definition 27 Let A be a DG R-algebra The derived category D(A) is formed from the category of DG A-modules by formally inverting the quasiisomorphisms; see [ Isomorphisms in D(A) are identified by the symbol The derived functor RHom A (M,N) is defined via a semi-projective resolution P M, as RHom A (M,N) Hom A (P,N) For each i Z, set Ext i A(M,N) : H i (RHom A (M,N)) 3 DG Ext vs Yoneda Ext We begin this section with examples of DG A-modules M and N such that Ext A(M,N) YExt A(M,N) These present two facets of the distinctness of Ext and YExt, as the first example has M and N both bounded, while the second one (from personal communication with Avramov) has M graded-projective Example 3 Let R k[x, and consider the following exact sequence of DG R-modules, ie, exact sequence of R-complexes: R R k R X R k R X R k This sequence does not split over R (it is not even degree-wise split) so it gives a non-trivial class in YExt R(k,R), and we conclude that YExt R(k,R) On the other hand, k is homologically trivial, so we have Ext R(k,R) since is a semi-free resolution of k Example 32 Let R k[x/(x 2 ) and consider the following exact gradedprojective DG R-module M R X R X Since M is exact, we X have Ext i R(M,M) for all i We claim, however, that YExt R(M,M) To see this, first note that M is isomorphic to the suspension ΣM and that M is not contractible Thus, the mapping cone sequence for the identity morphism id M is isomorphic to one of the form M X M and is not split

5 4 SAEED NASSEH AND SEAN SATHER-WAGSTAFF The definition of the isomorphism YExt i A (Q,N) Exti A (Q,N) for i in Theorem A is contained in the following construction The subsequent lemma and theorem show that Ψ is a well-defined isomorphism Construction 33 Let A be a DG R-algebra, and let N, Q be DG A-modules suchthatqisgraded-projective DefineΨ: YExt A (Q,N) H (Hom A (Q,N))as follows Note that if Q is semi-projective, then Ext A(Q,N) H (Hom A (Q,N)), which fits with what we have in Theorem A Let ζ YExt A (Q,N) be represented by the sequence N f X g Q (33) Since Q is graded-projective, this sequence is graded-split, that is there are elements h Hom A (X,N) and k Hom A (Q,X) with hf id N gk id Q hk fh+kg id X Thus, the sequence (33) is isomorphic to one of the form N i+ N i N i N i ǫ i X i+ N i Q i X i ǫ i N i Q i π i Q i+ Q i Q i π i Q i (332) N i X i Q i where ǫ j is the natural inclusion and π j is the natural surjection for each j Since this diagram comes from a graded-splitting of (33), the scalar multiplication on the middle column of (332) is the natural one a[ p q [ ap aq (We write elements of N i Q i as column vectors) The fact that (332) commutes implies that i X has a specific form: X i [ N i λ i Q i (333) Here, we have λ i : Q i N i, that is, λ {λ i } Hom R (Q,N) Since the horizontal maps in the sequence (332) are morphisms of DG A-modules, it follows that λ is a cycle in Hom A (Q,N) Thus, λ represents a homology class in H (Hom A (Q,N)), and we define Ψ: YExt A (Q,N) H (Hom A (Q,N)) by setting Ψ(ζ) equal to [λ the homology class of λ in H (Hom A (Q,N)) Lemma 34 Let A be a DG R-algebra, and let N, Q be DG A-modules such that Q is graded-projective Then the map Ψ: YExt A(Q,N) H (Hom A (Q,N)) from Construction 33 is well-defined

6 EXTENSION GROUPS FOR DG MODULES 5 Proof Let ζ YExt A (Q,N) be represented by the sequence (332), and let ζ be represented by another exact sequence N i+ N i N i N i ǫ i X i+ N i Q i X i ǫ i N i Q i π i Q i+ Q i Q i π i Q i (34) where N i X i [ i X N i λ i Q i Q i (342) We need to show that λ λ Im( HomA(Q,N) ) The sequences (332) and (34) are equivalent in YExt R(Q,N), so for each i there is a commutative diagram N i ǫ i N i Q i ui vi [ w i x i π i Q i N i ǫ i N i Q i π i Q i (343) where the middle vertical arrow describes a DG A-module isomorphism, and such that the following diagram commutes for all i [ N i λ i Q i N i Q i ui vi [ w i x i [ ui v i w i x i N i Q i N i Q i [ N i λ i Q i N i Q i (344) The fact that diagram (343) commutes implies that u i id Ni, x i id Qi, and w i Also, the fact that the middle vertical arrow in diagram (343) describes a DG A-module morphism implies that the sequence v i : Q i N i respects scalar multiplication, ie, we have v Hom A (Q,N) The fact that diagram (344) commutes implies that λ i λ i N i v i v i Q i We conclude that λ λ HomA(Q,N) (v) Im( HomA(Q,N) ), so Ψ is well-defined The next result contains the case i of Theorem A from the introduction, because if Q is semi-projective, then Ext A(Q,N) H (Hom A (Q,N)) Theorem 35 Let A be a DG R-algebra, and let N, Q be DG A-modules such that Q is graded-projective Then the map Ψ: YExt A (Q,N) H (Hom A (Q,N)) from Construction 33 is a group isomorphism Proof We break the proof into three claims

