Math. Z. 230, 545–567 (1999) Corrected version

DEPTH FOR COMPLEXES, AND INTERSECTION THEOREMS

S. IYENGAR

Abstract. This paper introduces a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier ex- tensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster’s big Cohen-Macaulay modules), and also a gen- eralization of the “depth formula” for tensor product of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules.

Introduction This paper studies homological invariants of modules and complexes over noe- therian local rings. We introduce a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Our interest in this approach arises from its efficacy in dealing with a class of problems centered around “intersection the- orems” in commutative algebra. On the one hand, it leads to a quick proof of an extension of the Improved New Intersection Theorem; this uses Hochster’s big Cohen-Macaulay modules, and hence is restricted to rings containing fields. On the other hand, it yields a generalization of the “depth formula” for tensor product of modules. For these and other applications, we extend the classical Auslander-Buchsbaum Equality: if a finitely generated module M, over a noetherian local ring R, has finite projective dimension, then depth R = depth M + pdR M. Generalizing this to complexes, in Section 2 we prove that for a complex M of finite projective dimension L depth N = depth(M⊗RN)+pdR M , L where N is any complex with Hi(N)=0for i  0, and M⊗RN stands for F ⊗R N, with F any free resolution of M. As a complement to this result we prove that in L certain cases the depth of M⊗RN can be computed in terms of the depth of its R homology modules Tori (M,N) . The basic technique employed in this paper is to L compare the depth of M⊗RN obtained by applying the two theorems. As direct corollaries of these results, we strengthen a theorem of Auslander, and extend a result of Foxby on the behavior of depth under flat base change. In Section 3, we use the same tools to prove, over a ring R containing a field, that if F is finite free complex with length Hi(F ) finite for i> 0, then length(F ) > dim R − dim(R/I) for each ideal I annihilating a minimal generator of H0(F ). This theorem is an avatar of a result of Bruns and Herzog, and reduces to the Improved New Intersec- tion Theorem of Evans and Griffith, when I is m-primary. In Section 4, we show that if M and N are Tor-independent modules, then

depth(M ⊗R N) + depth R = depth M + depth N

Date: November 3, 1999. 1991 Mathematics Subject Classification. 13C15, 13D25, 18G15. 1 2 S. IYENGAR provided one of them is of finite CI-dimension, in the sense of Avramov, Gasharov, and Peeva. This is both a common generalization and an extension of results by Auslander, and by Huneke and R. Wiegand. The results discussed thus far were for complexes over local rings. In fact, we define, for a finitely generated ideal I in a commutative ring R, the I-depth of a complex M to be the number

depthR(I,M)= n − sup{i | Hi(K ⊗R M) 6=0} where K is the Koszul complex on a set of n generators for I (the depth of complex over a noetherian local ring is the depth at the maximal ideal.) This definition is motivated by a classical result of Auslander and Buchsbaum: for a finitely generated module over a noetherian ring, this number is equal to the one obtained via regular sequences. Section 5 is devoted to proving that depth for complexes enjoys the usual prop- erties of depth for modules. Some of these extend results of Foxby, who defines depth for complexes with bounded above homology, using injective resolutions. In Section 6 we prove that for a large class of complexes, i i depthR(I,M) = inf{i | ExtR (R/I,M) 6=0} = inf{i | HI (M) 6=0} . i where HI (M) are the local cohomology modules of the complex M. In particular, our notion of depth coincides with that defined by Foxby, and Iversen. Moreover, when M is bounded above, any one of a multitude of complexes may be used to compute the depth of M. Noteworthy among these are finite free complexes with non-zero finite length homology. The arguments in this paper use a modicum of the homological theory of com- plexes, reviewed in Section 1.

1. Complexes Let R be a commutative ring. We consider complexes of the form

∂i+1 ∂i · · · −→ Mi+1 −−−→ Mi −→ Mi−1 −→· · · .

The k’th suspension of M is the complex M[k] with M[k]i = Mi−k and differential k given by ∂(m) = (−1) ∂i−k(m) for m ∈ M[k]i. For a complex M, we set

sup M = sup{i | Hi(M) 6=0} and inf M = inf{i | Hi(M) 6=0} . (so sup(0) = −∞ and inf(0) = ∞). A complex with sup M (respectively, inf M) finite is said to be bounded above (respectively, below). A bounded complex is one which is bounded both above and below. We write M \ for the graded module underlying a complex M. A quasi-isomorphism φ: M −→ N is a degree zero chain map such that H(φ) is an isomorphism. Complexes M and N are quasi-isomorphic (denoted M ' N) if they are linked by a chain of quasi-isomorphisms. A complex M with t = inf M finite (respectively, s = sup M finite) is quasi-isomorphic to τ> t (respectively, τ6 s), where

τ> t(M): · · · −→ Mt−1 −→ Mt −→ ∂(Mt) −→ 0 ,

τ6 s(M): 0 −→ ∂(Ms+1) −→ Ms −→ Ms−1 −→· · · . We use some basic facts concerning resolutions of complexes, for which we refer to [8], [19] or [22]. DEPTH FORMULAS 3

For a complex F of flat R-modules with F \ bounded below, if G and G0 are 0 quasi-isomorphic, then so are F ⊗R G and F ⊗R G . Every bounded below complex M has a flat resolution, that is, a complex F quasi-isomorphic to M such that each \ L Fi is a flat module, and F is bounded below. The symbol M⊗RN denotes any complex quasi-isomorphic to F ⊗R N, where N is a complex of R-modules. Set R L Tori (M,N) = Hi(M⊗RN) . When M and N are R-modules, this gives the classical notion. If a complex N has a flat resolution G, then

M ⊗R G ' F ⊗R G ' F ⊗R N . Thus, any one of the complexes above may be used to compute TorR (M,N) . A complex M has finite flat dimension if M ' F , where F \ is a bounded complex of flat R-modules. A projective resolution of M is a complex P quasi-isomorphic to M, with each \ Pi a projective R-module, and P bounded below. The projective dimension of M is defined by \ pdR M = inf{sup F | F a projective resolution of M} .

