LAYERED RESOLUTIONS OF COHEN-MACAULAY MODULES

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1 LAYERED RESOLUTIONS OF COHEN-MACAULAY MODULES DAVID EISENBUD AND IRENA PEEVA Abstract. Let S be a Gorenstein local ring and suppose that M is a finitely generated Cohen-Macaulay S-module ofcodimension c. Given a regular sequence f 1,...,f c in the annihilator of M we set R = S/(f 1,...,f c ) and construct layered S-free and R-free resolutions of M. The construction inductively reduces the problem to the case of a Cohen-Macaulay module of codimension c 1andleads to the inductive construction of a higher matrix factorization for M. Inthe case where M is a su ciently high R-syzygy of some module of finite projective dimension over S, the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP]. 1. Introduction Recall that if R is a local ring, then a finitely generated R-module N is called a maximal Cohen-Macaulay module (abbreviated MCM) if depth(n) = dim(r). Let S be a regular local ring and suppose that M is a finitely generated Cohen- Macaulay S-module of codimension c. Given a regular sequence f 1,...,f c in the annihilator of M, so that M is a MCM S/(f 1,...,f c )-module, we construct an S-free resolution L" S (M,f 1,...,f c ), and an R := S/(f 1,...,f c )-free resolution L# R (M,f 1,...,f c ) of M. These resolutions are constructed through an induction on the codimension, and each of them comes with a natural filtration by subcomplexes; we call them layered resolutions. The inductive construction of the resolutions follows a pattern often seen in results about complete intersections in singularity theory and algebraic geometry Mathematics Subject Classification. Primary: 13D02. Key words and phrases. Free resolutions, complete intersections, CI operators, Eisenbud operators, maximal Cohen-Macaulay modules. The work on this paper profited from the good conditions for mathematics at MSRI, and was partially supported by the National Science Foundation under Grant The authors are grateful to the National Science Foundation for partial support under Grants DMS and DMS

2 It allows us to exploit the fact that we can choose the regular sequence to be in general position with respect to M. In this way we achieve minimality for high R- syzygies, and we give necessary and su cient conditions for minimality in general. We now explain the inductive constructions. For brevity, we will always abbreviate the phrase maximal Cohen-Macaulay to MCM. In the base case of the induction, c = 0, M is 0 and the layered resolutions are trivial. For the inductive step we think of R as a quotient, R = R 0 /(f c ), where R 0 = S/(f 1,...,f c 1 ) and consider the MCM approximation : M 0 B 0 M of M as an R 0 -module, in the sense of Auslander-Buchweitz [AB]: here B 0 is a free R 0 -module, M 0 is an MCM R 0 -module without free summand and the kernel B 1 of the surjection has finite projective dimension. In our case B 1 is a free R 0 -module (Lemma 3.4) and we write B S for the complex of free S-modules B S : B S 1! B S 0 obtained by lifting the map B 1 b! B0 back to S. See Section 3 for details. Layered resolution over S. (Section 4) For the layered resolution of M over S we let K be the Koszul complex resolving R 0 as an S-module and let L 0 = L" S (M 0,f 1,...,f c 1 ), the layered resolution constructed earlier in the induction. There is an induced map B1 S! L 0 0 which, in turn, induces a map of complexes K B S! L 0 whose mapping cone we define to be the layered S-free resolution of M with respect to f 1,...,f c. Layered resolution over R. (Section 6) One way to construct the layered resolution of M over R, is to show (Section 9) that there is a periodic exact sequence!r B 1! R (M 0 B 0 )! R B 1! R (M 0 B 0 )! M! 0; this generalizes the periodic R-free resolution for a module over a hypersurface described in [Ei1] (Corollary 9.2). In the case c = 1, the module M 0 is zero, and the layered resolution is this periodic complex. As M 0 is an MCM module over R 0, the complex R L# R 0(M 0,f 1,...,f c 1 ) is an R-free resolution of R M 0. The layered resolution of M over R can be constructed from the double complex obtained by replacing R M 0 with R L# R 0(M 0,f 1,...,f c 1 ), but it is simpler to do something a little di erent, explained in Section 6: Set T 0 = L# R0 (M 0,f 1,...,f c 1 ), the layered resolution constructed earlier in the induction. The layered R-free resolution of M with respect to f 1,...,f c is obtained from T 0 by the Shamash construction applied to the box complex 2

3 T 0 :!T2 0 T1 0 T0 0 R 0 B 1 B 0, b where b and are the maps listed above. Filtrations and Layers. Each of the layered resolutions has a natural filtratration, whose subquotients are the layers; these will be described in Subsections 4.2 and 6.2. However the subcomplexes in the filtration are easy to describe: Let R(i) :=S/(f 1,...,f i ), and let M(i) be the essential MCM approximation of M over R(i) as defined in Section 3. The layered resolution L" S (M,f 1,...,f c ) is filtered by the sequence of sub-resolutions: L" S (M(1),f 1 ) L" S (M(2),f 1,f 2 ). Similarly, the layered resolution L# R (M,f 1,...,f c ) is filtered by the sequence of sub-resolutions: R L# R(1) (M(1),f 1 ) R L# R(2) (M(2),f 1,f 2 ). Minimality. Our criteria for the minimality of the layered resolutions is presented in Section 7. They imply that, when the residue field of S is infinite, the layered resolutions can be taken to be minimal for any su ciently high R-syzygy of a given R-module N. The precise statement is given in Section 8. Higher Matrix Factorizations. It is well-known that when R is a complete intersection of codimension 1 in a regular local ring, the MCM R-modules are described by matrix factorizations: Theorem 1.1. ([Ei1], see also [EP, Theorem 2.1.1]) Let 0 6= f be a non-zerodivisor in a regular local ring S. Set R = S/(f). A finitely generated R-module N is MCM if and only if it is a matrix factorization module: that is, N is the cokernel of a map d : U 1! U 0 of finitely generated free modules such that there exists a homotopy for f 1 on the complex 0! U 1 d! U0. This simply means that dh = f Id U0 and hd = f Id U1. The matrix factorization is called minimal if both d and h have entries in the maximal ideal of S. But to include all MCM modules in the result, we must allow non-minimal matrix factorizations (though only for modules with R/(f) as a summand.) In [EP] we introduced higher matrix factorizations, and showed that any su - ciently high syzygy module over a complete intersection is the module of a minimal 3

