Applications of Differential Graded Algebra Techniques in Commutative Algebra
aa r X i v : . [ m a t h . A C ] N ov APPLICATIONS OF DIFFERENTIAL GRADED ALGEBRATECHNIQUES IN COMMUTATIVE ALGEBRA
SAEED NASSEH AND SEAN K. SATHER-WAGSTAFF
To David Eisenbud on the occasion of his 75th birthday
Abstract.
Differential graded (DG) algebras are powerful tools from rationalhomotopy theory. We survey some recent applications of these in the realm ofhomological commutative algebra.
Contents
1. Introduction 12. Growth of Bass and Betti numbers 23. Friendliness And Persistence of local rings 74. Bass series of local ring homomorphisms of finite flat dimension 105. Ascent property of pd-test modules 126. A conjecture of Vasconcelos on the conormal module 137. A conjecture of Vasconcelos on semidualizing modules 148. Complete intersection maps and the proxy small property 169. Conjectures of Quillen on Andr´e-Quillen homology 1710. Finite Generation of Hochschild Homology Algebras 18Acknowledgments 19References 191.
Introduction
Throughout this paper, the term “ring” is short for “commutative noetherianring with identity.”In algebraic topology, it is incredibly useful to know that the singular cohomologyof a manifold has a natural algebra structure. Similarly, in commutative algebrathe fact that certain Ext and Tor modules carry algebra structures is a powerfultool. Both of these notions arise by considering differential graded (DG) algebrastructures on certain chain complexes. In short, a DG algebra is a chain complexthat is also a graded commutative ring, where the differential and multiplicationare compatible; see Section 2 for definitions and background material.Avramov, Buchsbaum, Eisenbud, Foxby, Halperin, Kustin, and others pioneeredthe use of DG algebra techniques in homological commutative algebra. The idea
Mathematics Subject Classification.
Primary: 13D02, 13D07; Secondary: 13B10, 13D03,13D05, 13D40. is to prove results about rings by broadening one’s context to include vast gen-eralizations. A deep, rich sample of the theory and applications can be found inAvramov’s lecture notes [14].The current paper is a modest follow-up to op. cit. , documenting a few appli-cations that have appeared in the twenty-some years since op. cit. appeared. Tobe clear, we focus on applications: results whose statements make no reference toDG algebras but whose proofs use them extensively. Furthermore, this survey isby no means comprehensive. We focus on small number of some of our favoriteapplications, limited by constraints of time and space.Most of the sections below begin by describing an application with little referenceto DG algebras. This is followed by a certain amount of DG background material,but generally only enough to give a taste for the material. The sections concludewith an indication of how the DG technology helps to obtain the application.As we noted above, the work of David Eisenbud is foundational in this area,especially the paper [39] with Buchsbaum; see 2.6. Those of us working in this areaowe him a huge debt of gratitude for this and other seminal work in the field.2.
Growth of Bass and Betti numbers
In this section, let ( R, m , k ) be a local ring with d := depth R . The embeddingcodepth of R , denoted c := ecodepth R , is defined to be e − d , where e := edim R isthe minimal number of generators of m . Cohen’s Structure Theorem states that the m -adic completion b R admits a minimal Cohen presentation, i.e., there is a completeregular local ring ( P, p , k ) and an ideal I ⊆ p such that b R ∼ = P/I . Note that theprojective dimension pd P ( b R ), i.e., the length of the minimal free resolution of b R over P , is equal to ecodepth R .Fundamental invariants of a finitely generated R -module M are the Bass andBetti numbers. These numerically encode structural information about the module M , e.g., the minimal number of generators and relations and higher degree versionsof these. A hot topic of research in commutative algebra is the growth of thesesequences. In this section, we describe some recent progress by Avramov on thissubject including how he uses DG techniques to get information about these invari-ants. Along the way, we also present foundational material about the DG context. Bass numbers, Betti numbers, and a question of Huneke.
Let M be a finitely generated R -module. For each integer i , the i th Bass number and the i th Betti number of M are defined to be, respectively µ iR ( M ) := rank k (cid:0) Ext iR ( k, M ) (cid:1) β Ri ( M ) := rank k (cid:0) Ext iR ( M, k ) (cid:1) = rank k (cid:0) Tor Ri ( M, k ) (cid:1) . The
Bass series and the
Poincar´e series of M are the formal power series I MR ( t ) := X i ∈ Z µ iR ( M ) t i P RM ( t ) := X i ∈ Z β Ri ( M ) t i . (2.0.1)In case that M = R , the Bass numbers and Bass series are denoted µ iR and I R ( t ).In this section we are concerned with the following unpublished question ofHuneke; see [105]. This question is motivated in part by the fact that R is Goren-stein if and only if its Bass numbers are eventually 0. PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 3
Question 2.1.
Let R be a Cohen-Macaulay local ring. If { µ iR } is bounded, must R be Gorenstein? If { µ iR } is bounded above by a polynomial in i , must R beGorenstein? If R is not Gorenstein, must { µ iR } grow exponentially?Very little progress has been made on this question. Christensen, Striuli, andVeliche [45] conduct a careful analysis of several special cases of this and otherrelated questions. Other progress comes from Jorgensen and Leuschke [75] andBorna, Sather-Wagstaff, and Yassemi [33, 105].In this section, we focus on work of Avramov [16] on this question for non-Gorenstein rings R with c = ecodepth( R ) Theorem 2.2 ([16, Theorem 4.1]) . If c and R is not Gorenstein, then thereis a real number γ R > such that for all i > µ d + iR > γ R µ d + i − R (2.2.1) with two exceptions for i = 2 : If I = ( wx, wy ) or I = ( wx, wy, z ) , where x, y ∈ P is a regular sequence, w ∈ P , and z ∈ p is a P/ ( wx, wy ) -regular element, then µ d +2 R = µ d +1 R = 2 . If R is Cohen-Macaulay, the inequality (2.2.1) holds for all i . DG algebra resolutions and DG modules.