For an element x ∈ R, we let K(x) denote the complex ∂ 0 −→ Rex −→ R −→ 0 with ∂(ex) = x, and Rex in degree 1. The Koszul complex on a set x = {x1,...,xn}⊆ R is the complex

K(x)= K(x1) ⊗R ···⊗R K(xn) .

If M is a complex of R-modules, then we set H(x; M) = H(K(x) ⊗R M). The following properties of Koszul complexes are well known:

1.1. If y = {x1,...,xn,y} and M is a complex, then there is a long exact sequence ±y ±y · · · −→ Hi(x; M) −−→ Hi(x; M) −→ Hi(y; M) −→ Hi−1(x; M) −−→ Hi−1(x; M) −→· · · .

Indeed, as K(y)= K(x) ⊗R K(y), there is a short exact sequence

0 −→ K(x) −→ K(x) ⊕ K(x) ⊗R Rey −→ K(x) ⊗R Rey −→ 0 . This sequence is split exact, and hence remains
exact upon tensoring with M. The corresponding homology exact sequence is the long exact sequence above.

1.2. If y ∈ (x1,...,xn) and M is a complex, then y H(x; M)=0.

It is easy to see that multiplication by xi is homotopic to zero on K(xi), so the same holds for K ⊗R M. Thus, xi H(x; M) = 0 for each xi, and hence for all y ∈ (x1,...,xn).

1.3. If y = {y1,...,ym} with (y1,...,ym) = (x1,...,xn) and M is a complex, then n − sup H(x; M)= m − sup H(y; M) .

Replacing y by {x1,...,xn,y1,...,ym}, we may assume that x ⊆ y. Inducing on m, we can further restrict our attention to y = {x1,...,xn,y}. Since y ∈ (x1,...,xn), 1.2 yields y H(x; M) = 0, and hence sup H(y; M) = sup H(x; M)+1, by 1.1. 4 S. IYENGAR

2. Depth of a complex Let I be an finitely generated ideal in a commutative ring R, and let K be the Koszul complex on a set of n generators for I. For a complex of R-modules M, we introduce the I-depth of M over R as the number

depthR(I,M)= n − sup(K ⊗R M) . By 1.3, it is independent of the choice of a generating set for I. When the ring R is clear from the context, we write depth(I,M) for the I-depth of M. Observation. We catalogue a few elementary properties of depth: 1. If M ' N, then depth(I,M) = depth(I,N). Indeed, since K is a complex of \ free modules with K bounded, (K ⊗R M) ' (K ⊗R N). 2. depth(I,M)= ∞ if and only if H(K ⊗R M)=0. 3. depth(I,M) = −∞ if and only if sup(K ⊗R M) = ∞. In particular, depth(I,M) > −∞ when M is bounded above. 4. depth M[k] = depth M − k 5. For a module M over a noetherian ring R, one has depthR(I,M) = 0 if and only if there is a prime ideal p ∈ Ass M with I ⊆ p, cf. [4, 9.1]. For a complex M over a noetherian local ring (R, m, k), the depth of M is defined to be depthR M = depthR(m,M) ; when the ring R is clear from the context, we write depth M. The following theorem extends a result of Foxby [8, 12.ai], where it is proved under the more restrictive hypothesis that N is bounded. Theorem 2.1. Let (R, m, k) be a noetherian local ring, and let N be a bounded above complex of R-modules. If a complex M has finite flat dimension, then L L L depthR N − sup(M⊗Rk) when H(M⊗Rk) 6=0 depthR(M⊗RN)= L (+∞ when H(M⊗Rk)=0 . If M is bounded below, and the homology modules of M are finitely generated, then there exists a complex of free R-modules F ' M such that ∂(F ) ⊆ mF and \ L sup F = pdR M, cf. [19, Chapter 2, 2.4]. In particular, sup(M⊗Rk)=pdR M. Therefore, the following is a special case of Theorem 2.1. Corollary 2.2. If M is a complex with finitely generated homology modules, and pdR M < ∞, then L depthR N = depthR(M⊗RN)+pdR M .  Note that, when M is an R-module, and N = R, the corollary reduces to the classical Auslander-Buchsbaum Equality: depth R = depth M + pdR M. L The ‘unknown’ entity in the expressions in 2.1 and 2.2 is depth(M⊗RN). Typically depth cannot be determined by the depth of the homology modules L R Hn(M⊗RN) = Torn (M,N) . The next result will allow us to do so in certain special cases. Theorem 2.3. Let L be a complex over a noetherian local ring (R, m, k). If sup L = s is finite, and depth Hs(L) − s 6 depth Hi(L) − i, for i 6 s, then

