SEQUENCES FOR COMPLEXES
LARS WINTHER CHRISTENSEN
Introduction
LetR be a commutative Noetherian ring and letM = 0 be a finite (that is, finitely generated)R-module. The concept ofM-sequencesis central for the study ofR-modules by methods of homological algebra. Largely, the useful- ness of these sequences is based on the following properties:
1◦ Whenᑾis an ideal inRandM/ᑾM =0, the number inf{∈Z|ExtR(R/ᑾ, M)=0},
the so-calledᑾ-depth ofM, is the maximal length of anM-sequence in ᑾ, and any maximalM-sequence inᑾis of this finite length.
2◦ If x1, . . . , xn is anM-sequence contained in ᒍ ∈ SuppRM, then the sequence of fractionsx1/1, . . . , xn/1, in the maximal ideal ofRᒍ, is an Mᒍ-sequence.
In commutative algebra, a wave of work dealing with complexes of modules was started by A. Grothendieck, see [9]. The underlying idea is the following:
Complexes (that is, complexes of modules) are tacitly involved whenever ho- mological methods are applied, and since hyperhomological algebra, that is, homological algebra for complexes, is a very powerful tool, it is better to work consistently with complexes. Modules are also complexes, concentrated in degree zero, so results for complexes yield results for modules as special cases.
Like most concepts for modules that of M-sequences can be extended to complexes in several non-equivalent ways; this short paper explores two such possible extensions: (ordinary)sequencesandstrong sequencesfor com- plexes. Ordinary sequences have a property corresponding to 1◦, at least over local rings where they coincide with theregular sequencessuggested by H.
-B. Foxby in [8, Sec. 12]. But ordinary sequences may fail to localize properly,
Received June 22, 1998; in revised form December 7, 1998.
whereas strong sequences not only enjoy the correspondent property of 2◦, but also that of 1◦in the special case whereRis local andᑾthe maximal ideal.
As a rule, the hyperhomological approach not only reproduces known results for modules, but also strengthens some of them. In this case we show, among other things, that also for a non-finite moduleMtheᑾ-depth is an upper bound for the maximal length of anM-sequence in ᑾ, and the ᑾ-depth of such a module may be finite even ifM/ᑾM =0.
1. Conventions, Notation, and Background
Throughout this paperR is a non-trivial, commutative, Noetherian ring. We work in the derived category of the category ofR-modules; this first section fixes the notation and sums up a few basic results.
Notation1.1. As usual, the set of prime ideals containing an idealᑾinR is written V(ᑾ); whenx = x1, . . . , xn is a sequence inRwe write V(x)for the set of prime ideals containingx. The set ofzero-divisorsfor anR-module M is denoted by zRM.
The ringRis said to belocalif it has a unique maximal idealᒊ, the residue fieldR/ᒊis then denoted byk. In general, forᒍ∈SpecRthe residue field of the local ringRᒍis denoted byk(ᒍ), that is,k(ᒍ)=Rᒍ/ᒍᒍ.
Complexes1.2. AnR-complexXis a sequence ofR-modulesXandR- linear maps, so-calleddifferentials,∂X : X → X−1, ∈ Z. Composition of two consecutive differentials always yields the zero map, i.e.∂X∂+X1= 0.
IfX = 0 for = 0, we identify X with the module in degree 0, and an R-moduleM is considered as a complex 0→M →0 withM in degree 0.
Amorphismα : X → Y ofR-complexes is a sequence ofR-linear maps α:X →Ysatisfying∂Yα−α−1∂X=0 for∈Z. We say that a morphism is aquasi-isomorphismif it induces an isomorphism in homology. The symbol is used to indicate quasi-isomorphisms while ∼= indicates isomorphisms of complexes (and hence modules). For an element r ∈ R the morphism rX:X→Xis given by multiplication byr.
The numbers supremum, infimum, and amplitude: supX = sup{ ∈ Z | H(X)=0}, infX= inf{∈Z|H(X)=0}, and ampX =supX−infX, capture the homological position and size ofX. By convention, supX= −∞
and infX= ∞ifX0.