7 6 SAEED NASSEH AND SEAN SATHER-WAGSTAFF Claim Ψisadditive Letζ,ζ YExt A (Q,N)berepresentedbyexactsequences N ǫ X π Q and N ǫ X π Q respectively, where X i N i Q i X i andthedifferentials X and X aredescribedasin(333)and(342), respectively We need to show that the Baer sum ζ +ζ is represented by an exact sequence N ǫ X π Q, where X [ i N i Q i and X N i i λ i+λ i, Q i with scalar multiplication a[ p q [ ap aq Note that it is straightforward to show that the sequence X defined in this way is a DG A-module, and the natural maps N ǫ X π Q are A-linear, using the analogous properties for X and X We construct the Baer sum in two steps The first step is to construct the pull-back diagram X X X π Q The DG module X is a submodule of the direct sum X X, so each X i is the submodule of (X X ) i X i X i N i Q i N i Q i consisting of all vectors [ x x such that π i (x ) π i (x), that is, all vectors of the form [p q p q T such that q q In other words, we have N i Q i N i X i (35) where the isomorphism is given by [p q p T [p q p q T The differential on X X is the natural diagonal map So, under the isomorphism (35), the differential on X has the form X i N i λ i Q i X i λ i N i N i Q i N i N i Q i N i X The next step in the construction of ζ +ζ is to build X, which is the cokernel of the morphism γ: N X given by p That is, since γ is injective, the complex X is determined by the exact sequence N γ X τ X It is straightforward to show that this sequence has the following form N i [ π [ p p N i Q i N i [ N i Q i i N i N i [ N i λ i Q i λ i N i [ N i λ i+λ i Q i [ N i Q i N i N i Q i By inspecting the right-most column of this diagram, we see that X has the desired form Furthermore, checking the module structures at each step of the construction, we see that the scalar multiplication on X is the natural one a[ p q [ ap aq This concludes the proof of Claim

8 EXTENSION GROUPS FOR DG MODULES 7 Claim 2 Ψ is injective Suppose that ζ Ker(Ψ) is represented by the displays (33) (333) The condition Ψ(ζ) says that λ Im( HomA(Q,N) ), so there is an element s Hom A (Q,N) such that λ HomA(Q,N) (s) Thus, for each i we have λ i i Ns i s i Q i From this, it is straightforward to show that the following diagram commutes: [ i N λ i Q i N i Q i [ si [ si N i Q i N i Q i [ N i Q i N i Q i From the fact that s is A-linear, it follows that the maps [ s i describe an A-linear isomorphism X N Q making the following diagram commute: N ǫ X π Q N ǫ N Q π Q In other words, the sequence (33) splits, so we have ζ, and Ψ is injective This concludes the proof of Claim 2 Claim 3 Ψ is surjective For this, let ξ H (Hom A (Q,N)) be represented by λ Ker( HomA(Q,N) ) Using the fact that λ is A-linear such that HomA(Q,N) (λ), one checks directly that the displays (332) (333) describe an exact sequence of DG A-module homomorphisms of the form (33) whose image under Ψ is ξ This concludes the proof of Claim 3 and the proof of the theorem Remark 36 After the results of this paper were announced, Avramov, et al [2 established the following generalization of Theorem 35 Proposition 37 Let A be a DG R-algebra, and let M and N be DG A-modules There is a monomorphism of abelian groups κ: H (Hom A (Σ M,N)) YExt U (M,N) with image equal to the set of equivalence classes of graded-split exact sequences of the form N X M To see how this generalizes Theorem 35, first note that if M is graded-projective, then the map κ is bijective, asin this caseeveryelement of YExt U(M,N) is gradedsplit; thus, we have H (Hom A (M,N)) H (Hom A (Σ M,N)) YExt U (M,N) Proof 38 (Proof of Theorem A) Using Theorem 35, we need only justify the isomorphism YExt i A (Q,N) Ext i A (Q,N) for i 2 Let L + L 2 L L π L Q be a resolution of Q by categorically projective DG A-modules Since each L j is categorically projective, we have YExt i A(L j, ) for all i and L j for each j, so we have Ext i A(L j, ) for all i Set Q i : Im L i for each i Each