Let S be a ring. A associative, commutative differential graded S -algebra ( DG S -algebra for short) is a chain complex A = · · · → A ∂ A −−→ A ∂ A −−→ A → A ♮ = L i > A i has the structure of a graded commutative S -algebra • for all a, b ∈ A the equality ab = ( − | a || b | ba holds, and a = 0 if the homologicaldegree | a | is oddthat satisfies the Leibniz rule • for all a, b ∈ A we have ∂ A ( ab ) = ∂ A ( a ) b + ( − | a | a∂ A b , i.e., the assignment a ⊗ b ab describes a chain map A ⊗ S A → A .The DG S -algebra A is called homologically degreewise noetherian if H ( A ) is noe-therian, and each H ( A )-module H i ( A ) is finitely generated.Examples of homologically degreewise noetherian DG S -algebras include S itself,considered as a complex concentrated in degree 0, and the Koszul complex K S ( x )over S on a sequence x = x , . . . , x n in S with the exterior algebra structure.A morphism of DG S -algebras is a chain map f : A → B such that for all a, a ′ ∈ A we have f ( aa ′ ) = f ( a ) f ( a ′ ) and f (1) = 1. A quasiisomorphism of DG algebras is amorphism that is a quasiisomorphism, i.e., such that the induced map on homologyis an isomorphism in each degree. A DG algebra resolution of an S -algebra T isa quasiisomorphism F ≃ −→ T of DG S -algebras such that each F i is free over S .Several examples of DG algebra resolutions are given below starting with 2.4.In case that ( S, n ) is a local ring, a DG S -algebra A is called local if it is homo-logically degreewise noetherian and H ( A ) is a local S -algebra. In this case, setting n to be the preimage of m H ( A ) in A , we let m A = · · · ∂ A −−→ A ∂ A −−→ n → augmentation ideal of A . By definition it is a subcomplex of A . Moreover, it isa DG ideal of A meaning that it absorbs multiplication by elements of A . In thissituation, we say that ( A, m A , A/ m A ) is a local DG S -algebra. As an example, if SAEED NASSEH AND SEAN K. SATHER-WAGSTAFF x ∈ n is a sequence of n elements, then K = K S ( x ) is a local DG S -algebra withthe augmentation ideal m K = 0 → S → · · · → S n → n → K/ m K ∼ = S/ n . A construction of Tate [111] (see also Avramov [14, Proposition 2.2.8]) guar-antees the existence of a DG algebra resolution F of b R over P , where each F i isfinitely generated and free over P and F i = 0 for all i > pd P ( b R ).Examples of DG algebra resolutions include the following. If I is generated by a P -regular sequence (that is, if R is a formal completeintersection), then the Koszul complex K P ( I ) on a minimal generating sequencefor I is a DG algebra resolution of b R over P . If pd P ( b R ) = 2, then it follows from the Hilbert-Burch Theorem [51, Theo-rem 20.15] that there is an element f ∈ P and a matrix Y of size n × ( n −
1) suchthat the minimal P -free resolution of b R can be chosen with the form0 → P ⊕ n − Y −→ P ⊕ n X −→ P → b R → X = f (cid:0) det( Y ) , . . . , ( − j − det( Y j ) , . . . , ( − n − det( Y n ) (cid:1) , where Y j is theminor obtained from Y by deleting the j -th row. Herzog [67] describes a DGalgebra structure on this resolution, as follows. Let { a , . . . , a n } be a basis for P ⊕ n and { b , . . . , b n − } be a basis for P ⊕ n − , and set( a i ) = 0 a i · a j = − a j · a i = n − X t =1 ( − i + j + t +1 det( Y ij,t ) f b t for i < j where Y ij,t denotes the minor obtained from Y by deleting rows i, j and column t . Assume pd P ( b R ) = 3. Buchsbaum and Eisenbud [39] show that the minimalfree resolution of b R over P has the structure of a DG algebra, though the explicitstructure of the resolution is not given.Let n denote the minimal number of generators for I , and assume that R isGorenstein (that is, I is a Gorenstein ideal). Then it is shown in op. cit. that theminimal free resolution of b R over P is of the form0 → P Z −→ P ⊕ n Y −→ P ⊕ n X −→ P → b R → n × n alternating matrix Y with entries in p and X = (cid:0) pf( Y ) , . . . , ( − j − pf( Y j ) , . . . , ( − n − pf( Y n ) (cid:1) where Y i denotes the matrix obtained from Y by deleting row i and column i .(Here, pf is the Pfaffian; see [38] for details.) Also, Z = Hom P ( X, P ).An explicit DG algebra structure on (2.6.1) is given by Avramov [9] as follows.Let { a , . . . , a n } be a basis for P ⊕ n in degree 1, let { b , . . . , b n } be a basis for P ⊕ n in degree 2, and let { c } be a basis for P in degree 3. Define( a i ) = 0 a i · b j = b j · a i = δ ij ca i · a j = − a j · a i = n X t =1 ( − i + j + t ρ ijt pf( Y ijt ) b t for i < j PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 5 where Y ijt is the matrix obtained from Y by deleting rows i, j, t and columns i, j, t ,and δ ij is the Kronecker delta, and ρ ijt = ( − i < t < j S = k [ x , . . . , x n ] be a polynomial ring over a field k , and let I be amonomial ideal in S , i.e., an ideal generated by monomials. A general DG algebraresolution of S/I is given by Taylor [113] and Bayer, Peeva, and Sturmfels [32], butit is not minimal in general. In the following cases the minimal free resolution of
S/I over S has a DG algebra structure: stable ideals (see Eliahou and Kervaire [52]or Peeva [98]), matroidal ideals (see Sk¨oldberg [109]), and ideals of the form I = f J , where J is a monomial ideal in S and f is the least common multiple of thegenerators of J (see Katth¨an [78]). It is important to note for 2.3 that the minimal free resolution of b R over P may not admit a DG algebra structure in general. Examples for this are given byKhinich (as documented in [8]), Avramov [10], and Katth¨an [78]. The example ofKatth¨an is generic and disproves a claim by Bayer, Peeva, and Sturmfels [32]. In contrast to 2.7, if R satisfies one of the following conditions, then theminimal free resolution of b R over P admits a DG algebra structure:(a) c
3, by 2.5–2.6;(b) c = 4 and R is Gorenstein, by Kustin and Miller [79, 82];(c) c = 4, R is Cohen-Macaulay, almost complete intersection, and 1 / ∈ R , byKustin [81];(d) R is complete intersection, by 2.4;(e) R is one link from a complete intersection, by Avramov, Kustin, and Miller [31];(f) R is two links from a complete intersection and is Gorenstein, by Kustin andMiller [83].The translation to DG algebras uses the following DG analogue of modules. Let B be a DG S -algebra. A DG B -module is an S -complex M such that M ♮ = L i M i is a graded A ♮ -module satisfying the Leibniz rule. For the case of S considered as a DG S -algebra, the DG S -modules are just the S -complexes.A DG B -module M is homologically bounded if H i ( M ) = 0 for all | i | ≫
0; it is homologically finite if L i H i ( M ) is a finitely generated H ( B )-module.Let M be a DG B -module. The trivial extension B ⋉ M is the DG algebra withthe underlying complex B ⊕ M equipped with the product that is given as follows:( b, m )( b ′ , m ′ ) := ( bb ′ , bm ′ + ( − | m || b ′ | b ′ m ) . Avramov’s machine.
Some of our favorite applications of DG techniques use the following tool whichKustin [80] calls
Avramov’s machine . Let x be a minimal generating sequence for m , and let y be a minimalgenerating sequence for the maximal ideal p . Since P is a regular local ring, the SAEED NASSEH AND SEAN K. SATHER-WAGSTAFF
Koszul complex K P ( y ) is a minimal free resolution of k over P . Since K P ( y ) ≃ k ,we obtain the following diagram of DG algebra quasiisomorphisms: K R ( x ) ≃ −→ K b R ( x b R ) ∼ = ←− K P ( y ) ⊗ P b R ≃ ←− K P ( y ) ⊗ P F ≃ −→ k ⊗ P F =: A. (2.10.1)The assumptions on F in 2.3 imply that A is a finite-dimensional DG k -algebra. Itfollows that Tor P ( b R, k ) inherits the structure of a finite-dimensional DG k -algebra;this is the Tor algebra . If F is minimal, e.g., in any of the cases from 2.8, thealgebra A has zero differential, so A ∼ = Tor P ( b R, k ).Rationality of Poincar´e series, which we discuss next, is an important applicationof Avramov’s machine.