Derived functors1.3. Thederived categoryof the category ofR-modules is the category ofR-complexes localized at the class of all quasi-isomorphisms (see [9] and [13]), we denote it byD(R). The symbolis used for isomorph- isms inD(R); a morphism of complexes is a quasi-isomorphism exactly if it
represents an isomorphism in the derived category, so this is in agreement with the notation introduced above.
The full subcategoriesD+(R),D−(R),Db(R), andD0(R)consist of com- plexesXwith H(X)=0 for, respectively,0,0,|| 0, and=0.
ByDf(R)we denote the full subcategory ofD(R)consisting of complexesX with H(X)a finiteR-module for all∈Z. We also use combined notations:
D−f(R)=D−(R)∩Df(R), etc. The category ofR-modules, respectively, finite R-modules, is naturally identified withD0(R), respectively,D0f(R).
The right derived functor of the homomorphism functor forR-complexes is denoted by RHomR(−,−), and − ⊗LR − is the left derived functor of the tensor product functor for R-complexes; by [2] and [12] no bounded- ness conditions are needed on the arguments. That is, forX, Y ∈ D(R)the complexesRHomR(X, Y ) andX ⊗LR Y are uniquely determined up to iso- morphism in D(R), and they have the expected functorial properties. Note that TorR(M, N)=H(M⊗LRN)and ExtR(M, N)=H−(RHomR(M, N)) forM, N ∈D0(R)and∈Z.
Letᒍ∈SpecR; by [2, 5.2] there are isomorphisms:(X⊗LRY )ᒍXᒍ⊗LRᒍYᒍ
andRHomR(Z, Y )ᒍRHomRᒍ(Zᒍ, Yᒍ)inD(Rᒍ). The first one always holds, and the second holds whenY ∈D−RandZ ∈D+f(R).
The next results are standard, cf. [6, (2.1)]. LetX∈D+(R)andY ∈D−(R), thenRHomR(X, Y )∈D−(R)and there is an inequality:
(1.3.1) supRHomR(X, Y )≤supY−infX.
Settingi=infXands=supY we have Hs−i(RHomR(X, Y ))= HomR(Hi(X),Hs(Y )); in particular,
(1.3.2)
supRHomR(X, Y )=supY −infX ⇐⇒ HomR(Hi(X),Hs(Y ))=0. LetX, Y ∈D+(R), thenX⊗LRY ∈D+(R)and there is an inequality (1.3.3) inf(X⊗LRY )≥infX+infY;
furthermore, withi=infXandj =infY we have (1.3.4) Hi+j(X⊗LRY )=Hi(X)⊗RHj(Y ).
Depth over local rings1.4. LetRbe local; in [7, Sec. 3] thedepthand (Krull)dimensionof anR-complexXare defined as follows:
depthRX= −supRHomR(k, X), for X ∈D−(R); and dimRX=sup{dimR/ᒍ−infXᒍ|ᒍ∈SpecR}.
Note that for modules these notions agree with the usual ones.
It follows immediately by (1.3.1) that−supX≤depthRXforX∈D−(R), and ifs=supX >−∞the next biconditional holds, cf. (1.3.2).
(1.4.1) depthRX= −supX ⇐⇒ ᒊ∈AssRHs(X).
ForX∈D−(R)andM ∈D0f(R)the next equality holds, cf. [7, 3.4].
(1.4.2) −supRHomR(M, X)=inf{depthRᒍXᒍ|ᒍ∈SuppRM}.
Let X ∈ D−f(R) and ᒍ ∈ SpecR; a complex version of [3, (3.1)], cf. [5, (13.13)], accounts for the inequality
(1.4.3) depthRX≤depthRᒍXᒍ+dimR/ᒍ.
Finally, letX 0 belong toD−f(R)and sets = supX; applying (1.4.3) to ᒍ∈AssRHs(X)with dimR/ᒍ=dimRHs(X)and using (1.4.1) we obtain the next inequalities.
(1.4.4) depthRX+supX≤dimRHs(X)≤dimR.
2. Ann, Supp, and Ass for Complexes
As for modules, regular elements for complexes are linked to concepts ofzero- divisorsandassociated prime ideals. These are introduced below within the relevant setting of support and annihilators.