4 higher matrix factorization; note that high syzygy modules are MCM. Using the theory in Section 4, we can extend this to arbitrary MCM modules and (notnecessarily minimal) higher matrix factorizations: in Section 10 we prove Theorem 10.5 which is our extension of Theorem 1.1. Remark. Though the case when S is regular is our primary interest, the constructions work more generally when S is a local Gorenstein ring; this is described in the rest of the paper. In some of the results one can also do without the local hypothesis; we leave this to the interested reader. Notation 1.2. Throughout the paper we will use the following conventions. Let (W,@ W ) and (Y,@ Y ) be complexes. Our sign conventions are as follows: We write W[ a] for the shifted complex with W[ a] i = W i+a and di erential ( 1) W, in particular the complex W[ 1] has di W. The complex W Y has di W Y q = X i+j=q ( 1) W i Id + Y j. If ' : W[ 1]! Y is a map of complexes, so that '@ W Y ',thenthe mapping cone Cone(') is the complex Cone(') = Y W with modules Cone(') i = Y i W i and di erential Y i W i Y i Y i ' i 1 W i 1 W i As is well-known, a free resolution over a local ring is minimal if its di erentials become 0 on tensoring with the residue class field k. We extend this definition and say that a map of (possibly non-free) modules is minimal if it becomes 0 on tensoring with k. Acknowledgements. We are grateful to Alexander Pavlov for helpful observations about MCM-approximations, and to Daniel R. Grayson and Mike Stillman, the makers of Macaulay2 [M2]; the packages CompleteIntersectionResolutions.m2 and MaximalCohenMacaulayApproximations.m2 implement the constructions in this paper. 4!.

5 2. Review of MCM Approximations For the reader s convenience we review the basic ideas of MCM approximations from [AB] (see also [Di] and [EP, Section 7.3]). For simplicity, we deal only with finitely generated modules over a local Gorenstein ring S. Note first that any MCM S-module P has a unique cosyzygy module Syz S 1(P ), which is also MCM, defined as the dual of the first syzygy of the dual of P. Further, the first syzygy module Syz S 1 (P ) cannot have free summands since P is MCM over a local Gorenstein ring, as one sees by reducing to the 0-dimensional case, and it follows from the description above that the cosyzygy module Syz S 1(P ) cannot have free summands either. The essential MCM approximation of a finitely generated S-module N is by definition an MCM module App S (N) without free summands together with a map : App S (N)! N determined as follows: choose an integer q>depth S depth N and set App S (N) :=Syz S q Syz S q (N), considered together with a map : App S (N)! N induced by the comparison map of the S-free resolutions of App S (N) and N. By the uniqueness of cosyzygies, this is independent of the choice of q. In particular, if N is an MCM module, we let : App S (N)! N be the inclusion of the largest non-free summand of N. The following result is [EP, Theorem and Corollary 7.3.4]; we recall the proof for the reader s convenience. Theorem 2.1. Let S R be a surjection of local Gorenstein rings, and suppose that R has finite projective dimension as an S-module. Let N be a finitely generated R-module. (1) For any i 0, App S Syz R i (N) =Syz S i App S (N). If N is an MCM module without free summands, then the statement is also true for i<0. (2) If j>depth S depth N, then (3) App S (App R (N)) = App S (N). App S Syz R j (N) =Syz S j (N). Proof: (1): It su ces to do the cases of first syzygies and cosyzygies. Let 0! N 0! F! N! 0 be a short exact sequence, with F free as an R-module. It su Syz S i (N 0 )=Syz S i+1(n) for some i. 5 ces to show that

6 We may obtain an S-free resolution of N as the mapping cone of the induced map from the S-free resolution of N 0 to the S-free resolution of F ;iftheprojective dimension of R as an S-module (and thus of F as an S-module) is u, it follows that Syz S u+1(n 0 )=Syz S u(n). Parts (2) and (3) follow easily from Part (1). ut 3. Codimension one MCM Approximations The constructions of our layered resolutions use the codimension one case of essential MCM approximations which we describe in this section. Assumptions 3.1. In the rest of the paper, we use the following notation. Let S R 0 R be surjections of local Gorenstein rings and suppose that R = R 0 /(f), with f a non-zerodivisor in R 0.Wewritek for the common residue field of R, R 0 and S. We consider a finitely generated MCM R-module M, and we may harmlessly assume that M has no free summand as an R-module Codimension one MCM approximations. We may construct the MCM approximation of M as an R 0 -module in the following way. Let M 0 2 be the second syzygy of M as an R 0 -module, and let M 0 be the minimal second cosyzygy of M 0 2 as an R 0 -module, which is well-defined, up to isomorphism and has no free summand because R 0 is local and Gorenstein. In the notation of Figure 1, G is the minimal R 0 -free resolution of M 0 and the module M 0 2 is the common kernel of F 0 1! F 0 0 and G 1! G 0. See Figure 1. G :...! G 2 G 1 G 0 M 0 0 F 0 1 F 0 0 M 0 Figure 1. Construction of M 0 from a minimal resolution of M over R 0. The module M 0, together with the induced map : M 0! M, is the essential MCM approximation App R 0(M) of M over R 0. Let : B 0 Coker be a surjection from a free R 0 -module of minimal rank to Coker : B 0! M 6, and let