depth L = depth Hs(L) − s. DEPTH FORMULAS 5

Before proving theorems 2.1 and 2.3, we present some corollaries. L The basic technique in this paper is to compare the depth of M⊗RN obtained via 2.2 and 2.3. The proofs of the next two results illustrate this procedure. The first was proved by Auslander, using different techniques, for finitely gener- ated N, cf. [2, 1.2]. Corollary 2.4. Let M be a finitely generated module over a local ring R such that R pdR M < ∞. Let N be an R-module, and set s = supTor (M,N) . If either s =0, R or depth Tors (M,N) 6 1, then R depthR N = depthR Tors (M,N) − s + pdR M . L R L Proof. Since H(M⊗RN) = Tor (M,N) , the hypotheses implies that M⊗RN sat- isfies the conditions of 2.3. Thus L R depth(M⊗RN) = depth Tors (M,N) − s , by 2.3 L depth N = depth(M⊗RN)+pdR M . by 2.2 Combining these two equalities gives the desired result.  An advantage in allowing N to be arbitrary is that one can deduce, using Hochster’s big Cohen-Macaulay modules, the Intersection Theorem for rings con- taining a field. This follows from the New Intersection Theorem, see remarks after 3.1, but we give a direct proof which offers a slightly different perspective on the result. 2.5. (Intersection Theorem.) Let R be a noetherian local ring containing a field. If M and N are finitely generated R-modules such that length(M ⊗R N) is finite, then dim N 6 pdR M. 0 Proof. The statement is trivial unless pdR M is finite. Let N be a big Cohen- Macaulay module over R/ ann N, so that depth N 0 = dim(R/ ann N) = dim N, cf. 3.2. Since M ⊗R N has finite length, one has Supp M ∩ Supp N = {m}, and hence Supp M ∩ Supp N 0 = {m}. A standard argument using localizations R 0 R 0 shows that Supp Tors (M,N ) = {m}, for s = supTor (M,N ) , and hence that R 0 depth Tors (M,N ) = 0. Consequently, 0 depth N = −s + pdR M 0 by 2.4. Therefore, dim N = depth N 6 pdR M.  The next corollary of Theorem 2.1 is well known for finitely generated modules, and extends a result in [8, Chapter 22], where it is proved for bounded complexes. Corollary 2.6. Let φ: (R, m, k) −→ (R0, m0, k0) be a local homomorphism, and let M be a bounded above complex of R-modules. If φ is flat, then 0 0 0 depthR0 (M ⊗R R ) = depthR M + depth(R /mR ) . Proof. Let K0 be the Koszul complex on a generating set for m0. Note that 0 depthR0 (M ⊗R R ) > −∞, since M is bounded above. Thus, the isomorphism 0 0 0 0 of complexes (M ⊗R R ) ⊗R0 K =∼ M ⊗R K implies that sup(M ⊗R K ) < ∞. Since K0 is a bounded complex of flat R-modules, by 2.1 0 0 depthR(M ⊗R K ) = depth M − sup(K ⊗R k) (*) 0 0 When M ⊗R K ' 0, one has depth M = ∞ by (*), and depthR0 (M ⊗R R ) = ∞ by definition; hence equality holds. 6 S. IYENGAR

0 In remains to consider the case where sup(M ⊗R K ) is finite. By 1.2, for each integer i 0 0 0 m Hi(M ⊗R K ) ⊆ m Hi(M ⊗R K )=0 . 0 0 In particular, depthR Hi(M ⊗R K ) = 0 whenever Hi(M ⊗R K ) 6=0, so by 2.3 0 0 depthR(M ⊗R K )= − sup(M ⊗R K ) 0 0\ = depthR0 (M ⊗R R ) − sup K 0 Combining this equation for depthR(M ⊗R K ) with the one in (*), and rearranging terms, yields 0 0 0 depthR0 (M ⊗R R ) = depth M + sup K − sup(K ⊗R k) . 0 0 0 0\ 0 0 But depth(R /m R ) = sup K − sup(K ⊗R k), as K ⊗R k is a Koszul complex on a set of generators of the maximal ideal of R0/mR0. This gives the required result.  Now we prove theorems 2.1 and 2.3. Proof of Theorem 2.1. To begin with, since s = sup N is finite, N is quasi- \ isomorphic to τ6 s(N). Thus, we may assume that N 6' 0, and that N itself is bounded above. \ L Let F ' M be a flat resolution of M with F bounded; thus M⊗RN ' F ⊗R N L and M⊗Rk ' F ⊗R k. Let K be the Koszul complex on a generating set for m, and set G = K ⊗R F ⊗R N. The spectral sequence of the filtration K ⊗R (F6 p)⊗R N of G converges to H(G). As F is a complex of flat modules, the spectral sequence has 1 2 Ep,q = Hq(K ⊗R N) ⊗R Fp and Ep,q = Hp(Hq(K ⊗R N) ⊗R F ).

By 1.2, each Hq(K ⊗R N) is a k-vector space, and so 2 Ep,q = Hq(K ⊗R N) ⊗k Hp(F ⊗R k) L 2 If either H(M⊗Rk) = 0, or H(K ⊗R N) = 0, then Ep,q = 0 for all p, q. This shows that H(K ⊗R N ⊗R F ) = 0, and hence that depth(N ⊗R F )= ∞; this is the desideratum. L 2 When H(M⊗Rk) 6= 0, the expression above for Ep,q shows that 2 Ep,q =0 for p> sup(F ⊗R k) and q> sup(K ⊗R N) ; 2 Ep,q 6=0 for p = sup(F ⊗R k) and q = sup(K ⊗R N) . Therefore, by a standard “corner” argument,

sup G = sup(K ⊗R N) + sup(F ⊗R k) , which is equivalent to L depth(F ⊗R N) = depth N − sup(F ⊗R k) = depth N − sup(M⊗Rk) . This is the required equality.  Proof of Theorem 2.3. Let K be the Koszul complex on a generating set for m. \ Since L ' L6 s, we may assume that L is bounded above. The spectral sequence of the filtration K6 p ⊗R L of G = K ⊗R L converges to H(G) with 1 2 Ep,q = Kp ⊗R Hq(L) and Ep,q = Hp(K ⊗R Hq(L)). DEPTH FORMULAS 7

Rewriting the hypotheses in the form

sup(K ⊗R Hq(L)) 6 sup(K ⊗R Hs(L)) + s − q for q 6 s we get the first formula below (the second one is tautological) : 2 Ep,q =0 for p + q> sup(K ⊗R Hs(L)) + s ; 2 Ep,q 6=0 for p = sup(K ⊗R Hs(L)) and q = s .

2 Furthermore, Ep,q = 0 for q>s. The standard “corner argument” shows that

sup(K ⊗R L) = sup(K ⊗R Hs(L)) + s , and this implies that depth L = depth Hs(L) − s. 

Information on the depth of Li, or on the depth of the homology modules Hi(L), leads to useful lower bounds on the depth of L: Proposition 2.7. Let L 6' 0 be a complex over a noetherian local ring (R, m, k).

(1) depthR L > inf{ depthR Li − i | Li 6=0}.