Weak notions2.1. Weak notions ofsupport and annihilatorsforX ∈ D(R)are defined by uniting/intersecting the corresponding sets for the homo- logy modules H(X), cf. [7, Sec. 2] and [1, Sec. 2]:
SuppRX =
∈Z
SuppRH(X)= {ᒍ∈SpecR|Xᒍ0}; and AnnRX =
∈Z
AnnRH(X)= {r ∈R|H(rX)=0}.
These are complemented by the next definitions. ForX0 inD−(R)we set assRX=AssRHsupX(X) and zRX=zRHsupX(X), cf. [8, Sec. 12], and forX0 we set assRX = ∅and zRX= ∅.
The small support 2.2. The small, or homological, support for X ∈ D+(R)was introduced in [7, Sec. 2]:
suppRX= {ᒍ∈SpecR|Xᒍ⊗LRᒍk(ᒍ)0}.
Its principal properties developed ibid. are as follows:
LetX∈D+(R). Then
(2.2.1) X0 ⇐⇒ suppRX= ∅;
there is an inclusion
(2.2.2) suppRX⊆SuppRX,
and equality holds whenX ∈ D+f(R). For X, Y ∈ D+(R)the next equality holds.
(2.2.3) suppR(X⊗LRY )=suppRX∩suppRY.
IfRis local, the next biconditional holds forX∈Db(R). (2.2.4) ᒊ∈suppRX ⇐⇒ depthRX <∞.
Definitions2.3. LetX∈D−(R); we say thatᒍ∈SpecRis anassociated prime idealforXif and only if depthRᒍXᒍ= −supXᒍ<∞, that is,
AssRX= {ᒍ∈SuppRX|depthRᒍXᒍ+supXᒍ=0}
= {ᒍ∈SuppRX|ᒍᒍ∈assRᒍXᒍ},
cf. (1.4.1). The union of the associated prime ideals forms the set of zero- divisorsforX:
ZRX=
ᒍ∈AssRX
ᒍ.
Observations2.4. LetX ∈ D−(R),ᒍ ∈ SuppRX, and sets = supXᒍ
(∈Z); then
ᒍ∈AssRX ⇐⇒ ᒍᒍ∈assRᒍXᒍ ⇐⇒ ᒍ∈AssRHs(X).
That is, ᒍ ∈ AssRX if and only if there exists an m ∈ Z such that ᒍ ∈ AssRHm(X)andᒍ∈SuppRH(X)for > m. In particular there is an inclu- sion
(2.4.1) assRX⊆AssRX;
and since zRX= ∪ᒍ∈assRXᒍ, also the next inclusion holds.
(2.4.2) zRX⊆ZRX.
We also note that AssRXis a finite set forXinDbf(R).
Modules 2.5. For M ∈ D0(R) the weak notions in 2.1 agree with the classical notions for modules; furthermore, assRM = AssRM and zRM = ZRM, but suppRM and SuppRMmay differ ifM is not finite.
Proposition2.6.LetX ∈D−(R); every minimal prime ideal inSuppRX belongs toAssRX, that is,
MinRX⊆AssRX;
and forX ∈Db(R)also the next inclusion holds.
AssRX⊆suppRX.
Proof. Let X ∈ D−(R) and assume that ᒍ is minimal in SuppRX. As SuppRᒍXᒍ= {ᒍᒍ}it follows thatᒍᒍ∈assRᒍXᒍand henceᒍ∈AssRX.
LetX∈Db(R); the first biconditional in the next chain is (2.2.4).
ᒍ∈AssRX ⇒ depthRᒍXᒍ<∞ ⇐⇒ ᒍᒍ∈suppRᒍXᒍ
⇐⇒ Xᒍ⊗LRᒍ k(ᒍ)0 ⇐⇒ ᒍ∈suppRX.
Lemma2.7.Let Sbe a multiplicative system inR; the following hold for ᒍ∈SpecRwithᒍ∩S= ∅:
(a) ᒍ∈suppRX⇐⇒S−1ᒍ∈suppS−1RS−1X, ifX∈D+(R); and (b) ᒍ∈AssRX⇐⇒S−1ᒍ∈AssS−1RS−1X, ifX∈D−(R).