7 be a lift of this map, so that :=, : M 0 B 0 M is a surjection. The MCM approximation of M over R 0 is defined to be the module M 0 B 0 or, more properly, the map. Let : B 1! M 0 B 0 be the kernel of. Wewrite : B 1! M 0 b : B 1! B 0 for the components of. Thus we have the short exact sequence, which we call the MCM-approximation sequence over R 0, (3.3) 0! B 1 = b! Lemma 3.4. B 1 is a free R 0 -module. Proof:! M 0 B 0 = By the diagram in Figure 1, Tor R0 i,! M! 0. (M 0,k) = Tor R0 i (M,k) for i>1, so the (B 1,k)=0 long exact sequence in Tor R0 (,k) obtained from (3.3) shows that Tor R0 i for i>1 and it follows that B 1 is an R 0 module of finite projective dimension. Since the depth of M is one less than the depth of the MCM R 0 -module M 0 B 0, the short exact sequence (3.3) implies that B 1 is an MCM R 0 -module. It follows from the Auslander-Buchsbaum formula that B 1 is free. ut We will use the following proposition to derive minimality criteria for the layered resolutions. Proposition 3.5. The map b is minimal. The map is minimal if and only if the induced map k : k M 0! k M is a monomorphism. Proof: The short exact sequence (3.3) yields a right exact sequence k B 1! k k b! k M 0 k B 0 (k, k )! k M! 0. By construction, k M is the direct sum of the image of k and k, and k is a monomorphism. Thus the kernel of (k, k ) is contained in k M 0, and k b = 0. It follows that k = 0 if and only if k is a monomorphism. ut 7

8 4. The layered S-free resolution of M We let M be a Cohen-Macaulay S-module of codimension c, and we suppose that f 1,...,f c is a regular sequence in the annihilator of M. We will now construct the layered S-free resolution L" S (M,f 1,...,f c ) of M. For simplicity we work in the case where M has finite projective dimension over S. See Remark 4.3 for the changes necessary in the general case. We do this by an induction on c. In the case c =0themoduleM is 0 since we have assumed that M is an MCM R-module without free summands, and we take the resolution to be 0. For simplicity, let R 0 = S/(f 1,...,f c 1 ) and let f = f c. We now describe the inductive step. Given an S-free resolution L 0 of the essential MCM approximation M 0 of M over R 0, we construct an S-free resolution L" S (L 0,f) of M. In the induction, we will take L 0 = L" S (M 0,f 1,...,f c 1 ) and L" S (M,f 1,...,f c )=L" S (L 0,f). With notation as in Section 3, we use the MCM approximation sequence (3.3): 0! B 1 = b! Denote by B S the two-term complex! M 0 B 0 = B S : B S 1 b S! B S 0,,! M! 0. where B S 1 and B S 0 are free S-modules such that B S 1 R 0 = B 1 and B S 0 R 0 = B 0, and b S is any lift to S of the map b : B 1! B 0. Let K be the Koszul complex resolving R 0 over S. Let S : B S [ 1]! L 0 be the map of complexes whose component S 0 : B S 1! L 0 0 is a lift of the map : B 1! M 0. Choose a map of complexes S : K S B S [ 1]! L 0 extending the map S : B S [ 1]! L 0. We define L" S (L 0,f) to be the mapping cone of. Theorem 4.1. The complex L" S (L 0,f) is an S-free resolution of M. Itisminimal if and only if L 0 is minimal and the induced map is a monomorphism. k : M 0 k! M k Proof: Neither the homology nor the minimality of the mapping cone changes if we replace S with a homotopic map of complexes, and any two liftings of S are homotopic. 8

9 Minimality: Because M 0 is an R 0 -module there is a map µ : K L 0! L 0 inducing the multiplication map R 0 M 0! M 0. We take S to be the composition K B S [ 1] 1 S! K L 0 µ! K L 0. Since S is 0 on B0 S, it follows that S is zero on K B0 S. The mapping cone of S is minimal if and only if 1 S is minimal. By Lemma 3.5, thisistrueif and only if k is a monomorphism. Exactness: Because S vanishes on K B0 S, the mapping cone M( S ) is isomorphic to the mapping cone of the map of free resolutions,! S Id b S = S : K (B1 S [ 1])! K (B0 S [ 1]) L 0, S K (B 1 [ 1]) which extends the maps b S : B1 S! B0 S and ( S ) 0 = 0 S : B1 S! L 0 0. It follows from the long exact sequence of the mapping cone that M( S ) is a minimal S-free resolution of M. ut 4.2. Layers of the S-resolution. Let R(i) =S/(f 1,...,f i ) and set M(i 1) = App R(i 1) M(i) = App R(i 1) (M). It is clear from the construction that the layered resolution L" S (M,f 1,...,f c )is filtered by the sequence of sub-resolutions L" S (M(i 1),f 1,...,f i 1 ) L" S (M(i),f 1,...,f i ). We define the i-th layer to be the quotient, L" S (M(i),f 1,...,f i ) L" S (M(i 1),f 1,...,f i 1 ) = K(f 1,...,f i 1 ) B S (i), where B S (i) isthes-free complex lifting the two-term complex B(i) : B 1 (i)! B 0 (i) derived from the MCM approximation sequence for M(i) as an R(i 0! B 1 (i)! M(i 1) B 0 (i)! M(i)! 0. 1)-module, Remark 4.3. When M does not have finite projective dimension over S the MCM approximation of M over S is not free, and the inductive construction must start with a given free resolution P S of the essential MCM approximation M S of M over S. In this case we write L" S (P S,f 1,...,f c ) for the layered resolution over S. By Part (3) of 2.1, the essential MCM approximation of M 0 over S is the same as that of M. Given this, we may simply replace L" S (M,f 1,...,f c )byl" S (P S,f 1,...,f c ) and L" S (M 0,f 1,...,f c 1 )byl" S (P S,f 1,...,f c 1 ) in the proof above. Thus in the base case, c = 0, we take L" S (P S,f 1,...,f c )tobep S itself. 9