(2) depthR L > inf{ depthR Hi(L) − i | Hi(L) 6=0}.

Proof. Let K be the Koszul complex on a generating set for m, and set G = K ⊗R L. (1) The spectral sequence of the filtration K ⊗R L6 p converges to H(G) with 1 Ep,q = Hq(K ⊗R Lp) . If d is the right hand side of the expression in (1) above, then \ sup H(K ⊗R Lp) 6 sup K − d − p for each p , 2 \ and this forces Ep,q = 0 for p+q> sup K −d. As the spectral sequence converges, ∞ the same holds for Ep,q. In particular \ sup(K ⊗R L) 6 sup K − d . This is the desired result. (2) Consider the spectral sequence associated to the filtration K6 p ⊗R L of G, used in the proof of 2.3, and argue as in the proof of (1) above. 

The next example shows that the inequalities in the proposition may be strict: Example. Let R = k[[x, y]] be the power series ring over a field k, and consider 0 ∂2 3 ∂1 L: 0 −→ R −→ R −→ R −→ 0 with ∂2 = 0 , ∂1 = (x, y, 0) . 1   Clearly, H0(L)= k, H1(L) =∼ R and H2(L) = 0, hence

inf{ depthR Li − i | Li 6=0} = −1 ;

inf{ depthR Hi(L) − i | Hi(L) 6=0} =0

(x,y) On the other hand, L is quasi-isomorphic to the complex M : 0 −→ R2 −−−→ R −→ 0. Since pdR M = 1, one has depth L = depth M = 1, by 2.2. 8 S. IYENGAR

3. Intersection theorems The purpose of this section is to prove a theorem that contains the equicharac- teristic case of a series of “intersection theorems” in commutative algebra. When I is an m-primary, it specializes to the Improved New Intersection Theorem of Evans and Griffith, see [6, 1.6] and [10, 2.6]; it is not known whether that result holds for all local rings. When H0(F ) 6= 0 has finite length, this is the New Intersection The- orem, proved by Hochster [11, 6.1], Peskine and Szpiro [17], and Paul Roberts [18]; Roberts [20] extended the New Intersection Theorem to all local rings. Recall that a minimal generator of an R-module C is an element e ∈ C \ mC. Theorem 3.1. Let (R, m, k) be a noetherian local ring containing a field, and let

F : 0 −→ Fn −→ . . . −→ F0 −→ 0 be a complex of finitely generated free R-modules. If length Hi(F ) is finite for each i> 0, and I is an ideal which annihilates a minimal generator of H0(F ), then n > dim R − dim(R/I) . The result above is a variant of a theorem of Bruns and Herzog, cf. 3.5 below. Their proof uses Hochster’s big Cohen-Macaulay modules and the Buchsbaum- Eisenbud Acyclicity Criterion. Our proof of Theorem 3.1, based on techniques developed in Section 2, uses only big Cohen-Macaulay modules. 3.2. Big Cohen-Macaulay modules. An R-module N is said to be big Cohen- Macaulay if depth N = dim R. Such modules were constructed by Hochster [9] when R contains a field. Hochster [11, 5.4] has shown that such a ring also has a balanced big Cohen-Macaulay module N, that is, one for which every system of parameters of R is N-regular. The reader is referred to [4, Chapter 8] for details on the existence and properties of big Cohen-Macaulay modules. R. Sharp [21, 3.2] has shown that if N is a balanced big Cohen-Macaulay module, then dim Rp 6 depth Np for each prime ideal p ∈ Spec R; in particular, if depth Np is finite, then depth Np = dim Rp. When I is m-primary, the following lemma is proved in [6, 1.5]. Lemma 3.3. Let I be a ideal in a noetherian local ring R. Let C be an R-module, and and let e ∈ C \ mC be such that Ie =0. If N is an R-module with N 6= mN, then depth(C ⊗R N) 6 dim R/I.

Proof. Set D = C ⊗R N. The natural surjection D −→ (C/mC) ⊗k (N/mN) maps e ⊗R N onto a submodule isomorphic to N/mN. In particular, there is an element f ∈ D \ mD, with If = 0. We induce on the depth of D: the case where depth D = 0 is trivial. Suppose that depth D > 0. By going to a faithfully flat extension (with a zero dimensional fiber), of R, we may assume that there is an x ∈ m, which is D-regular; see [4, 9.1.3]. Set J = ann(f), and note that I ⊆ J. As x is D-regular, x 6∈ p, for each prime ideal p ∈ Min(J). Furthermore, (J, x) annihilates the image of e in D/xD. As depth(D/xD) = depth D − 1, the induction step yields depth D − 1 = depth(D/xD) 6 dim R/(J, x) 6 dim(R/J) − 1 6 dim(R/I) − 1. This is the desired inequality.  Proof of Theorem 3.1. Let N be a balanced big Cohen-Macaulay R-module, cf. 3.2, set C = H0(F ) and s = sup H(F ⊗R N). DEPTH FORMULAS 9

Take a prime p ∈ Ass Hs(F ⊗R N), so that depthRp Hs(Fp ⊗Rp Np) = 0. Clearly

L = Fp ⊗Rp Np satisfies the condition in 2.3, so depthRp (Fp ⊗Rp Np)= −s. This, in conjunction with 2.2, yields

dim Rp = depthRp Np = depthRp (Fp ⊗Rp Np)+pdRp Fp (*)

= −s + pdRp Fp .

If s > 0, then we claim that p = m. Indeed, if p 6= m, then Fp ' Cp, and the expression above, along with the classical Auslander-Buchsbaum Equality yields a contradiction: 6 dim Rp = −s + pdRp Cp −s + depth Rp < depth Rp . Thus, p = m, hence (*) reduces to dim R = −s + pd F 6 −s + n

dim R = depthR N = depthR(F ⊗R N)+pdR F by 2.2

= depthR(C ⊗R N)+pdR F 6 dim(R/I)+ n which is the desired result. 