Consider the notation from 2.10. In this paragraph, assume that one ofthe conditions (a), (b), (e), or (f) in 2.8 holds. Using the fact that the minimalfree resolution of b R over P has a DG algebra structure, Avramov, Kustin, andMiller [31] give a factorization P ϕ −→ Q ψ −→ b R of the canonical map P → b R such that ϕ is complete intersection and ψ is Golod (see, e.g., [12] for the definition). Thenthey invoke a result of Levin [85] to conclude the following;( ∗ ) the Poincar´e series of every finitely generated R -module is rational with com-mon denominator.In case (d) of 2.8 where R is a complete intersection, conclusion ( ∗ ) was provedfor P Rk ( t ) by Tate [111] and, in general, by Gulliksen [64] and Avramov [13]. Incase (c) of 2.8, conclusion ( ∗ ) was proved by Kustin and Palmer [84]. Growth rates in embedding codepth at most . With these tools in hand, the proof of Theorem 2.2 proceeds in the followingsteps. First, consider the following structure result for the Tor algebra.
Assume that c
3. Using the notation from 2.10, we know that A is afinite-dimensional DG algebra with zero differential. In this case, by [31] and [122],the ring R belongs to one of the following classesClass c A B C D C ( c ) B V k Σ k c S B ⋉ W k T B ⋉ W C ⋉ Σ ( C/C > ) V k Σ k B B ⋉ W C ⋉ Σ C + V k Σ k G ( r ) 3 B ⋉ W C ⋉ Hom k ( C, Σ k ) k ⋉ Σ k r H ( p, q ) 3 B ⋉ W C ⊗ k D k ⋉ ( Σ k p ⊕ Σ k q ) k ⋉ Σ k where W is a finitely generated positively graded k -vector space with B + W = 0and ⋉ designates the trivial extension from 2.9. The ring R is in class S (that is, A is of the form k ⋉ W ) if and only if R is Golod; see [61]. If R is in class C ( c ),then R is a complete intersection.The next step in the proof of Theorem 2.2 is to connect the Poincar´e and Bassseries of R to analogous series for A : I R ( t ) = t e · I A ( t ) P Rk ( t ) = (1 + t ) e · P Ak ( t )where I A ( t ) and P Ak ( t ) are the Bass series and the Poncar´e series for A which aredefined in the DG setting as in (2.0.1). These equalities are based on work in [9, 23]. PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 7
The third step in the proof of Theorem 2.2 is to analyze the Poincar´e and Bassseries of A to draw the following conclusions about the corresponding series for R ;the proof then concludes from an analysis of the coefficients in the displayed series. Theorem 2.13 ([16, Theorem 2.1]) . Use the notation from 2.12. Assume that c and set l := rank k A − , n := rank k A , p := rank k ( A ) , q := rank k ( A · A ) ,and r := rank k ( δ ) , where δ : A → Hom k ( A , A ) is defined by δ ( a )( a ) := a a for all a ∈ A and a ∈ A . Then the following equalities hold for the Poincar´eseries and Bass series of R : P Rk ( t ) = (1 + t ) e − g ( t ) and I RR ( t ) = t d · f ( t ) g ( t ) where f ( t ) , g ( t ) ∈ Z [ t ] are described as follows, where p + q > :Class g ( t ) f ( t ) C ( c ) (1 − t ) c (1 + t ) c − (1 − t ) c (1 + t ) c − S − t − lt t − t T − t − lt − ( n − t − t n + lt − t − t + t B − t − lt − ( n − t + t n + ( l − t − t + t G ( r ) 1 − t − lt − nt + t n + ( l − r ) t − ( r − t − t + t H (0 ,
0) 1 − t − lt − nt n + lt + t − t H ( p, q ) 1 − t − lt − ( n − p ) t + qt n + ( l − q ) t − pt − t + t We end this section with the discussion of some properties of the class G ( r )including recent counterexamples to a conjecture of Avramov [16]. Consider thenotation from 2.12. Let n denote the minimal number of generators for I , andassume that c = 3. In case that R is a Gorenstein ring which is not completeintersection, it is known from work of J. Watanabe [121] that n > n is odd.Also, in this case, R belongs to the class G (2 i + 1) for some i > R belongs to the class G ( n ).Conversely, Avramov op. cit. conjectured that if R is in the class G ( r ) with r >
2, then R is Gorenstein and therefore, the classes G (3) and G (2 i ) for all i > S is the powerseries algebra in three variables over a field, then for every r > I of S with type( S/I ) = 2 such that
S/I belongs to G ( r ). For counterexamples toAvramov’s conjecture of arbitray type, see VandeBogert [114].3. Friendliness And Persistence of local rings
In this section, let ( R, m , k ) be a local ring. Vanishing of Ext and Tor, and finiteness of homological dimensions.
Let
M, N be finitely generated R -modules. Following Avramov, Iyengar, Nasseh,and Sather-Wagstaff [29], R is called Tor-friendly if Tor Ri ( M, N ) = 0 for all i ≫ R M < ∞ or pd R N < ∞ . We say that R is Tor-persistent ifTor Ri ( M, M ) = 0 for all i ≫ R M < ∞ . The ring R is Ext-friendly if Ext iR ( M, N ) = 0 for all i ≫ R M < ∞ or id R N < ∞ , where idis the injective dimension. Finally, R is Ext-persistent if Ext iR ( M, M ) = 0 for all i ≫ R M < ∞ or id R M < ∞ . SAEED NASSEH AND SEAN K. SATHER-WAGSTAFF
Friendliness and persistence have been studied in numerous works; see for in-stance [17, 18, 29, 70, 71, 73, 74, 76, 77, 92, 93, 94, 95, 106, 107]. The mainmotivation for this section is the following result in which the proofs of parts (a),(b), (c), (e), and (f) use DG algebra techniques.
Theorem 3.1 ([29, Theorem 5.1, Lemmas 5.7 and 5.9]) . Assume there exist a localhomomorphism R → R ′ of finite flat dimension and a deformation R ′ և Q , i.e., alocal surjection with kernel generated by a Q -regular sequence, where Q satisfies atleast one of the conditions (a) edim Q − depth Q . (b) Q is Gorenstein and edim Q − depth Q = 4 . (c) Q is Cohen-Macaulay, almost complete intersection, edim Q − depth Q = 4 ,and ∈ Q . (d) Q is complete intersection. (e) Q is one link from a complete intersection. (f) Q is two links from a complete intersection and is Gorenstein. (g) Q is Golod. (h) Q is Cohen-Macaulay and mult Q .Then R is Tor- and Ext-persistent. Moreover, Q can be chosen to be complete, withalgebraically closed residue field, and with no embedded deformation; in this case, Q is Tor-friendly. One of the most important motivations for working on friendliness and persis-tence is the following conjecture that is known as the
Auslander-Reiten Conjec-ture [7]. This conjecture stems from work of Nakayama [88] and Tachikawa [110]on the representation theory of Artin algebras.