10 Corollary 4.4. If the ring S is regular and the layered resolution L" S (M,f 1,...,f c ) S c is minimal, then the Betti numbers of M satisfy i (M) i for all i Review of CI operators and the Shamash construction We will make use of the CI operators ( Complete Intersection operators) introduced in [Ei1, Section 1] (see also [EP, Section 4.1]) and the Shamash construction [Sh] (see also Construction in [EP]). For the reader s convenience we provide a summary CI operators. Suppose that f 1,...,f c 2 S is a regular sequence and (V,@) is a complex of free modules over R = S/(f 1,...,f c ). Suppose that V e is a lifting of V to S, that is, a sequence of free modules V e i and e i+1 : V e i+1! V e i such = e 2 = 0 we can choose maps et j : 1 apple j apple c, such e 2 = P c j=1 f j e t j. We set t j := R et j. Vi+1 e! V e i 1,where By [Ei1], the t j are maps of complexes V[ 2]! V that are functorial (and thus, in particular commutative) up to homotopy. If (V,@) is the minimal free resolution of a finitely generated R-module N then, writing k for the residue field of S, the CI operators t j induce well-defined, commutative maps j on Ext R (N,k), and thus make Ext R (N,k) intoamodule over the polynomial ring k[ 1,, c], where the variables j have degree 2. The j are also called CI operators. A version of the following result was first proved in [Gu] by Gulliksen, who used a di erent construction of operators on Ext. The relations between the CI operators and various constructions of operators on Ext were explained by Avramov and Sun [AS], who also proved Theorem 5.2. For a short proof see [EP, Theorem 4.2.3]. Theorem 5.2. Let f 1,...,f c be a regular sequence in a local ring S with residue field k, and set R = S/(f 1,...,f c ).IfN is a finitely generated R-module with finite projective dimension over S, then the action of the CI operators makes Ext R (N,k) into a finitely generated k[ 1,..., c]-module Higher homotopies and the Shamash construction. We need only the version for a single element, due to Shamash [Sh]; the more general case of a collection of elements is treated by Eisenbud in [Ei1]. Definition 5.4. Let G be a complex of finitely generated free R 0 -modules. system of higher homotopies for f 2 R 0 on G is a collection of maps A j : G! G[ 2j + 1] for j =0, 1,... of the underlying modules such that 10

11 0 is the di erential on G. The map is multiplication by f on G. For every j 2wehave P j q=0, q j q = 0. Proposition 5.5. [Ei2, Sh] If G is a free resolution of an R 0 -module annihilated by elements f, then there exists a system of higher homotopies on G for f. Construction 5.6. ([Ei1, Sh]) Suppose that (G,@) is a free complex over R 0 with asystem = { j } of higher homotopies for f 2 R 0. We will define a new complex over R := R 0 /(f). We write R{y} for the divided power algebra over R on one variable y; that is, R{y} = Hom graded R-modules (R[t],R)= i Ry (i) where the y (i) form the dual basis to the basis t i of the polynomial ring R[t]. The graded module R{y} G, where y has degree 2, becomes a free complex over R when equipped with the di erentials := X t j j R. This complex is called the Shamash complex of (G, ) and denoted Sh(G, ) or simply Sh(G). We now record the properties of the Shamash construction that we will use. The minimality was first proven by Avramov-Gasharov-Peeva in [AGP, Proposition 6.2]. See also [EP, Corollary 4.3.5], where a di erent proof is given. Proposition 5.7. Let G be an R 0 -free resolution of a finitely generated module Nannihilated by a non-zerodivisor f. The Shamash complex Sh(G) is a free resolution of N over R = R 0 /(f), and is minimal if and only if the CI-operator corresponding to f acts as a monomorphism on Ext R (N,k). This happens if and only if Ext R 0(N,k) = Ext R(N,k) Ext R (N,k). 6. The layered R-free resolution of M We let M be a Cohen-Macaulay S-module of codimension c, and we suppose that f 1,...,f c is a regular sequence in the annihilator of M. We will now construct the layered R-free resolution L# R (M,f 1,...,f c ) of M. For simplicity we work in the case where M has finite projective dimension over S. See Remark 6.3 for the changes necessary in the general case. We do this by induction on c. In the case c =0themoduleM is 0 since we have assumed that M is an MCM R-module without free summands, and we take the resolution to be 0. 11

12 For simplicity, let R 0 = R/(f 1,...,f c 1 ) and let f = f c. We now describe the inductive step. Given an R 0 -free resolution L 0 of the essential MCM approximation M 0 of M over R 0, we construct an R-free resolution L# R (L 0,f) of M. In the induction, we will take L 0 = L# R 0(M 0,f 1,...,f c 1 ). With notation as in Section 3, we use the MCM approximation sequence (3.3): We write 0! B 1 = b!! M 0 B 0 =,! M! 0. B : B 1! B 0 for the R 0 -free 2-term complex with di erential b. The map : B 1 a map 0 : B 1! L 0 0, which in turn defines a map of complexes! M 0 lifts to : B[ 1]! L 0. Let C( 0,b) be the mapping cone of, as shown in Figure 2. C( 0,b):!L 0 2 L 1 1 L b B 1 B 0. Figure 2. The box complex. We call C( 0,b)thebox complex. WedefineL# R (L 0,f) to be the Shamash complex Sh(C( 0,b)) defined in Construction 5.6. Theorem 6.1. With notation as above, the box complex C( 0,b) is an R 0 -free resolution of M. Thus the complex L# R (L 0,f) is an R-free resolution of M. Further C( 0,b) is minimal if and only if L 0 is minimal and the induced map k : k M 0! k M is a monomorphism. Thus L# R (L 0,f) is minimal if, in addition, the CI operator induced by the expression R = R 0 /(f c ) is a monomorphism on Ext R (M,k). Proof: Using the notation in Figure 2, 0 0 b is the first di erential of C( 0,b). We have Coker = Coker = M. b 12