In order to discuss the relation between Theorem 3.1 and the result of Bruns and Herzog mentioned above, we recall: 3.4. Buchsbaum-Eisenbud Acyclicity Criterion For a complex of finitely gen- erated free R-modules

φn φ1 F : 0 −→ Fn −→ . . . −→ F1 −→ F0 −→ 0 , n j−i and 1 ≤ i ≤ n, set ri = j=i(−1) rank Fj and let Iri (φi) be the ideal of ri × ri minors of φi. The following conditions on an R-module N are equivalent: P (a) F ⊗ N is acyclic;

(b) depth(Iri (φi),N) ≥ i for i =1,...,n. For a complex F as above, Bruns and Herzog introduce

codim F = inf{dim R − dim(R/Iri (φi)) − i | for 1 ≤ i ≤ n} .

If codim F ≥ 0, and N is a balanced big Cohen-Macaulay module, then Iri (φi) contains a part of a system of parameters x1,...,xi, and so by the Acyclicity Criterion, F ⊗ N is acyclic, (the preceding remark is Lemma [4, 9.1.8]). Thus, the argument in the last paragraph of the proof of Theorem 3.1 yields: 3.5. Theorem [4, 9.4.1]. Let (R, m, k) be a noetherian local ring containing a field, and let F be a complex of finitely generated free R-modules. If codim F ≥ 0, and I is an ideal which annihilates a minimal generator of H0(F ), then n > dim R − dim(R/I) .  10 S. IYENGAR

Next, we deduce Theorem 3.1 from the theorem above. It suffices to assume that n ≥ dim R. Let 1 ≤ i ≤ n be an integer, and let p ∈ Spec R be a prime ideal with height Iri (φi) = height p. If height p = dim R, then height Iri (φi) ≥ dim R ≥ i. If height p < dim R, then

height Iri (φi) ≥ depth(Iri (φi) ⊗R Rp, R) = depth(Iri (φi ⊗R Rp), R) ≥ i . where the last inequality is by 3.4, since Fp is acyclic. In either case, it follows that

dim R − dim(R/(Iri (φi)) ≥ height Iri (φi) ≥ i , which shows that codim F ≥ 0. Now 3.5 gives n > dim R − dim(R/I).

4. Depth of tensor products

In this section, we explore the relation between the depths of M, N, and M ⊗RN. The following consequence of Theorem 2.1 was pointed out to us by Foxby. Theorem 4.1. Let (R, m, k) be a noetherian local ring, and let N be a complex this is bounded above. If a complex M has finite flat dimension, then L depth(M⊗RN) + depth R = depth M + depth N . L Proof. By 2.1, for any bounded above complex N, if H(M⊗Rk) = 0, then L depth(M⊗RN)= ∞. In particular, with N = R, one has depth M = ∞. Since N is bounded above, depth N > −∞, and the desired equality holds trivially. L Suppose that H(M⊗Rk) 6= 0. In this case, one has L L depth(M⊗RN) = depth N − sup(M⊗Rk) L depth M = depth R − sup(M⊗Rk) , where the first equality is by 2.1, and the second equality is the special case N = R of the first. The desired expression is obtained by subtracting the second equality from the first, and rearranging terms.  R Modules M and N over a ring R are Tor-independent if Tori (M,N) = 0 for i > 0. This condition is equivalent to the fact that the canonical augmentation L M⊗RN −→ M ⊗R N is a quasi-isomorphism. Thus, Theorem 4.1 specializes to Corollary 4.2. If M and N are Tor-independent modules over a local ring R, and M has finite flat dimension, then

depth(M ⊗R N) + depth R = depth M + depth N . 

In the case where M and N are finitely generated, this can also be deduced from a theorem of Auslander [2, 1.2] (see Corollary 2.4 above). Recently, Huneke and Wiegand [12, 2.5] proved that this equation holds for any two finitely generated, Tor-independent modules over a complete intersection ring. Following [1], we say that a module M 6= 0 over a noetherian local ring R has finite CI-dimension if there exists a diagram of local homomorphisms R −→ R0 ←− Q such that R −→ R0 is flat, R0 ←− Q is surjective with kernel generated by a regular 0 sequence, and the flat dimension of Q(M ⊗R R ) over Q is finite. Evidently, if pdR M < ∞, then M has finite CI-dimension. Furthermore, it is easy to see that all modules over a complete intersection have finite CI-dimension. Hence, the following theorem extends both results above. DEPTH FORMULAS 11

Theorem 4.3. Let M and N be Tor-independent modules over a noetherian local ring R. If M has finite CI-dimension, then

depth(M ⊗R N) + depth R = depth M + depth N . Remark . The statement of the theorem imposes two homological conditions on the modules M and N: the first is that they are Tor-independent, and the second is that one of them has finite CI-dimension. When M and N are not Tor-independent, the equation above may fail to hold, even over a regular ring R. Indeed, by the Auslander-Buchsbaum Equality the assertion in 4.3 is equivalent to pd(M ⊗R N)=pd M + pd N. When dim R> 0, for M = N = k one has

pd(M ⊗R N) = dim R< 2 dim R = pd M + pd N . On the other hand, since R is a UFD, if an ideal J is two-generated, then pd(R/J) 6 2. Burch [5] constructs an ideal I = (a,b,c) with pd(R/I) = dim R. Thus, for M = R/(a,b) and N = R/c, we get pd(M ⊗R N) > pd M + pd N, whenever dim R> 3.