Proof. S−1ᒍis a prime ideal inS−1Rand
k(S−1ᒍ)=(S−1R/S−1ᒍ)S−1ᒍ∼=k(ᒍ), so (S−1X)S−1ᒍ⊗L(S−1R)S−1ᒍk(S−1ᒍ)Xᒍ⊗LRᒍk(ᒍ);
and
RHom(S−1R)S−1ᒍ(k(S−1ᒍ), (S−1X)S−1ᒍ)RHomRᒍ(k(ᒍ), Xᒍ).
(a) follows directly from the first isomorphism, and (b) follows from the second by the definition of depth.
3. Three Types of Sequences
We are now ready to define sequences – and strong and weak ones – for com- plexesY ∈D−(R). The main results of this section are that strongY-sequences localize properly, and that forM ∈ D0(R)the notions ofM-sequences and strongM-sequences both agree with the classical notion for modules.
Koszul complexes 3.1. Forx ∈ R the complex K(x) = 0 → R −→x R→0, concentrated in degrees 1 and 0, is called theKoszul complexofx. Let x=x1, . . . , xnbe a sequence inR, the Koszul complex K(x)=K(x1, . . . , xn) ofxis the tensor product K(x1)⊗R· · · ⊗RK(xn). The Koszul complex of the empty sequence isR.
ForY ∈D(R)we set K(x;Y )= Y ⊗RK(x), and form∈ {1, . . . , n}we write K(xm;Y )for the complex K(x1, . . . , xm;Y ). We also set K(x0;Y )=Y, corresponding to the empty sequence.
Observations3.2. In the followingx=x1, . . . , xnis a sequence inRand Y ∈D(R).
Form∈ {0, . . . , n−1}we have
(3.2.1) K(x;Y )=K(xm+1, . . . , xn;K(xm, Y )),
by associativity of the tensor product. Letᒍ∈SpecRand denote by
x1/1, . . . , xn/1 the sequence of fractions inRᒍ corresponding tox. There is an isomorphism:
(3.2.2) K(x1, . . . , xn;Y )ᒍ∼=K(x1/1, . . . , xn/n;Yᒍ).
For eachjthe Koszul complex K(xj)is a complex of finite free, in particular flat, modules, and hence so is K(x). Thus, we can identify K(x)with K(x1)⊗LR
· · · ⊗LRK(xn)and K(x;Y )withY ⊗LRK(x). It follows by (1.3.3) and (1.3.4) that
(3.2.3) inf K(x)≥0 and H0(K(x))=R/(x).
It is well-known (see [1, Sec. 2] or [11, 16.4]) that (3.2.4) (x1, . . . , xn)⊆AnnRK(x;Y ).
It is easy to see that SuppRK(xj) = V(xj), and it follows by (2.2.2) and (2.2.3) that SuppRK(x) = suppRK(x) = V(x). If Y ∈ D+(R)it follows, also by (2.2.3), that
(3.2.5) suppRK(x;Y )=suppRY ∩V(x).
Finally, it follows by the definition of tensor product complexes that (3.2.6)ifY belongsD−(R), respectively,D−f(R)then also K(x;Y )∈D−(R),
respectively, K(x;Y )∈D−f(R); and
(3.2.7)ifY belongsDb(R), respectively,Dbf(R)then also K(x;Y )∈Db(R), respectively, K(x;Y )∈Dbf(R).
In view of (3.2.6) the next definitions make sense.
Definitions3.3. LetY ∈D−(R). An elementx ∈Ris said to beregular forY if and only ifx∈zRYandstrongly regularforY if and only ifx∈ZRY.
Letx=x1, . . . , xnbe a sequence inR. We say that
• x is aweakY-sequence if and only ifxj is regular for K(xj−1;Y )for eachj ∈ {1, . . . , n};
• xis aY-sequenceif and only ifxis a weakY-sequence, and K(x;Y )0 orY 0; and
• xis astrongY-sequenceif and only ifxjis strongly regular for K(xj−1;Y ) for eachj ∈ {1, . . . , n}, and K(x;Y )0 orY 0.