13 Also, for i 2wehaveH i (C( 0,b)) = H i (L 0 ) = 0, so it is enough to show that C( 0,b) is exact at L 0 1 B 1. Suppose that (x, y) 2 Ker. It follows that by = 0 and 0 y 1 (L 0 1). Composing with the surjection L 0 0 B 0! M 0 B 0 we see that the image z of y in M 0 B 0 is zero. Since z = b y and the map b : B 1! M 0 B 0 is a monomorphism by (3.3), it follows that y = 0. 1 x = 0. Since L 0 is acyclic, x 2 2. Thus C( 0,b) is exact at L 0 1 B 1. The box complex C( 0,b) is minimal if and only if L 0 is minimal and the maps 0 and b are minimal. By Lemma 3.5, 0 and b are minimal if and only if k is a monomorphism. Since C( 0,b) is an R 0 -free resolution of M the complex Sh(C( 0,b)) is an R-free resolution of M. By[AGP, Proposition 6.2] (see [EP, Corollary 4.3.5] for a second proof), the minimal R-free resolution of M is obtained by applying the Shamash construction to the minimal R 0 -free resolution of M if and only if the CI operator :Ext R (M,k)! Ext R (M,k)[2] is injective. ut 6.2. Layers of the R-resolution. We use the same notation as in subsection 4.2. It is clear from the construction that the layered resolution L# R (M,f 1,...,f c )is filtered by the sequence of sub-complexes R L# R(i 1) (M(i 1),f 1,...,f i 1 ) R L# R(i) (M(i),f 1,...,f i ), which are themselves resolutions because f i+1,...,f c is a regular sequence on M(i) for each i. We define the i-th layer to be the quotient, R L# R(i) (M(i),f 1,...,f i ) R L# R(i 1) (M(i 1),f 1,...,f i 1 ). To describe this quotient we begin with the complexes and L 0 := L# R(i 1) (M(i 1),f 1,...,f i 1 ), B(i) : B 1 (i)! B 0 (i), corresponding to the essential MCM approximation M(i 1) of M over R(i 1). With notation as in Figure 2, the homotopy for f i on the box complex C( 0,b) induces a map h from L 0 0 to B 1, and from this we get the complex L 00 :!R L 0 1! R L h! R B1 (i) b! R B0 (i).

14 From the inductive construction we see that the i-th layer of L# R (M,f 1,...,f c )is R{y} R L 00. Remark 6.3. The situation is analogues to that in Remark 4.3. When M does not have finite projective dimension over S the essential MCM approximation of M over S is not 0, and the inductive construction must start with a given free resolution P S of the essential MCM approximation M S of M over S. In this case we write L# R (P S,f 1,...,f c ) for the layered resolution over R. We note that the essential MCM approximation of M 0 over S is the same as that of M by Part (3) of Theorem 2.1. Given this, we may simply replace L# R (M,f 1,...,f c ) by L# R (P S,f 1,...,f c ) and L# R 0(M 0,f 1,...,f c 1 )byl# R 0(P S,f 1,...,f c 1 )inthe proof above. Thus in the base case, c = 0, we take the layered resolution to be P S itself. 7. When is k a monomorphism? Theorem 7.1. Let P be an MCM R-module, and let M =Syz R 2 (P ). Let be the CI operator on Ext R (P, k) derived from the expression R = R 0 /(f). If the CI operator :Ext j R (P, k)! Extj+2 R (P, k) is injective for j =0, 1, then the essential MCM approximation : M 0! M of M over R 0 induces a monomorphism k : k M 0,! k M. Proof: Figure 3 exhibits the modules and maps that will be used. Let F be a minimal R-free resolution of P, so that M is the image 2 : F 2! F 1. Let F 0 be a lifting of F to R 0, and let t : F! F[ 2] denote the CI operator derived from the expression R = R 0 /(f). We may define maps t 0 : Fj+2 0! Fj 0 for j apple 1 by the 02 = ft 0. From the assumption that : Ext j R (P, k)! Extj+2 R (P, k) is a monomorphism, we see using Nakayama s Lemma that the maps t : F j+2! F j and t 0 : Fj+2 0! Fj 0 are surjections for j apple 1. 14

15 0 M G 2 F 0 3 G 1 = F1 F 0 2 t 0 t 0 G 0 = F 0 0 P 0 @ F 2 F 1 t t F 0 P 0 M Figure 3. For j =0, 1wesetG j = F 0 j, and we define G 2 =Ker(F 0 2 t 0! F 0 0) which is free because t 0 is surjective. Let : G 2! G 1 be the map induced 0 : F2 0! F1. 0 It follows at once that G : G 2! G 0! G 0 is a minimal R 0 -free complex. Let M 0 be the image of : G 2! G 1, and write : M 0! M for the induced map. We will show that M 0 is the essential MCM approximation of M over R 0. First we prove that G is the beginning of an R 0 -free resolution of P. 02 = ft 0 : F2 0! F0, 0 we see that the cokernel 0 : F1 0! F0 0 is annihilated by f. After tensoring with R, the cokernel is P. Thus the cokernel 0 : F1 0! F0 0 itself is P. Next we prove the exactness of G at G 1. Suppose `1 2 G 1 = F1 0 goes to 0 in G 0 = F0. 0 It follows from the exactness of F at F 1 that there is an element `2 2 F2 0 such that 0`2 = fm 0 1 for some m F1. 0 The surjectivity of t 0 : F3 0! F1 0 shows that we may write fm m 0 3 for some m F3.Thus`1 0 0 (`2 0 m 0 3). 02 (`2 0 m 0 3)=@ 0`1 = 0 by hypothesis, we see that `2 0 m G 2,proving the exactness at G 1. This shows that G 2! G 1! G 0 is the beginning of the minimal R 0 -free resolution of P. It follows that M 0 =Syz R0 2 P. Because depth R 0P =depthr 0 1 and M = Syz R 2 P, it follows from the construction in Subsection 3.2 and Theorem 2.1(3) that : M 0! M is the essential MCM approximation of M over R 0. Since G 2 is a direct summand of F2, 0 we see that the induced map k : k M 0! k M is injective. ut 15