Conjecture 3.2 ([7, p. 70]) . Let M be a finitely generated R -module that satisfiesthe condition Ext iR ( M, M ⊕ R ) = 0 for all i > . Then M is a free R -module. It is straightforward to show that if R is Ext-persistent, then it satisfies theAuslander-Reiten Conjecture 3.2.By [29, Proposition 6.5], Tor-friendliness implies Ext-friendliness. (In the contextof complexes, these two notions are equivalent; see [29, Propositions 3.2 and 6.5].)The question of whether all rings are Tor-persistent is open. However, examplesof rings that are not Ext-persistent (hence, not Ext-friendly nor Tor-friendly) arestraightforward to construct: for instance, ( k [ x, y ] / ( x, y ) ) ⊗ k ( k [ u, v ] / ( u, v ) ).Next, we describe some DG methods from [29, 30] used to prove Theorem 3.1. Perfect DG modules, trivial extensions, and DG syzygies.
In order to apply DG techniques in the above setting, the first tool we need is thefollowing DG analogue of finitely generated module of finite projective dimension.
Assume that ( B, m B ) is a local DG algebra. A homologically finite DG B -module M is called perfect is it satisfies one of the following equivalent conditions(see [30] or [103]):(i) M is quasiisomorphic to a DG B -module F such that the underlying graded B ♮ -module F ♮ has a finite basis.(ii) For all homologically bounded DG B -modules N , one has Tor Bi ( M, N ) = 0for all i ≫ Bi ( M, B/ m B ) = 0 for all i ≫ PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 9
The approach described below to understanding friendliness and persistence ismotivated in parts by work of Nasseh and Yoshino [95] who prove that the trivialextension R ⋉ k is Tor-friendly. See 2.9 for the definition of trivial extensions. Thisresult is generalized to the DG setting as follows. Theorem 3.5 ([30, Theorem 4.1]) . Let A be a DG algebra that is quasiisomorphicto B ⋉ W , where B is a homologically bounded local DG algebra, and W is ahomologically bounded DG k -module with H( W ) = 0 . If M, N are homologicallyfinite DG A -modules with Tor Ai ( M, N ) = 0 for all i ≫ , then M or N is perfect. The proof of Theorem 3.5 is similar to that of [95, Theorem 3.1]. In order totranslate loc. cit. to the DG setting, a DG version of the important notion of asyzygy was needed. This is the DG module N in the following result which weexpect to be useful for other applications. Proposition 3.6 ([30, Proposition 4.2]) . Let ( A, A + ) be a local DG R -algebra. Let M be a homologically finite DG A -module. Then there exists a short exact sequence → N α −→ F → f M → of morphisms of DG A -modules such that (1) M ≃ f M ; (2) the underlying graded A ♮ -module F ♮ has a finite basis; and (3) Im( α ) ⊆ A + · F . Friendliness and persistence.
An important consequence of Theorem 3.5 is the following result that is a bridgebetween Ext vanishing over R and its corresponding DG algebra. Theorem 3.7 ([30, Theorem 6.3]) . Assume there exists a minimal Cohen presenta-tion b R ∼ = P/I such that the minimal free resolution of b R over P has the structure ofa DG algebra and the k -algebra A = Tor P ( b R, k ) is isomorphic to the trivial exten-sion B ⋉ W of a graded k -algebra B by a graded B -module W = 0 with B > · W = 0 .Then R is Tor-friendly. The proof of this result, which we outline next, relies on Avramov’s machine 2.10whence we also take our notation. To prove Theorem 3.7, one transfers Tor-vanishing over R to Tor-vanishing over the Koszul complex K = K R ( x ) by basechange. Then using the quasiisomorphisms (2.10.1), one transfers Tor-vanishingover K to Tor-vanishing over A . Since the property of being perfect transfers from A to K , then to R , the DG result Theorem 3.5 gives us the desired conclusion.Next we sketch the proof of Theorem 3.1. Using standard base-change tech-niques, one can assume without loss of generality that R = R ′ and hence, R and Q have a common residue field k . Furthermore, we can assume that Q is complete, k is algebraically closed, and Q does not admit embedded deformation; see [29,Lemma 5.7]. It suffices by [29, Theorems 2.2 and 6.3] and 3.3 to show that Q is Tor-friendly. Let b Q ∼ = P/J be a minimal Cohen presentation, and let F be aminimal free resolution of b Q over P . If Q satisfies one of the conditions (a)–(g) inTheorem 3.1, then F admits a DG-algebra structure as we mentioned in 2.8. Forsome of these cases, the Tor algebra Tor P ( b Q, k ) satisfies the assumptions of Theo-rem 3.7. Hence, Q is Tor-friendly in those cases by Theorem 3.7. In the remainingcases other methods are used to conclude that Q is Tor-friendly.Geller[59] and Morra [87] are working to apply Theorem 3.7 to other rings. Bass series of local ring homomorphisms of finite flat dimension
In this section, let ϕ : ( R, m , k ) → ( S, n , ℓ ) be a local ring homomorphism. Relations among Bass series.
Assume in this paragraph that ϕ is flat. Then many properties of S are controlledby the corresponding properties for R and the closed fibre S/ m S . For instance, S is Gorenstein if and only if R and S/ m S are both Gorenstein. More generally, theBass series of S is related to the Bass series for R and S/ m S by the formula I S ( t ) = I R ( t ) I S/ m S ( t ) . (4.0.1)In particular, for each i ∈ Z , we have µ i +depth RR µ i +depth SS . If S/ m S is Gorenstein,then Grothendieck says that ϕ is Gorenstein [63, 7.3.1–7.3.2].When ϕ is not flat, the properties in the previous paragraph can fail, e.g., forthe natural surjection R → k when R is not regular, i.e., when pd R k is not finite.However, Avramov, Foxby, and Lescot [19, 20, 23] recognized that the full strengthof flatness is not needed: Theorem 4.1 ([23, Theorems A, B, C]) . Assume that ϕ is of finite flat dimension,i.e., the R -module S has a bounded resolution by flat modules. For instance, thisholds if S = R/I , where I is an ideal of R with finite projective dimension. (a) There is a formal Laurent series I ϕ ( t ) with non-negative integer coefficientssuch that I S ( t ) = I R ( t ) I ϕ ( t ) . (4.1.1)(b) For each i ∈ Z , the following inequality holds: µ i +depth RR µ i +depth SS . (c) Assume further that the closed fibre S/ m S is artinian, and either ϕ is not flator S/ m S is not a field. Then the following coefficient-wise inequality holds: I S ( t ) I R ( t ) − (1 + t ) + P fd R ( S ) i =0 len S (cid:0) Tor Ri ( k, S ) (cid:1) t − i t − P fd R ( S ) i =0 len S (cid:0) Tor Ri ( k, S ) (cid:1) t i +1 . (4.1.2) Equality in (4.1.2) holds if and only if ϕ is Golod; see 2.11. Here is some perspective on Theorem 4.1(c). If the closed fibre S/ m S isartinian, then the following coefficient-wise inequality holds: P Sℓ ( t ) P Rk ( t )1 + t − P fd R ( S ) i =0 len S (cid:0) Tor Ri ( k, S ) (cid:1) t i +1 . (4.2.1)The ring homomorphism ϕ is called a standard Golod homomorphism if equalityholds in (4.2.1).Assume either ϕ is not flat or S/ m S is not a field. Then ϕ is a Golod homomor-phism if and only if it is a standard Golod homomorphism; see Avramov [12]. Hence,in the finite flat dimension setting, Theorem 4.1(c) says that equality in (4.1.2) holdsif and only if equality in (4.2.1) holds if and only ϕ is Golod.The proof of Theorem 4.1 uses the DG fibre introduced by Avramov [11]. or “fiber,” depending on your preference PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 11