9 8 SAEED NASSEH AND SEAN SATHER-WAGSTAFF L i is graded-projective, so the fact that Q is graded-projective implies that each Q i is graded-projective Now, a straightforward dimension-shifting argument explains the first and third isomorphisms in the following display for i 2: YExt i A (Q,N) YExt A (Q i,n) Ext A (Q i,n) Ext i A (Q,N) ThesecondisomorphismisfromTheorem35sinceeachQ i isgraded-projective The next example shows that one can have YExt A(Q,N) Ext A (Q,N), even when Q is semi-free Example 39 Continue with the assumptions and notation of Example 3, and set Q N R It is straightforward to show that the morphisms R R are precisely given by multiplication by fixed elements of R, so we have the first step in the next display: YExt A (R,R) R Ext A (R,R) The third step follows from the condition R Remark 3 It is perhaps worth noting that our proofs can also be used to give the isomorphisms from Theorem A when Q is not necessarily semi-projective, but N is semi-injective 4 YExt and Truncations For our work in [2, we need to know how YExt respects the following notion Definition 4 Let A be a DG R-algebra, and let M be a DG A-module Given an integer n, the nth soft left truncation of M is the complex τ(m) (n) : M n /Im( M n+) M n M n 2 with differential induced by M In other words, τ(m) (n) is the quotient DG A-module M/M where M is the following DG submodule of M: M M n+2 M n+ Im( M n+) Note that we have M if and only if n sup(m), so the natural morphism ρ: M τ(m) (n) of DG A-modules yields an isomorphism in D(A) if and only if n sup(m) Proposition 42 Let A be a DG R-algebra, and let M and N be DG A-modules Assume that n is an integer such that N i for all i > n Then the natural map YExt A (τ(m) (n),n) YExt A (M,N) induced by the morphism ρ: M τ(m) (n) from Definition 4 is a monomorphism Proof Let Υ denote the map YExt A (τ(m) (n),n) YExt A (M,N) induced by ρ Let α Ker(Υ) YExt A(τ(M) (n),n) be represented by the exact sequence N f X g τ(m) (n) (42) Note that, since N i (τ(m) (n) ) i for all i > n, we have X i for all i > n Our assumptions imply that Υ([α) [β where β comes from the following

10 EXTENSION GROUPS FOR DG MODULES 9 pull-back diagram: K K β : N f h X g h M (422) α : N f ρ X g ρ τ(m) (n) The middle row β of this diagramis split exact since [β, so there is a morphism F: X N of DG A-modules such that F f idn Note that K has the form K M n+2 M n+ M n+ Im( n+ M ) (423) because of the right-most column of the diagram We claim that F h It suffices to check this degree-wise When i > n, we have N i, so F i, and F i h i When i < n, the display (423) shows that K i, so h i, and F i h i For i n, we first note that the display (423) shows that n+ K is surjective In the following diagram, the faces with solid arrows commute because h and F are morphisms: K n+ hn+ F n+ K n+ X n+ N n K n F n hn Xn X n+ Since n+ K is surjective, a simple diagram chase shows that F n h n This establishes the claim To conclude the proof, note that the previous claim shows that the map K is a left-splitting of the top row of diagram (422) that is compatible with the left-splitting F of the middle row It is now straightforward to show that F induces a morphism F: X N of DG A-modules that left-splits the bottom row of diagram (422) Since this row represents α YExt A (τ(m) (n),n), we conclude that [α, so Υ is a monomorphism

11 SAEED NASSEH AND SEAN SATHER-WAGSTAFF The next example shows that the monomorphism from Proposition 42 may not be an isomorphism Example 43 Continue with the assumptions and notation of Example 3 The following diagram describes a non-zero element of YExt R (M,N): N R M R X R R π π R k It is straightforward to show that τ(m) (), so we have YExt A (τ(m) (),N) YExt A (M,N) thus this map is not an isomorphism Proposition 44 Let A be a DG R-algebra, and let C be a semi-projective DG A-module such that Ext R (C,C) For n sup(c), one has YExt A(C,C) YExt A(τ(C) (n),τ(c) (n) ) Proof From Theorem 35, we have YExt A (C,C) Ext A (C,C) For the remainder of the proof, assume without loss of generality that sup(c) < Another application of Theorem 35 explains the first step in the next display: YExt A (C,τ(C) (n)) Ext A (C,τ(C) (n)) Ext A (C,C) The second step comes from the assumption n sup(c) which guarantees that the natural morphism C τ(c) (n) represents an isomorphism in D(A) Proposition 42 implies that YExt A(τ(C) (n),τ(c) (n) ) is isomorphic to a subgroup of YExt A (C,τ(C) (n)), so YExt A (τ(c) (n),τ(c) (n) ), as desired Acknowledgments We are grateful to Luchezar Avramov for helpful discussions about this work References L L Avramov, Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 996), Progr Math, vol 66, Birkhäuser, Basel, 998, pp 8 MR 99m:322 2 L L Avramov, H-B Foxby, and S Halperin, Differential graded homological algebra, in preparation 3 L L Avramov, H-BFoxby, and J Lescot, Bass series of local ring homomorphisms of finite flat dimension, Trans Amer Math Soc 335 (993), no 2, MR 93d:326 4 R Baer, Erweiterung von Gruppen und ihren Isomorphismen, Math Z 38 (934), no, MR

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