16 8. High syzygies and the criterion for minimality Throughout this section, N denotes a finitely generated Cohen-Macaulay S-module of codimension c that has finite projective dimension as an S-module. We suppose that f = f 1,...,f c is a regular sequence in the annihilator of N and write R = S/(f 1,...,f c ) as usual. For i =0,...,c we set R(i) :=S/(f 1,...,f i ); in particular, R = R(c). Let R(i) = k[ 1,..., i] be the ring of CI operators corresponding to f 1,...,f i. To prove the minimality of the layered resolutions, we will need the i to form a quasi-regular sequence on Ext R (N,k). This can always be achieved when k is infinite. We review the relevant ideas: A sequence of elements h c,...,h 1 in a ring T is said to be quasi-regular on a T -module E if, for each i, the annihilator of h i in the module E/(h c,...,h i+1 )E has finite length. The case of interest for us is that of the finitely generated graded module Ext R (N,k) over the polynomial ring R(c). In addition to the hypotheses of Section 3, we now suppose that S contains an infinite field k. Then, any su ciently general choice of the variables i forms a quasi-regular sequence on Ext R (N,k). More precisely, for g 2 GL c (k), let f g := (f 1,...,f c )g be the sequence of k-linear combinations of the f i corresponding to g. Sincethe i form a dual basis to the f i, there is an open subset U GL c (k) such that for g 2 U the sequence of CI operators ( c ) g,...,( 1 ) g corresponding to f g is a quasi-regular sequence on Ext R (N,k); see for example [EP, Lemma 6.1.9]. For the minimality criteria we will make use of the Castelnuovo-Mumford regularity of Ext R (N,k) as an R-module, defined in the usual way in terms of the top degrees of nonvanishing components of the local cohomology with respect to ( 1,..., c) R. As R-modules we have Ext R (N,k) =Ext even R (N,k) Ext odd R (N,k), so the regularity is the maximum of the regularities of these two submodules (where Ext odd R (N,k) inherits its grading from Ext R (N,k)). In particular, if N is not R-free then reg Ext R (N,k) 1sinceExt R (N,k) is not generated in degree 0. For example, if c =0thenR = S and R = k. In this case, reg R Ext S (N,k) =reg k Ext S (N,k) = max{i Ext i S(N,k) 6= 0} = c, since we have assumed that N is Cohen-Macaulay of codimension c. In general, the invariant we will use is r(f,n) := max 2appleiapplec reg R(i) Ext R(i) App R(i) (N),k. 16

17 Theorem 8.1. Let N be a finitely generated Cohen-Macaulay S-module of codimension c that has finite projective dimension as an S-module, and let f = f 1,...,f c be a regular sequence in the annihilator of N. Suppose that the sequence of CI operators c, c 1,..., 1 on Ext R (N,k) corresponding to f is quasi-regular. If n 3 + max c 2, r(f,n), and M is the n-th syzygy of N over R, then the layered resolutions of M with respect to f, both over S and over R, areminimal. Proof: First, by a descending induction on i we will prove that i,..., 1 is a quasi-regular sequence on Ext R(i) (N,k). For i = c this is part of our hypothesis. We may assume, by induction, that i+1,..., 1 is a quasi-regular sequence on Ext R(i+1) (N,k). Choose a q such that i+1 is a non-zerodivisor on Ext >q R(i+1) (N,k). Let U be the q-th syzygy of N over R(i + 1). By Proposition 5.7, we get Ext R(i) (U, k) = Ext R(i+1) (U, k)/ i+1 Ext R(i+1) (U, k). By Theorem 2.1, Ext >m R(i) (U, k)[ q] =Ext>m R(i) (N,k) for m 0. Thus the R(i)- modules Ext R(i) (N,k) and Ext R(i+1) (N,k)/ i+1 Ext R(i+1) (N,k) become isomorphic after a su ciently high truncation, completing the induction. As the modules N and App R(i) (N) have a common syzygy over R(i), we see that the modules Ext R(i) (N,k) and Ext R(i) (App R(i) (N),k) become isomorphic after a su ciently high truncation. Therefore, i,..., 1 is a quasi-regular sequence on Ext R(i) (App R(i) (N),k) as well. Since n>1, the module M is an MCM module over R with no free summands. For i =0...,c let M(i) = App R(i) (M). For i>0wehavem(i 1) = App R(i 1) (M(i)) by Theorem 2.1, and we write i : App R(i 1) (M(i))! M(i) for the essential MCM approximation map. We will show that k i is a monomorphism. Suppose i = 1. Since both N and R have finite projective dimension over S, it follows that M has finite projective dimension as well. Therefore, M(0) = 0, so 1 = 0. Next, for i n 2, we will show that the inequality n o 3 + max c 2, reg R(i) Ext R(i) App R(i) (N),k implies that k i is a monomorphism. Let P be the minimal (n 2)-th syzygy of N over R(i). Since n 2 1+c i, the module P is an MCM module over R(i) without free summands. Note that Syz R(i) 2 (P )=Syz R(i) n (N). By Theorem 2.1(2) this is the module M(i). 17

18 We have shown, above, that the element Ext R(i) (App R(i) (N),k). Since n 2 1+reg R(i) Ext R(i) (App R(i) (N),k), the largest submodule of Ext R(i) App R(i) (N),k i is quasi-regular on the module of finite length does not meet Ext >n 2 R(i) App R(i) (N),k =Ext R(i) (P, k)[ n + 2], and thus i is a non-zerodivisor on Ext R(i) (P, k). From Theorem 7.1 we conclude that the map k i : k M(i 1)! k M(i) is a monomorphism. We now prove the minimality of the layered resolutions of M(i) overs and over R(i) by induction on i. The case i = 0 is obvious. By Theorems 4.1 and 6.1, the minimality for M(i) follows from the fact that k i is a monomorphism and the minimality of the layered resolutions of M(i 1). ut We can state a version of Theorem 8.1 that does not depend on information about the approximations App R(i) (N) at the expense of a slight weakening of the bound. We let r 0 (f,n) = max 2appleiapplec {reg R(i) Ext R(i) (N,k). Corollary 8.2. We have r 0 (f,n) replace r by r 0 in the hypothesis. r(f,n), and thus Theorem 8.1 still holds if we Corollary 8.2 follows at once from a more precise result: Proposition 8.3. Let L be a finitely generated R-module, and suppose that = depth R depth R L. We have for some b 0 while Ext > R (L, k) =k( )b Ext > R (App R(L),k) Ext > +1 R (L, k) =Ext > +1 R (App R (L),k), and the inclusion and equality respect the structure of Ext as a module over the ring of CI operators R = k[ 1,..., c] corresponding to f. We have In particular, if reg R Ext R (App R (L),k) apple max 1, reg R Ext R (L, k). =1then reg R Ext R (App R (L),k) apple reg R Ext R (L, k). Proof: It follows from the construction given in Section 3 that Syz R (L) is equal to the direct sum of Syz R App S (L) and a (possibly trivial) free summand. The inclusion and equality of the Ext-modules follows, and the R-structure is preserved. It follows that the supremum of degrees of socle elements of Ext R (App R (L),k) is at most the maximum of 1 and the supremum of the degrees of the socle 18