The proof of Theorem 4.3 (which is new also when R is a complete intersection) starts with the following:

Computation 4.4. Let x = x1,...,xc be a regular sequence in a commutative ring Q. The Koszul complex K(x) is a Q-free resolution of R = Q/(x). Thus, if M is a Q-module annihilated by (x), then Q q c Torq (M, R) = Hq(x; M)= ∧ (R ) ⊗R M, for all q > 0 . This generalizes to:

Lemma 4.5. Let x = x1,...,xc be a regular sequence in a commutative ring Q. If M and N are Tor-independent modules over R = Q/(x), then Q n c Torn (M,N) = ∧ (R ) ⊗R (M ⊗R N) for all n > 0 . Proof. Consider the standard change of rings spectral sequence with 2 R Q Q Ep,q = Torp Torq (M, R) ,N =⇒ Torp+q (M,N) . Using 4.4, we get
2 q c R Ep,q = ∧ (R ) ⊗R Torp (M,N) . 2 Since M and N are Tor-independent over R, we have Ep,q = 0 for p > 0, and Q 2 hence Torn (M,N) = E0,n, hence the required equality.  Proof of Theorem 4.3. Consider a diagram of local homomorphisms R −→ R0 ←− Q given by the hypothesis that the CI-dimension of M is finite. As R −→ R0 is flat, 0 R 0 0 ∼ R 0 Torp (M ⊗R R ,N ⊗R R ) = Torp (M,N) ⊗R R , for all p> 0. Thus, M 0 and N 0 are Tor-independent R0-modules, as M and N are Tor- independent. Furthermore, 0 0 0 depthR0 (L ⊗R R ) = depthR L + depth(R /mR ) for any R-module L, cf. 2.6. Thus, it is enough to prove the depth formula for 0 0 0 0 the R -modules M ⊗R R and N ⊗R R . Replacing R by R, we may assume that R = Q/(x), with x = x1,...,xc a Q-regular sequence, and that the flat dimension of M over Q is finite. 12 S. IYENGAR

Note that depthR L = depthQ L for any R-module L, since Q −→ R is surjective. Hence, it is enough to establish that

depthQ(M ⊗R N) + depthQ R = depthQ M + depthQ N Q If M ⊗R N = 0, then 4.5 shows that Torn (M,N) =0, for all n, so M and N are Tor-independent over Q. Therefore, depthQ M + depthQ N = depthQ(M ⊗Q N)= ∞, by 4.2, and the depth formula holds. Consider the case where M ⊗R N 6= 0. Lemma 4.5 tells us two things: first, Q Q that sup Tor (M,N) = c; second that depthQ Torp (M,N) = depthQ(M ⊗R N), L for all 0 6 p 6 c. Thus, the complex H(M⊗QN) satisfies the conditions of 2.3, since L Q H(M⊗QN) = Tor (M,N), and hence L Q depthQ(M⊗QN) = depthQ Torc (M,N) − c = depthQ(M ⊗R N) − c . Furthermore, 4.1 applies, as the flat dimension of M over Q is finite, so L depthQ(M⊗QN) = depthQ M + depthQ N − depth Q . Combining the two equalities, and rearranging terms yields

depthQ(M ⊗R N) + depth Q − c = depthQ M + depthQ N .

As x is a regular sequence, depthQ R = depth Q−c. This is the required result. 

5. Properties of Depth In this section we show that many of the properties of depth for modules extend to identical statements for complexes. The first result captures the behavior of depth on short exact sequences. Proposition 5.1. Let R be a commutative ring. Consider a short exact sequence of complexes of R-modules: 0 −→ L −→ M −→ N −→ 0 . For each finitely generated ideal I of R, there are inequalities depth(I,M) > min{depth(I,L), depth(I,N)} , depth(I,N) > min{depth(I,L) − 1, depth(I,M)} , depth(I,L) > min{depth(I,M), depth(I,N)+1} . Proof. Let K be the Koszul complex on a system of generators for I. As K is a complex of free modules, the sequence of complexes

0 −→ K ⊗R L −→ K ⊗R M −→ K ⊗R N −→ 0 , is exact. A routine analysis of the homology long exact sequence yields the required inequalities.  The next proposition tracks depth under change of rings or ideals. Proposition 5.2. Let φ: R −→ R0 be a homomorphism of rings, and let I and I0 be finitely generated ideals in R and R0, respectively. For a complex of R0-modules N

0 (1) depthR(I,N) = depthR0 (IR ,N) 0 0 (2) depthR(I,N) 6 depthR0 (I ,N) if φ(I) ⊆ I . DEPTH FORMULAS 13

For a complex of R-modules M

0 0 (3) depthR(I,M) 6 depthR0 (IR ,M ⊗R R ) if φ is flat. 0 0 (4) depthR(I,M) = depthR0 (IR ,M ⊗R R ) if φ is faithfully flat. 0 0 Proof. (1) If K is the Koszul complex on a set of generators for I, then K = K⊗RR is the Koszul complex on a set of generators for IR0. The equality that we seek follows from 0 K ⊗R N = K ⊗R S ⊗S N = K ⊗S N . 0 0 (2) By (1), it is enough to show that depthR0 (IR ,N) 6 depthR0 (I ,N) Let x be a finite set of generators for IR0, and take a generating set for I0 of the form x ∪{y1,...,ym}. By inducing on m, we may restrict ourselves to the case where m = 1. The inequality follows from the long exact sequence in 1.1. (3) and (4) Let K and K0 be as above. Since 0 0 0 K ⊗R0 (M ⊗R R ) = (K ⊗R M) ⊗R R 0 0 0 and R −→ R is flat, sup(K ⊗R0 (M ⊗R R )) 6 sup(K ⊗R M), with equality when R −→ R0 is faithfully flat.  For a module over a noetherian ring R, it is well known that I-depth may be computed locally. The next proposition extends this to complexes, and character- izes those primes at which the I-depth is attained. In it, we denote by V(I) the set of prime ideals containing I. Proposition 5.3. Let R be a noetherian ring, and let M be a bounded above com- plex of R-modules. Let I be an ideal in R, and let K be the Koszul complex on a finite set x of generators for I, and set s = sup H(K ⊗R M)

(1) The R-module Hs(K ⊗R M) is independent up to isomorphism of the choice of x. (2) If depthR(I,M)= d is finite, then

{p ∈ V(I) | depthRp Mp = d} = AssR Hs(K ⊗R M) .