The DG fibre of ϕ . Assume that ϕ is of finite flat dimension. Let G ≃ −→ k and L ≃ −→ S be DG algebraresolutions over R . (Note that the free modules in L will not be finitely generatedover R in general.) The DG fibre of ϕ is defined to be the local DG algebra F ( ϕ ) := G ⊗ R S ≃ G ⊗ R L ≃ k ⊗ R L where the quasiisomorphisms come from the balance property for Tor R ( k, S ). Themultiplication on F ( ϕ ) is inherited from G , S , k , and L . The degree 0 homologymodule of F ( ϕ ) is the closed fibre S/ m S . In case that ϕ is flat, F ( ϕ ) ≃ S/ m S .The Bass series of ϕ , denoted I ϕ ( t ), is the Bass series I F ( ϕ ) ( t ) of the DG algebra F ( ϕ ), which by [23, Theorem A] is a formal Laurent series.In the case where ϕ is flat, the formulas (4.0.1) and (4.1.1) are the same. In thiscase, they are a particular instance of the formula I M ⊗ R SS ( t ) = I MR ( t ) I S/ m S ( t )where M is finitely generated over S ; one verifies this formula using the isomorphismExt S ( ℓ, M ⊗ R S ) ∼ = Ext R ( k, M ) ⊗ k Ext S/ m S ( ℓ, S/ m S )In the general finite flat dimension case, Theorem 4.1(a) follows from a similarisomorphism. The innovative point in [23] that we want to emphasize here is thereplacement of the usual closed fibre S/ m S by the DG fibre F ( ϕ ).It is worth noting that Avramov and Foxby [21] established the conclusions ofTheorem 4.1 for a larger class of local ring homomorphisms using relative dualizingcomplexes, but this work does not use DG techniques. Gorenstein homomorphisms.
As we mentioned above, if ϕ is flat with Gorenstein closed fibre, then S is Goren-stein if and only if R is Gorenstein. In case ϕ has finite flat dimension, one shouldnot expect Gorensteinness of the closed fibre to guarantee the same conclusion. Inpart to remedy this, Avramov and Foxby [19, 20] extend Grothendieck’s aforemen-tioned notion of a Gorenstein homomorphism:The local ring homomorphism ϕ is called Gorenstein if there is an integer a such that for all i we have µ iR = µ i + aS . In particular, if ϕ is Gorenstein, then S is Gorenstein if and only if R is Gorenstein. If ϕ has finite flat dimension,Gorensteinness of ϕ is equivalent to having the equality µ iR = µ i +depth S − depth RS forall i by Theorem 4.1(a).In case that ϕ is flat, Gorensteinness of ϕ is equivalent to the Gorensteinness ofthe closed fibre S/ m S ; see [20, (4.2) Proposition]. Hence, this notion of Gorensteinhomomorphisms is a generalization of Grothendieck’s Gorenstein homomorphisms.The result op. cit. can be extended to the following characterization of Goren-stein homomorphisms in terms of their DG fibres. Theorem 4.3 ([20, (4.4) Theorem]) . Assume that ϕ has finite flat dimension.Then ϕ is Gorenstein if and only if the DG fibre F ( ϕ ) is a Gorenstein DG algebra(that is, I ϕ ( t ) = t d for some integer d ). As one might imagine, given the usefulness of the Gorenstein property for lo-cal rings, Gorenstein DG algebras have been investigated separately; see Frankild,Iyengar, and Jørgensen [56, 57]. Ascent property of pd-test modules
In this section, let ϕ : ( R, m , k ) → ( S, n , ℓ ) be a flat local ring homomorphism. Pd-test modules.
A useful, classical result states that the residue field k has the ability to test forfinite projective dimension: a finitely generated R -module N has finite projectivedimension if and only if Tor Ri ( k, N ) = 0 for i ≫
0. According to the followingdefinition, which was coined by O. Celikbas, Dao, and Takahashi [41], this saysthat k is a pd-test R -module.A finitely generated R -module M is called a pd-test module if for every finitelygenerated R -module N with Tor Ri ( M, N ) = 0 for i ≫ R N < ∞ .It is natural to ask how the pd-test property for a finitely generated R -module M behaves under completion. This is related to the well-known fact that R isregular if and only if b R is regular. It is straightforward to show that if c M is pd-testover b R , then M is pd-test over R . That is, the pd-test property descends from thecompletion. The question of ascent is more subtle. It was posed in [41] and answeredby O. Celikbas and Sather-Wagstaff [42] using derived category techniques. Thefollowing more general ascent result is proved by Sather-Wagstaff [103]. Theorem 5.1 ([103, Theorem 4.8]) . Assume that the closed fibre S/ m S of ϕ isregular and the induced field extension k → ℓ is algebraic. If a finitely generated R -module M is pd-test over R , then S ⊗ R M is a pd-test module over S . Theorem 5.1 is proved using the following DG techniques.
Pd-test DG modules.
A homologically finite DG module M over a local DG algebra B is a pd-test DGmodule if every homologically finite DG B -module N with Tor Bi ( M, N ) = 0 for all i ≫ Theorem 5.2 ([103, Theorem 4.6]) . Let A be a finite-dimensional DG k -algebrawith A = k and H ( A ) = 0 . Let k → ℓ be an algebraic field extension, and set B = ℓ ⊗ k A . If M is pd-test over A , then B ⊗ A M is pd-test over B . Before applying Theorem 5.2, we sketch its proof. Assume that N is a homolog-ically finite DG B -module such that Tor Bi ( B ⊗ A M, N ) = 0 for all i ≫
0. In casethat k → ℓ is a finite field extension, the assertion follows from a standard argumentusing 3.4. Now consider the general case, where k → ℓ is algebraic. By truncatingan appropriate resolution of N over B one can assume that N is finite-dimensionalover ℓ . It then follows that the differential and scalar multiplication on N arerepresented by matrices consisting of finitely many elements of ℓ . Adjoining thesealgebraic elements to k , one obtains an intermediate field extension k → k ′ → ℓ such that k → k ′ is finite. By construction of k ′ , with A ′ = k ′ ⊗ k A , there is abounded DG A ′ -module L such that N ∼ = B ⊗ A ′ L . At this point, the assumptionof Tor Bi ( B ⊗ A M, N ) = 0 for all i ≫ A ′ i ( A ′ ⊗ A M, L ) = 0 for all i ≫
0. Since k → k ′ is finite, it follows that L is perfect over A ′ , so N ∼ = B ⊗ A ′ L is perfect over B . PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 13
Outline of the proof of Theorem 5.1.