19 elements of Ext R (L, k), while the higher local cohomology of Ext R (App R (L),k) and Ext R (L, k) coincide, proving the regularity formula. ut In order to make use of the sharper estimate involving the r(f,n) and thus depending on reg R(i) Ext R(i) (App R(i) (N),k), we would like to understand the relationship between reg R(i) Ext R(i) (App R(i) (N),k) and reg R(j) Ext R(j) (App R(j) (N),k). In the examples we have tried using the Macaulay 2 package MCMApproximations, the following question has a positive answer: Question 8.4. With hypotheses as in Theorem 8.1, is it true that applereg R(i 1) Ext R(i 1) (App R(i 1) (N),k) apple reg R(i) Ext R(i) (App R(i) (N),k) apple so that the maximum is attained by reg R(c) Ext R(c) (N,k)? Since App R(0) (N) = 0 and App R(1) (N) is by definition an MCM R(1)-module without free summands, we have at least 0=reg R(0) Ext R(0) (App R(0) (N),k) apple reg R(1) Ext R(1) (App R(1) (N),k) apple 1. The answer to Question 8.4 is also positive for high syzygies: Corollary 8.5. With hypotheses as in Theorem 8.1, if M(i) 6= 0then for every i 1. reg Ext R(i) (M(i),k)=1 Proof: We will prove the corollary by a descending induction on i. Suppose M(i) 6= 0. The proof of Theorem 8.1 shows that i is regular on Ext R(i) (M(i),k) since M(i) =Syz R(i) 2 (P ). By Proposition 5.7 it follows that Ext R(i 1) (M(i),k) = Ext R(i)(M(i),k) iext R(i) (M(i),k). Therefore, reg Ext R(i 1) (M(i),k)=regExt R(i) (M(i),k) = 1 by induction hypothesis. By Proposition 8.3 we have reg Ext R(i 1) (M(i 1),k) apple reg Ext R(i 1) (M(i),k)=1. The regularity in the lefthand-side vanishes if and only if M(i 1) = 0. ut In general, we can establish a weaker inequality than the one in Question 8.4: Proposition 8.6. With hypotheses as in Theorem 8.1, reg R(i 1) Ext R(i 1) (App R(i 1) (N),k) apple 2+reg R(i 1) Ext R(i) (App R(i) (N),k). 19

20 Proof: We may assume i = c, which simplifies the notation: we write R 0 for R(i 1) and N 0 for N(i 1). Set r =reg R Ext R (N,k), and let T be the (r + 1)-st R-syzygy of N. The operator c is a non-zerodivisor on Ext R (T,k) so, by Proposition 5.7, Ext R 0(T,k) = Ext R(T,k) cext R (T,k). The module Ext R (T,k) = Ext >r+1 R (N,k)[r + 1] has regularity 1 over R, hence reg 0 R Ext R 0(T,k) = 1. Since T is an MCM module over R we may apply Proposition 8.3 to get reg R 0 Ext R 0(App R 0(T ),k) apple reg R 0 Ext R 0(T,k)=1. By Theorem 2.1, App R 0(T ) = Syz R0 r+1(n 0 ) and so Ext R 0(App R 0(T ),k)[ r 1] = Ext >r+1 R 0 (N 0,k). We conclude reg Ext >r+1 R 0 (N 0,k) apple r + 2. From the exact sequence 0! Ext >r+1 R 0 (N 0,k)! Ext R 0(N 0,k)! Ext 6r R 0 (N 0,k)! 0 we see that Ext R 0(N 0,k) has regularity at most r + 2, as required. ut 9. Generalized Matrix Factorization of an Element As explained in the introduction, an alternate presentation of the layered resolution over R could be deduced from the following generalization of a result on periodic resolutions over hypersurfaces in [Ei1]. Theorem 9.1. Let f 2 A be an element of a commutative ring, and let 0! N 1 d! N0! P! 0 be a short exact sequence of A-modules. If f is a non-zerodivisor on N 0 and on N 1 but fp =0, then there is a unique map h : N 0! N 1 such that dh = f Id. The map h is a monomorphism and satisfies hd = f Id. Further, if we write for A/(f), then the complex is exact.!n 1 d! N 0 h! N 1 d! N 0! P! 0 Proof. From the left exactness of the functor Hom we see that 0! Hom(N 0,N 1 )! Hom(N 0,N 0 )! Hom(N 0,P) is exact. Since f Id 2 Hom(N 0,N 0 ) goes to 0 in Hom(N 0,P), it comes from a unique map h 2 Hom(N 0,N 1 ) with the property that dh = f Id. 20

21 We claim that hd = f Id as well. Since f is a non-zerodivisor on N 1 it su ces to prove this after inverting f. However, if f is a unit then the equation dh = f Id shows that d is surjective. Since d is a monomorphism, it follows that d, and therefore also h, become isomorphisms on inverting f, soh can be cancelled on the right from the expression yielding hd = f Id as required. The right exactness A/(f) hdh = h(f Id) = (f Id)h, shows that N 1 d! N 0! P! 0 is exact. To show that the infinite sequence is exact at N 1, suppose that da = 0 for some a 2 N 1.Thenda = fe for some e 2 N 0, and so da = fe = dhe, whichimplies a = he. A similar argument proves exactness at N 0. ut Theorem 9.1 applies to the setting of MCM approximations, and yields: Corollary 9.2. Suppose that R 0 is a Gorenstein ring, f 2 R 0 a non-zerodivisor, and M an MCM module over R := R 0 /(f)-module. Let (3.3) be the corresponding MCM-approximation sequence over R 0. There is a unique map h : M 0 B 0! B 1 such that dh = f Id. The map h is a monomorphism and satisfies hd = f Id. Further, the complex! B 1 R d R! (M B 0 ) R h R! B 1 R d R! (M 0 B 0 ) R! M! 0 of R-modules is exact. 10. Maximal Cohen-Macaulay Modules from Matrix Factorization In this section we provide a description of all MCM modules over a complete intersection. In keeping with the inductive nature of layered resolutions, we give an inductive definition of a CI matrix factorization essentially equivalent to the corresponding definitions in [EP]; see Remark We write K(c 1) for the Koszul complex K(f 1,...,f c 1 )overs on f 1,...,f c 1. be its di erential, and let {e i } be a basis of K(c 1) 1 such i )=f i 2 K(c 1) 0 = S. Definition By an initial homotopy h for f 2 S on a 3-term complex U 2 d! U1 d! U0 21