Proof. (1) Suppose that y = {y1,...,yn} generates I. As in the proof of 1.3, one can reduce to the case where y = x ∪ y. Since y ∈ (x), one has y H(x; M)=0, by 1.2, and hence Hs(x; M) =∼ Hs+1(y; M), by 1.1. \ (2) Since I ⊂ p, the complex Kp is minimal, and so pdRp Kp = sup K . In particular \ depthRp Mp = depthRp (Kp ⊗Rp Mp) + sup K (*) by 2.2. Note that p ∈ Ass Hs(K ⊗R M) if and only if depth Hs(Kp ⊗Rp p)=0.

If depthRp Hs(Kp ⊗Rp Mp) = 0, then L = Kp ⊗Rp Mp satisfies the condition in

2.3. This implies that depthRp (Kp ⊗Rp Mp)= −s, and so equation (*) reduces to \ depthRp Mp = −s + sup K = depthR(I,M) .

On the other hand, if depthRp Hs(Kp ⊗Rp Mp) > 0, then 6 depthRp Hi(Kp ⊗Rp Mp) − i> −s for i sup(Kp ⊗Rp Mp) 6 since sup(Kp ⊗Rp Mp) s. Thus, depthRp (Kp ⊗Rp Mp) > −s, by 2.7.2, and so (*) yields 14 S. IYENGAR

\  depthRp Mp > −s + sup K = depthR(I,M) .

Proposition 5.4. Let I be an ideal in a noetherian ring R. If a complex M is bounded above, then

depthR(I,M) = inf{depthRp Mp | p ∈ V(I)} .

Proof. By 5.2.2, the inclusion IRp ⊆ pRp yields the first inequality below:

depth Mp > depth(Ip,Mp) > depth(I,M) ; the second inequality holds as the homomorphism R −→ Rp is flat, cf. 5.2.3. Now we may assume that depth(I,M) is finite. Thus, there exists a prime ideal p ∈ V(I) such that depth(I,M) = depth Mp, by 5.3. 

Proposition 5.5. Let I and J be finitely generated ideals in a commutative ring R. For a complex M that is bounded above (1) depth(IJ,M) = min{depth(I,M), depth(J, M)}. (2) depth(IJ,M) = depth(I ∩ J, M) if I ∩ J is finitely generated. (3) depth(J, M) = depth(I,M) if rad(J) = rad(I) and R is noetherian. If, in addition, R is a noetherian local ring, and M has finitely generated homol- ogy modules, then (4) depth M 6 depth(I,M) + dim R/I . Proof. (1) To begin with, depth(IJ,M) 6 min{depth(I,M), depth(J, M)}, by 5.2.2. Now, we may assume that depth(IJ,M) is finite. Furthermore, in view of 5.2.1, a standard argument allows us to assume that R is noetherian. By Proposition 5.4, there exists a prime ideal p ⊇ IJ with depth Mp = depth(IJ,M). As IJ ⊆ p, it must be that I ⊆ p or J ⊆ p. Assume, without loss of generality, that I ⊆ p. By 5.4, depthR(I,M) 6 depth Mp . Thus depth(I,M) 6 depth(IJ,M), and we are done. (2) When I ∩ J is finitely generated, the required result follows from (1) and the obvious inequalities depth(IJ,M) 6 depth(I ∩ J, M) 6 min{depth(I,M), depth(J, M)} . (3) Clearly, we may assume that J = rad(I). As R is noetherian, we can choose an integer n such that J n ⊆ I. The result follows from depth(J, M) = depth(J n,M) 6 depth(I,M) 6 depth(J, M) , where the equality comes from (1), and the inequalities from 5.2.2. (4) Since R is a noetherian local ring, we can pick elements y = {y1,...,yn}⊂ R, whose image under the canonical surjection R −→ R/I forms a system of parameters for R/I. In particular, rad(I, y)= m and n = dim R/I. Since Hi(M) is finitely generated for each i, a standard argument using 1.1 shows that depth((I,y1),M) 6 depth(I,M)+1, and hence

depth(m,M) = depth((I,y1,...,yn),M) 6 depth(I,M)+ n , where the equality follows from rad(I, y) = m and (3) above. Since depth M = depth(m,M), we are done.  DEPTH FORMULAS 15

6. Depth sensitivity For modules over a noetherian ring, it is well known that the depth with respect to an ideal may be computed via injective resolutions. In this section, we extend this to complexes, under certain conditions on the homology modules. (Refer to 6.5 for a recap on injective resolutions of complexes.) Theorem 6.1. Let I be an ideal in a noetherian ring R. If M is a bounded above complex of R-modules, then i depthR(I,M) = inf{i | ExtR (R/I,M) =0} .

Moreover, when depthR(I,M)= d is finite d {p ∈ V(I) | depthRp Mp = d} = AssR ExtR (R/I,M) . In particular, with I = m, the theorem shows that our notion of depth for complexes, given in Section 1, extends that of Foxby [8, Chapter 12], and Iversen [14]. The next result relates depth to local cohomology modules, cf. [22]. Theorem 6.2. Let I be an ideal in a noetherian ring R. If M is a bounded above complex of R-modules, then i depthR(I,M) = inf{i | HI (M) 6=0}.

Moreover, when depthR(I,M)= d is finite d {p ∈ V(I) | depthRp Mp = d} = AssR HI (M) .

Note . When depthR(I,M) is finite, the theorem above, and Proposition 5.3 yield: d AssR HI (M) = AssR Hs(K ⊗R M), where K is the Koszul complex on a generating set for I, and s = sup(K ⊗R M). In particular, if the R-module Hi(M) is finitely d generated for all i, then Ass HI (M) is finite. This fact is folklore, at least when M is a finitely generated R-module. We deduce the preceding theorems from the result below. It identifies a large family of complexes which may be used, in lieu of the Koszul complex, to measure depths of complexes. Lemma 6.3. Let I be an ideal in a noetherian ring R, M a complex of R-modules, and let F be a complex of flat R-modules

∂n 0 → Fn −→ Fn−1 −→· · · \ such that Fn is faithfully flat and ∂n(Fn) ⊆ IFn−1. If either M is bounded above and (F ⊗R M)p is exact for each prime ideal p ∈ Spec(R) \ V(I), or M is bounded \ above, F is bounded, and Fp is exact for each prime ideal p ∈ Spec(R) \ V(I), then

depthR(I,M)= n − sup(F ⊗R M) .