Assume that M is a pd-test module over R . We need to show that S ⊗ R M is apd-test module over S . Assume that Tor Si ( S ⊗ R M, N ) = 0 for i ≫
0, where N is afinitely generated S -module. Standard techniques reduce to the case where R and S are complete with S/ m S = ℓ . Using the notation from 2.10 and applying [22,(1.6) Theorem] we have a minimal Cohen presentation P ′ τ ′ −→ S and a commutativediagram of local ring homomorphisms P α / / τ (cid:15) (cid:15) P ′ τ ′ (cid:15) (cid:15) R ϕ / / S such that α is flat, τ ′ is surjective, P ′ / p P ′ ∼ = ℓ , and S ∼ = R ⊗ P P ′ . The lastisomorphism implies that F ′ := F ⊗ P P ′ ≃ −→ S is a DG algebra resolution of S over P ′ . Note that ϕ ( x ) minimally generates n . Following the process of 2.10 for thering S , we get the next commutative diagram of morphisms of DG algebras R / / ϕ (cid:15) (cid:15) K R (cid:15) (cid:15) K P ⊗ P R ∼ = o o (cid:15) (cid:15) K P ⊗ P F ≃ o o (cid:15) (cid:15) ≃ / / k ⊗ P F = A (cid:15) (cid:15) S / / K S K P ′ ⊗ P ′ S ∼ = o o K P ′ ⊗ P ′ F ′≃ o o ≃ / / ℓ ⊗ P ′ F ′ in which K R = K R ( x ), K S = K S ( ϕ ( x )), K P = K P ( y ), and K P ′ = K P ′ ( α ( y )).Note that the DG algebra ℓ ⊗ P ′ F ′ is isomorphic to ℓ ⊗ k A . Now, the pd-testproblem between R and S can be translated through the rows of this diagram toa DG pd-test problem between A and ℓ ⊗ k A . At this point the assertion followsfrom Theorem 5.2.In case that ℓ = k ( x ) is a transcendental extension of k , the same conclusion asin the statement of Theorem 5.1 holds by a result of Tavanfar [112].6. A conjecture of Vasconcelos on the conormal module
Throughout this section, let I be an ideal of a ring R , and set S = R/I .Ferrand [54] and Vasconcelos [115] show that properties of the ring S are oftenreflected in the properties of the conormal module I/I over S . This section focuseson the following conjecture of Vasconcelos [117]. Conjecture 6.1 ([117, ( C )]) . If pd R S and pd S I/I are finite, then I is locallygenerated by a regular sequence. This conjecture was settled in the affirmative for some special cases by Vasconce-los [118], Gulliksen and Levin [65], and Herzog [68]. The following major progresson this conjecture was made by Avramov and Herzog [25] using Andr´e-Quillenhomology and DG homological methods.
Theorem 6.2 ([25, Theorem 3]) . Let k be a field of characteristic , and assume R is a positively graded polynomial ring over k and I is homogeneous. Then thefollowing are equivalent: (i) S is complete intersection; (ii) I/I is a free S -module; (iii) pd S I/I < ∞ . In a recent paper, Briggs [34] establishes Conjecture 6.1 in its full generality.
Theorem 6.3 ([34, Theorem A]) . Conjecture 6.1 holds in general.
Briggs’ proof for Theorem 6.3 relies on methods pioneered by Avramov andHalperin [11, 24] on homotopy Lie algebras π ∗ ( ϕ ) arising from DG constructions.Assume without loss of generality that ( R, m , k ) and ( S, n , k ) are local. Let ϕ : R → S be the natural surjection. Fix a minimal model for ϕ which is a factor-ization R → A ≃ −→ S , where ( A, m A ) is a local DG R -algebra such that:(a) The underlying algebra A ♮ = R [ X , X , . . . ] is the free graded commutative R -algebra, where each X i is a set of variables of degree i ; and(b) ∂ ( m A ) ⊆ m + m A .The DG algebra A is also denoted R h X i .A graded basis for each π i ( ϕ ) is dual to X i , and each element z ∈ π ( ϕ ) corre-sponds to a derivation θ z : A → m A of degree − θ z : A → n be the composition of θ z and the surjective quasiisomorphism m A → n .Under the assumptions of Conjecture 6.1, one can find a certain factorization of θ z which implies that z is radical in π ( ϕ ); see [34, proof of Lemma 2.6 and Theo-rem 2.7]. Now [24, Theorem C] implies that ϕ is complete intersection, as desired.7. A conjecture of Vasconcelos on semidualizing modules
In this section, ( R, m , k ) is a local ring.Here we discuss a class of modules that are particularly well-suited for creat-ing dualities. They were originally introduced by Foxby [55] who called them PGmodules of rank
1. They are useful, e.g., for understanding Gorenstein dimensions,in particular, Avramov and Foxby’s composition question for local ring homomor-phisms of finite G-dimension [21, 104].
Semidualizing modules.
A finitely generated R -module C is called semidualizing if the homothety mor-phism χ RC : R → Hom R ( C, C ) is an isomorphism and Ext iR ( C, C ) = 0 for all i > dualizing module .Let S ( R ) be the set of isomorphism classes of semidualizing R -modules.This section is centered on the following conjecture posed by Vasconcelos [116]. Conjecture 7.1 ([116, p. 97]) . If R is Cohen-Macaulay, then S ( R ) is finite. Note that if R is Ext-persistent, then R satisfies this conjecture. Moreover, inthis case, the only semidualizing R -modules are the free module of rank 1 and adualizing module, if one exists.Christensen and Sather-Wagstaff [43] answered Conjecture 7.1 in the case where R contains a field. Their proof reduces to the case of a finite-dimensional algebra,then implicitly uses the following technology from geometric representation theory. Assume that R is a finite-dimensional k -algebra, where k is algebraicallyclosed. The R -modules of a fixed length r are parametrized by an algebraic va-riety Mod Rr . One can define an action of the general linear group GL kr on Mod Rr .The isomorphism class of an R -module M is the orbit GL kr · M , and the tangentspace T GL kr · MM to the orbit GL kr · M at M is identified with a subspace of the tangentspace T Mod Rr M . A result of Voigt [120] (see also Brion [37] or Gabriel [58]) provides PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 15 an isomorphism Ext R ( M, M ) ∼ = T Mod Rr M / T GL kr · MM . As in work of Happel [66], itfollows that if Ext R ( M, M ) = 0 (e.g., if M is a semidualizing module), then theorbit GL kr · M is open in Mod Rr . Since Mod Rr is quasi-compact, it can contain onlyfinitely many open orbits, hence, S ( R ) is finite.Using a modification of these ideas, Nasseh and Sather-Wagstaff [91] establishConjecture 7.1 in total generality with no Cohen-Macaulay hypothesis. Theorem 7.3 ([91, Theorem A]) . For the local ring R , the set S ( R ) is finite. A DG version of Voigt’s theorem and the proof of Theorem 7.3.