22 we mean a map of degree 1 with components h : U i! U i+1 such that dh : U 0! U 0 and dh + hd : U 1! U 1 are both multiplication by f. Definition Let S be a local ring. A CI matrix factorization complex with initial homotopies with respect to a regular sequence f 1,...,f c in S is defined as a 3-term complex of free finitely generated S-modules U(c) : U 2 d! U1 d! U0 with initial homotopies h i for f i on U(c), such that: If c =1thenU(1) has the form U(1) : 0! B 1 (1) b 1! B 0 (1) with a homotopy h 1 for multiplication by f 1. (This structure is the same as that of a matrix factorization introduced in [Ei1].) If c>1then (1) U(c) has a subcomplex U(c 1) : U 0 2! U 0 1! U 0 0 with initial homotopies h 0 1,...,h 0 c 1 that is a CI matrix factorization complex with respect to f 1,...,f c 1. Furthermore, U(c) has a quotient complex U(c)/U(c 1) of the form KB : K(c b 1) S 0! B 1 (c) c! B 0 (c) apple2 for some complex of finitely generated free S-modules 0! B 1 (c) b c! B 0 (c). (2) With this decomposition, U(c) is isomorphic to the mapping cone of a map of complexes c : KB[ 1]! U(c 1) that vanishes on K(c 1) B 0 (c), while c restricted to the summand e i B 1 (c) is equal to h 0 i c,where c is the component of c from B 1 (c) to U0 0 = c i=1b 1 0 (c) (see the diagram below). (3) For p<c, the initial homotopy h p is equal to h 0 p when restricted to U(c 1) and is equal to ( 1) s+1 e p Id when restricted to K(c 1) B s (c). (4) There exists an initial homotopy h c for f c on U(c). We define the CI matrix factorization module M of U(c) tobe M = Coker (U 1 d! U0 ). The resulting CI matrix factorization with respect to f 1,...,f c is the pair (d, h), c where d is the component of the di erential in U(c) mapping p=1b 1 (p)! U 0 = 22

23 c p=1b 0 (p) (thus,d is the collection of maps b i and i ), and h is the collection of i the components of the initial homotopies h i mapping p=1b 0 (p)! i p=1b 1 (p). The following diagram may help to visualize the definition. Here U r (c) isthe direct sum of the modules in the r-th column: U 0 2 U 0 1 U 0 0 = c 1 p=1b 0 (p) e i b 7! h 0 i c (b) B 1 (c) c b c B 0 Id c 1 i=1e i B 1 (c) Id b c c 1 i=1e i B 0 Id L 1applei<japplec 1 e i ^ e j B 0 (c), is the di erential in the Koszul complex. Remark The construction above is consistent with the construction before Theorem 4.1. The complex U(c) is the beginning of the layered resolution described in Theorem 4.1. Remark Our concepts of matrix factorizations here and in [EP] are equivalent in the sense that the following three properties are equivalent: (1) M is the module of a CI matrix factorization. (2) M is the module of a higher matrix factorization (introduced in [EP, Definition 1.2]). (3) M is the module of a strong matrix factorization (introduced in [EP, Definition 1.2.3]). It is immediate that (1) implies (2), and that (3) implies (2). By [EP, Theorem 5.3.1], (2) implies (3). Furthermore, (2) implies that M is a MCM R-module by [EP, Corollary 3.11], and then Theorem 4.1 implies (1). We can now state a complete analogue of Theorem 1.1: Theorem Let f 1,...,f c be a regular sequence in a regular local ring S. Set R = S/(f 1,...,f c ). A finitely generated R-module N is MCM if and only if it is a CI matrix factorization module for the sequence f 1,...,f c. 23

24 Proof. Suppose that N is a CI matrix factorization module. Then it is a higher matrix factorization module in the sense of [EP, Definition 1.2]. By [EP, Corollary 3.11], it follows that N is a MCM R-module. Suppose that N is MCM. The free resolution in Theorem 4.1 implies that N is a CI matrix factorization module. ut As far as minimality goes, we have Theorem [EP, Theorem 1.4] Let S be a regular local ring with infinite residue field, and let I S be an ideal generated by a regular sequence of length c. Set R = S/I, andsupposethatw is a finitely generated R-module. Let f 1,...,f c be a generic choice of elements minimally generating I. IfM is a su ciently high syzygy of W over R, then M is the module of a minimal CI matrix factorization (d, h) with respect to f 1,...,f c. Moreover d R and h R are the first two di erentials in the minimal free resolution of M over R. References [AB] M. Auslander, R.-O. Buchweitz: The homological theory of maximal Cohen-Macaulay approximations, Colloque en l honneur de Pierre Samuel (Orsay, 1987), Mem. Soc. Math. France (N.S.) No. 38, (1989),5 37. [AS] L. Avramov, L.-C. Sun: Cohomology operators defined by a deformation, J. Algebra 204 (1998), [AGP] L. Avramov, V. Gasharov, and I. Peeva: Complete intersection dimension, Publ. Math. IHES 86 (1997), [Di] S. Ding: Cohen-Macaulay approximations over a Gorenstein local ring, Ph.D. Thesis, Brandeis University, [Ei1] D. Eisenbud: Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), [Ei2] D. Eisenbud: Enriched free resolutions and change of rings, Séminaire d Algèbre Paul Dubreil (Paris, ), Lecture Notes in Math. 586 (1977), Springer, 1 8. [EP] D. Eisenbud, I. Peeva: Minimal Free Resolutions over Complete Intersections, Springer Lecture Notes in Math (2016), Springer. [Gu] T. Gulliksen: A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), [M2] Macaulay2 a system for computation in algebraic geometry and commutative algebra programmed by D. Grayson and M. Stillman, [Sh] J. Shamash: The Poincaré series of a local ring, J. Algebra(1969), Mathematics Department, University of California at Berkeley, Berkeley, CA 94720, USA address: de@msri.org Mathematics Department, Cornell University, Ithaca, NY 14853, USA 24