Remarks3.4. ForM ∈ D0(R)regular and strongly regular elements are the same, cf. 2.5, and the definition agrees with the usual definition ofM- regular elements, cf. [11, Sec. 16]. In 3.8 we prove that also the definition of M-sequences agrees with the classical one.
LetY ∈D−(R). By (2.4.2) a strongly regular element forY is also regular forY; hence any strongY-sequence is aY-sequence and, thereby, a weak one.
The empty sequence is a strongY-sequence for any complexY ∈D−(R). A unitu ∈Ris a strongly regular element for any complexY ∈D−(R)and constitutes a weakY-sequence,ucan, however, not be part of aY-sequence if Y 0. On the other hand, ifY 0 then any sequence is a strongY-sequence.
Later we supply an example – 3.13 – to show that aY-sequence need not be a strong one.
Observation3.5. Let Y ∈ D−(R), letx = x1, . . . , xn be a sequence in R, and letm ∈ {1, . . . , n−1}. It follows by (3.2.1) thatxis aY-sequence, respectively, a weak or a strong one, if and only ifx1, . . . , xmis aY-sequence, respectively, a weak or strong one, andxm+1, . . . , xnis a K(xm;Y )-sequence, respectively, a weak or a strong one.
Lemma3.6.The following hold forx ∈RandY 0inD−(R): (a) sup K(x;Y )≤supY +1;
(b) sup K(x;Y )=supY+1if and only ifx∈zRY; and
(c) sup K(x;Y )≥supY ifxHsupY(Y )=HsupY(Y ).
Proof. It is easy to see that K(x;Y )is the mapping cone for the morphism xY, multiplication by x on Y. Thus, K(x;Y ) fits in the exact sequence of complexes
0→Y →K(x;Y )→Y[1]→0,
where Y[1] is a shift of Y: Y[1] = Y−1 and ∂Y[1] = −∂−Y 1. Now, set s = supY and examine the corresponding long exact sequence of homology modules:
0→Hs+1(K(x;Y ))→Hs(Y )−−−−→xHs (Y ) Hs(Y )→Hs(K(x;Y ))→ · · ·. Parts (a) and (b) have the following immediate consequence:
Corollary 3.7. Let Y ∈ D−(R); a sequence x = x1, . . . , xn in R is a weakY-sequence if and only if sup K(xj;Y ) ≤ sup K(xj−1;Y ) for each j ∈ {1, . . . , n}.
Sequences for modules3.8.LetM be anR-module; the following hold for a sequencex=x1, . . . , xninR:
(a) H0(K(xj;M))=M/(x1, . . . , xj)Mforj ∈ {1, . . . , n}. (b) The next three conditions are equivalent.
(i) xis a weakM-sequence.
(ii) K(xj;M)M/(x1, . . . , xj)M for eachj ∈ {1, . . . , n}. (iii) xj ∈zRM/(x1, . . . , xj−1)M for eachj ∈ {1, . . . , n}1. (c) The next three conditions are equivalent.
(i) xis a weakM-sequence, andM/(x1, . . . , xn)M =0orM =0.
(ii) xis anM-sequence.
(iii) xis a strongM-sequence.
Proof. All three assertions are trivial ifM = 0, so we assume thatM is non-zero and letx=x1, . . . , xnbe a sequence inR.
(a): Considering, as always,Mas a complex concentrated in degree 0, we see that
(∗) inf K(xj;M)≥0 for j ∈ {1, . . . , n},
cf. (3.2.3) and (1.3.3), and H0(K(xj;M))=M⊗RR/(x1, . . . , xj), cf. (1.3.4).
1Forj=1 this meansx1∈zRM.
(b): For eachj ∈ {1, . . . , n}we have inf K(xj;M)≥0, cf. (∗), so by 3.7 it follows thatx is a weakY-sequence if and only if K(xj;M)∈ D0(R)for eachj, that is (by (a)), if and only if K(xj;M)M/(x1, . . . , xj)M for each j. This proves the equivalence of (i) and (ii); that of (ii) and (iii) follows from (a), (3.2.1), and 3.6 by induction onn.