To prove Theorem 7.3, we work with the following DG version of semidulazingmodules due to Christensen and Sather-Wagtaff [44].Let A be a homologically degreewise noetherian DG R -algebra. A homologi-cally finite DG A -module C is semidualizing if the homothety morphism χ AC : A → R Hom A ( C, C ) is an isomorphism in the derived category D ( A ). If A = R , a semid-ualizing DG R -module C is called a semidualizing R -complex . A semidualizing R -complex of finite injective dimension is called a dualizing complex . Let S ( A ) de-note the set of shift-isomorphism classes of semidualizing DG A -modules in D ( A ).Theorem 7.3 is a consequence of the following result because S ( R ) ⊆ S ( R ). Theorem 7.4 ([91, 4.2 and Theorem A]) . Consider the notation of 2.10. The sets S ( A ) and S ( R ) are finite. Using Grothendieck [62, Proposition (0.10.3.1)], we can assume in Theorem 7.4that R is complete with algebraically closed residue field. Because of Avramov’smachine 2.10, it suffices to show that S ( A ) is finite. To establish this finiteness,one uses the following DG version of 7.2 above.The set of finite-dimensional DG A -modules M with fixed underlying graded k -vector space W is parametrized by an algebraic variety Mod A ( W ). A productGL( W ) of general linear groups acts on Mod A ( W ) and the isomorphism class of M is the orbit GL( W ) · M under this action. See [91] for more details.The DG version of Voigt’s result from 7.2 that enables us to prove Theorem 7.4is the following. Theorem 7.5 ([91, Theorem B]) . Let W be a finite-dimensional graded k -vectorspace. Given an element M ∈ Mod A ( W ) , there is an isomorphism T Mod A ( W ) M / T GL( W ) · MM ∼ = YExt A ( M, M ) where YExt A ( M, M ) denotes the Yoneda Ext group defined as the set of equivalenceclasses of short exact sequences → M → L → M → . As in 7.2, it follows from Theorem 7.5 that if YExt A ( M, M ) = 0, then the orbitGL( W ) · M is open in Mod A ( W ). Since Mod A ( W ) is quasi-compact, it followsthat there are only finitely many open orbits in it. Thus, it remains to show thata each semidualizing DG A -module C satisfies YExt A ( C, C ) = 0. This vanishingfollows from work of Nasseh and Sather-Wagstaff [90].One can actually obtain a very tight connection between the sizes of S ( A ) and S ( R ) using a lifting result in [89] that generalizes results of Auslander, Ding,and Solberg [6] and Yoshino [123]. See Nasseh and Yoshino [96] and Ono andYoshino [97] for more general lifting results. Also, Altmann and Sather-Wagstaff [1]utilize Avramov’s machine to extend results of Gerko [60] from the realm of finite-dimensional algebras to arbitrary local rings. Complete intersection maps and the proxy small property
In this section, let ϕ : R → S be a surjective ring homomorphism.Here, we outline results of Briggs, Iyengar, Letz, and Pollitz [36] on questionsmotivated by work of Dwyer, Greenlees, and Iyengar [50] and Pollitz [101].A triangulated subcategory X of the derived category D ( R ) is called thick if itis closed under direct summands and satisfies the following two-of-three property:for each exact triangle L → M → N → in D ( R ) if two of the objects are in X ,then so is the third. The thick subcategory of D ( R ) generated by an R -complex M is the smallest thick subcategory of D ( R ) (with respect to inclusion) that contains M . Note that an R -complex is perfect if and only if it is in the thick subcategorygenerated by R . If an R -complex N is in the thick subcategory generated by another R -complex M , we say that N is finitely built from M .A triangulated subcategory of D ( R ) is called localizing if it is closed under ar-bitrary coproducts. Note that a localizing subcategory is thick. The localizingsubcategory of D ( R ) generated by an R -complex M is the smallest localizing subcat-egory of D ( R ) that contains M . If an R -complex N is in the localizing subcategorygenerated by another R -complex M , we say that N is built from M .A small complex M over a ring R is an R -complex such that Hom D ( R ) ( M, − )commutes with arbitrary direct sums. Note that the perfect R -complexes are pre-cisely the small R -complexes (or the small objects in D ( R )).In [49], an R -complex M is proxy small if there exists a small R -complex N suchthat N is finitely built from M , and M is built from N . Note that every small R -complex is proxy small. Other examples of proxy small complexes include theresidue field of a local ring and modules of finite complete intersection dimensionover a local ring.Let R be a local ring. The famous result of Auslander-Buchsbaum and Serre [4,108] says that R is regular if and only if every homologically bounded R -complexis small. The paper [50] contains a partial analogue of this statement for completeintersection rings: if R is complete intersection, then every homologically bounded R -complex is proxy small. Pollitz [101] proved the converse of this by showingthat if every homologically bounded R -complex is proxy small, then R is completeintersection. Pollitz’s proof heavily uses DG methods relying on his version [100]of Avramov and Buchweitz’s [17] support varieties over Koszul complexes. Due tospace restrictions here, we do not provide further details of this construction.In the not necessarily local setting, [50] includes a more general statement thanthe one mentioned above: if ϕ is complete intersection, then proxy smallness ascendsalong ϕ , i.e., any S -complex that is proxy small over R is proxy small over S . Briggs,Iyengar, Letz, and Pollitz [36] prove the following converse of this statement. Theorem 8.1 ([36, Theorem B]) . Assume that ϕ has finite projective dimension.If proxy smallness ascends along ϕ , then ϕ is complete intersection. A consequence of this theorem [36, Corollary 4.1] is another proof of a funda-mental result of Avramov [15, (5.7.1) Lemma] used in his solution to Quillen’sconjecture discussed in Section 9 below. More precisely, if R ϕ −→ S ψ −→ T are surjec-tive local homomorphisms such that fd S T < ∞ , then ψ ◦ ϕ is complete intersectionif and only if ϕ and ψ are complete intersection.The proof of Theorem 8.1 reduces to the case where ( R, m ) and ( S, n ) are local.Set e S := R/I , where I is an ideal generated by a maximal R -regular sequence in PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 17
Ker ϕ \ m Ker ϕ . The surjection R → S is the composition of the natural surjections R e ϕ −→ e S ˙ ϕ −→ S . To complete the proof it suffices to show that S is small over e S ;indeed, then [38, Corollary 1.4.7] implies ϕ = e ϕ is complete intersection, as desired.To show that S is small over e S , let K = K S ( n ) be the Koszul complex ona minimal generating set of n , and consider the restriction ˙ ϕ ∗ : D ( S ) → D ( e S ).By [50, Remark 5.6], it suffices to prove that ˙ ϕ ∗ ( K ) is a small e S -complex. Thissmallness follows from the next lemma which uses Hochschild cohomology for DGalgebras as constructed by Avramov, Iyengar, Lipman, and Nayak [28]. Lemma 8.2 ([36, Lemma 2.5]) . Let A be a DG R -algebra, and let M and N beDG A -modules. Let α be an element of the graded Hochschild cohomology algebra HH ∗ ( A | R ) . If N is (finitely) built from M , then the mapping cone N//α of aninduced morphism N χ N ( α ) −−−−→ Σ | α | N is (finitely) built from M//α . In particular, if M is proxy small then so is M//α . Conjectures of Quillen on Andr´e-Quillen homology
In this section, let ϕ : R → S be a ring homomorphism.Here, we describe Avramov’s solution [15] to a famous conjecture of Quillen [102]and Avramov and Iyengar’s significant progress [27] on a second one. Quillen’s conjectures.