Moreover, if depthRp (I,M)= d is finite, then

{p ∈ V(I) | depthRp Mp = d} =AssR Hs(F ⊗R M)

where s = sup(F ⊗R M) . 16 S. IYENGAR

\ Proof. When F is bounded, if s = sup M is finite, then F ⊗R M is quasi-isomorphic to F ⊗R τ6 s(M). Moreover, if Fp exact, then (F ⊗R M)p is exact. Thus, it suffices \ to consider the case where M is bounded above and (F ⊗R M)p is exact for each prime ideal p ∈ Spec(R) \ V(I). Let K be the Koszul complex on a finite set of generators for I, and set G = \ K ⊗R F ⊗R M. The hypotheses ensure that G is bounded above. The spectral sequence of the filtration K ⊗R (F6 p) ⊗R M converges to H(G). Arguing as in the proof of 2.1 (with m replaced by I), one sees that

sup G = sup(K ⊗R M)+ n . (*)

A similar argument with the spectral sequence of the filtration K6 p ⊗R F ⊗R M, this time using the fact that (F ⊗R M)p is exact for each p ∈ Spec(R) \ V(I), yields \ sup G = sup(F ⊗R M) + sup K . \ Thus, we have an equality: sup(F ⊗R M) + sup K = sup(K ⊗R M)+ n, and this gives the expression for depth M, upon rearrangement. The proof of the statement regarding the associated primes is along the lines of that of 5.3. Note that (*) is equivalent to depthR(I,M) = depthR(I, F ⊗R M)+ n. Thus, for any prime ideal p ∈ V(I), since ∂n((Fn)p)) ⊆ p(Fn−1)p, one has

depthRp Mp = depthRp (Fp ⊗Rp Mp)+ n .

For the complex L = Fp ⊗Rp Mp, by 2.3, we get depth L > −s, with equality if and only if p ∈ AssR Hs(F ⊗R M). This, and the equality above, give the required result. 

A special case of the lemma above is given in Corollary 6.4. Let (R, m, k) be a noetherian local ring, and let

F : 0 −→ Fn −→· · · −→ F0 −→ 0 be a complex of free R-modules with ∂(F ) ⊆ mF and Hi(F ) of finite length, for each integer i. If M is a bounded above complex, then

depth M = n − sup(F ⊗R M).  Before proving Theorem 6.1, we recall some basic facts concerning injective res- olutions of complexes, cf. [22]. 6.5. Injective resolutions. When E is a complex of injective modules with E\ 0 0 bounded above, if P ' P , then HomR (P, E) ' HomR (P , E). A bounded above complex M has an injective resolution, that is, a complex E ' M, with each Ei an \ injective module, and E bounded above. The symbol RHomR (N,M) denotes any complex quasi-isomorphic to HomR (N, E), for a complex of R-modules N. Set i ExtR (N,M) = H−i(RHomR (N,M)) . When M and N are R-modules, this gives the classical notion. If P ' N is is a projective resolution of N then,

HomR (N, E) ' HomR (P, E) ' HomR (P,M) .

Thus, any one of the complexes above may be used to compute ExtR (N,M) . DEPTH FORMULAS 17

Proof of Theorem 6.1. This equality holds trivially when M ' 0. Consider the case where sup M = s is finite; thus M ' M6 s. Since depth(I,M) = depth(I,M6 s) ∼ and ExtR (R/I,M) = ExtR (R/I,M6 s) , we may replace M with M6 s and assume that M 6' 0 and M \ is bounded above. Let P be a minimal free resolution of R/I, and set F = HomR (P, R). Since ExtR (R/I,M) p = 0 for each prime ideal p ∈ V(I), the complexes F and M satisfy the conditions of Lemma 6.3. Thus, \ depthR(I,M) = sup F − sup(F ⊗R M)= − sup(F ⊗R M) . ∼ i Note that F ⊗R M = HomR (P,M), and hence ExtR (R/I,M) = H−i(F ). The assertion on the associated primes is also contained in 6.3. 

Let x1,...,xn be a set of generators for the ideal I. For U = {s1,...,si}, we set

RU = Rxs1 ···xsi . The complex ∂ C : · · · −→ RU −→ RV −→· · · U⊆{M1,...,n} V ⊆{M1,...,n} card(U)=i card(V )=i+1 j where the component of ∂ on RU −→ RV is (−1) times the canonical localization Cechˇ complex map RU −→ (RV )xsj if U = V \{sj }, and 0 otherwise, is called the on I. More often than not, it is defined as a cohomological complex (with upper indices). The notation used here is in line with that of the rest of the paper. Proof of Theorem 6.2. Let C be the complex described above. It is well known that local cohomology modules may be computed via Cechˇ complexes; for example, see [4, §3.5], or [15, §3]: i ∼ HI (M) = Hn−i(C ⊗R M) . Since C satisfies the hypotheses of Lemma 6.3, one has i depthR(I,M)= n − sup(C ⊗R M) = inf{i | HI (M) 6=0} , and also the statement regarding the associated primes.  Acknowledgement . This paper has evolved over numerous conversations with my thesis advisor Luchezar L. Avramov. I should like to thank him for raising the right questions. I also thank Hans-Bjørn Foxby for valuable suggestions on an earlier version, and the referee for drawing my attention to Theorem 3.5. Anders Frankild pointed out the inaccuracy in the published version of Lemma 6.3; I thank him for the same.

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Department of Mathematics, Purdue University, W. Lafayette, IN 47907, U.S.A. E-mail address: [email protected]