The n th Andr´e-Quillen homology of the R -algebra S with coefficients in an S -module N is D n ( S | R, N ) = H n (L( S | R ) ⊗ S N ), where L( S | R ) is the cotangentcomplex of ϕ ; see Andr´e [2], Iyengar [72], and Quillen [102] for definitions andfoundational properties.The first of Quillen’s conjectures that we consider deals with locally completeintersection homomorphisms. This notion was originally defined for maps that areessentially of finite type or flat. Avramov’s solution of this conjecture hinges onthe following generalization of this notion.Assume in this paragraph that ϕ : R → ( S, n ) is a local ring homomorphism,and let ` ϕ : R → b S be the composition of ϕ with the natural completion map S → b S . A Cohen factorization of ` ϕ is a factorization into local ring homomorphisms R ˙ ϕ −→ R ′ ϕ ′ −→ b S such that ˙ ϕ is flat with regular closed fibre, ϕ ′ is surjective, and R ′ is complete. If there is a Cohen factorization R → R ′ ϕ ′ −→ b S of ` ϕ in which Ker ϕ ′ isgenerated by an R ′ -regular sequence, then ϕ is called complete intersection at n .In general, the (not necessarily local) ring homomorphism ϕ : R → S is called locally complete intersection if it is complete intersection at all prime ideals q of S ,i.e., for all such q , the induced local ring homomorphism ϕ q : R q ∩ R → S q is completeintersection at q S q . Also, ϕ is locally of finite flat dimension if fd R S q < ∞ for allprime ideals q of S . In case that R has finite Krull dimension this condition isequivalent to fd R S < ∞ ; see Auslander and Buchsbaum [5].Now we can state the conjectures of Quillen [102] that we are concerned with. Conjecture 9.1 ([102, (5.6) and (5.7)]) . Assume ϕ is essentially of finite type. (a) If ϕ is locally of finite flat dimension and D n ( S | R, − ) = 0 for all n ≫ , thenit is locally complete intersection. (b) If D n ( S | R, − ) = 0 for all n ≫ , then D n ( S | R, − ) = 0 for all n > . Avramov’s solution of Conjecture 9.1(a) via DG techniques.Theorem 9.2 ([15, (1.3)]) . Conjecture 9.1(a) holds without the essentially of finitetype assumption.
The proof of Theorem 9.2 reduces to the case where ϕ is surjective and local. Inthis case, the proof hinges on the following spectral sequence [15, (4.2) Theorem] E p,q = π p + q (cid:16) Sym ℓq ( Σ L( S | R ) ⊗ S ℓ ) (cid:17) = ⇒ ℓ h X i p + q where ℓ is the residue field of S , and the other notation including the DG algebra ℓ h X i is from 6.4.Very recently Briggs and Iyengar [35] improved upon Theorem 9.2 with thefollowing. The proof of this result also uses DG technology, but we do not discussit because of space constraints. Theorem 9.3 ([35, Theorem A]) . If ϕ is locally of finite flat dimension and onehas D n ( S | R, − ) = 0 for some n > , then ϕ is locally complete intersection. Conjecture 9.1(b) for algebra retracts.
Avramov and Iyengar [27] proved Conjecture 9.1(b) in the case where S is an algebra retract of R , that is, where there is a ring homomorphism ψ : S → R suchthat ϕ ◦ ψ = id S . Theorem 9.4 ([27, Theorem I]) . Assume that S is an algebra retract of R . Thenthe following conditions are equivalent. (i) D n ( S | R, − ) = 0 for all n ≫ . (ii) D n ( S | R, − ) = 0 for all n > . (iii) D ( S | R, − ) = 0 . (iv) D n ( S | R, − ) = 0 for some n > such that ⌊ n − ⌋ ! is invertible in S . Conjecture 9.1(b) fails in the non-noetherian case; see Andr´e [3] and Planas-Vilanova [99]. This conjecture is still open in general for noetherian rings.In the proof of Theorem 9.4, the following notion plays an essential role. Alocal homomorphism ϕ : ( R, m , k ) → ( S, n , ℓ ) is almost small if the kernel of thehomomorphism Tor ϕ ( ϕ, ℓ ) : Tor R ( k, ℓ ) → Tor S ( ℓ, ℓ ) of graded algebras is generatedby elements of degree 1.DG techniques are crucial in the proof of Theorem 9.4. Key to this is a structuretheorem [27, 4.11 Theorem] for surjective almost small homomorphisms in termsof DG algebra homomorphisms. From this one concludes [27, 5.6. Theorem] thatalmost small homomorphisms have finite weak category ; a notion motivated bythe works of F´elix and Halperin [53]. As a result, information on the positivityand growth of deviations of almost small homomorphisms is revealed by [27, 5.4.Theorem]. The local version of Theorem 9.4 follows from this via a characterizationof complete intersection local homomorphisms having finite weak category in termsof the vanishing of the Andr´e-Quillen homology with coefficients in the residue field;see [27, 6.4. Theorem]. A reduction to the local case then finishes the proof.10. Finite Generation of Hochschild Homology Algebras
Throughout this section, let ϕ : R → S be a ring homomorphism. PPLICATIONS OF DG ALGEBRA TECHNIQUES IN COMMUTATIVE ALGEBRA 19
We discuss work of Avramov and Iyengar [26] on finite generation of Hochschildhomology algebras. In it, they prove the converse of the Hochschild-Kostant-Rosenberg Theorem using DG methods and Andr´e-Quillen homology; see [40, 72,86] for definitions and facts that are used in this section.The Hochschild homology algebra, denoted HH ∗ ( S | R ), is a graded commutativealgebra defined using shuffle products on the Hochschild complex. This satisfiesHH ( S | R ) = S , and HH ( S | R ) = Ω S | R is the S -module of K¨ahler differentials.Recall that the R -algebra S is called regular if ϕ is flat and S ⊗ R k is regularfor each homomorphism R → k from R to a field k . Hochschild, Kostant, andRosenberg [69] proved that if R is a perfect field and S is smooth over R (that is, S is a regular R -algebra and essentially of finite type), then HH ∗ ( S | R ) is a finitelygenerated S -algebra. Here is the aforementioned converse. Theorem 10.1. [26, Theorem (5.3)]
Assume that ϕ is flat and essentially of finitetype. If the S -algebra HH ∗ ( S | R ) is finitely generated, then S is smooth over R . This result settles a conjecture of Vigu´e-Poirrier [119] who already establishedit in the case where S = R [ x , . . . , x n ] /I , and R is a field of characteristic 0, and I is generated by a regular sequence. It was also known for positively graded S suchthat S = R is a field of characteristic 0 by Dupont and Vigu´e-Poirrier [48].The DG techniques used in the proof of Theorem 10.1 are confined to thecharacteristic-0 case. Here Avramov and Iyengar use a version of Avramov’s ma-chine [26, 4.2] which gives a DG algebra A where H( A ) is the Tor algebra Tor R ( S, S ). Acknowledgments
We are grateful to Josh Pollitz and Keller VandeBogert for helpful suggestionsabout this survey.
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Department of Mathematical Sciences, Georgia Southern University, Statesboro,GA 30460, USA
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