(c): First note that (i) ⇒ (ii) by (a); it is then sufficient to prove that (ii) implies (iii): Supposex is an M-sequence; forj ∈ {1, . . . , n}we have xj ∈zRK(xj−1;M), and K(xj−1;M) ∈ D0(R)by (b), so zRK(xj−1;M) = ZRK(xj−1;M), cf. 2.5, whencexis a strongM-sequence.
Remark 3.9. Let M be a non-zero R-module and let x = x1, . . . , xn
be a sequence in R. Classically, cf. [11, Sec. 16], x is said to be an M- sequence if and only if (1)xj ∈ zRM/(x1, . . . , xj−1)M forj ∈ {1, . . . , n}, and (2)M/(x1, . . . , xn)M =0. A sequence satisfying only the first condition is called a weak M-sequence, cf. [4, 1.1.1]. It follows by (b) and (c) in 3.8 that the notions of (weak)M-sequences defined in 3.3 agree with the classical ones.
Observation3.10. LetY 0 belong toD−(R); it follows by 3.6 that a sufficient condition for x ∈ R to be aY-sequence is that x is a HsupY(Y )- sequence. This condition is, of course, not necessary, see 5.3 for an example.
Theorem3.11.LetY ∈ Db(R)andᒍ ∈ suppRY; if x = x1, . . . , xn is a strongY-sequence inᒍ, thenx1/1, . . . , xn/1in the maximal ideal of Rᒍis a strongYᒍ-sequence.
Proof. Let x/1 = x1/1, . . . , xn/1 denote the sequence of fractions in Rᒍ corresponding to x. Since ᒍ ∈ suppRK(x;Y ) by (3.2.5), it follows by 2.7 (a) and (3.2.2) thatᒍᒍ∈suppRᒍK(x/1;Yᒍ); in particular, K(x/1;Yᒍ)0.
We are now required to prove that xj/1 ∈ ZRᒍK(x1/1, . . . , xj−1/1;Yᒍ)for j ∈ {1, . . . , n}. This follows by the lemma below asᒍ ∈SuppRK(xj−1;Y ), xj ∈ZRK(xj−1;Y ), and ZRᒍK(xj−1;Y )ᒍ=ZRᒍK(x1/1, . . . , xj−1/1;Yᒍ).
Lemma 3.12. Let Y belong to D−(R)and ᒍ ∈ SuppRY; if x ∈ ᒍ and x∈ZRY thenx/1∈ZRᒍYᒍ.
Proof. We assume thatx/1∈ZRᒍYᒍand want to prove thatxbelongs to ZRY. By assumptionx/1 belongs to a prime ideal in AssRᒍYᒍ, that is,x/1∈ᒎᒍ
for someᒎ∈SpecRcontained inᒍ. Thenx∈ᒎ, andᒎ∈AssRY by 2.7 (b), sox∈ZRY as wanted.
As the next example demonstrates, aY-sequence does not necessarily loc- alize properly, not even ifRis local andY ∈Dbf(R).
Example3.13. LetRbe a local ring, assume that there existᒍ,ᒎ∈SpecR such thatᒍ ⊆ ᒎ andᒎ ⊆ ᒍ, and consider the complexY = 0 → R/ᒎ −→0
R/ᒍ → 0. Let x be an element in ᒍ not in ᒎ; it follows by 3.6 thatx is a Y-sequence, but the localization ofY atᒍis the fieldk(ᒍ), andx/1 ∈ Rᒍis certainly not ak(ᒍ)-sequence.
Note that ifᒍ∩ᒎ=0, then there is no non-empty strongY-sequence inᒍ. 4. Length of Sequences and Depth of Complexes
In this section we prove that any (strong) sequence can be extended to a max- imal (strong) sequence, and we discuss various upper bounds for the length of such sequences.
Maximal sequences 4.1. LetY 0 belong to D−(R)and letᑾ be an ideal inR. A sequencex = x1, . . . , xn inᑾis said to be amaximal(strong) Y-sequenceinᑾif and only if it is a (strong)Y-sequence and not the first part of a longer (strong)Y-sequence inᑾ.
Lemma4.2.LetY 0belong toD−(R); ifx=x1, . . . , xnis aY-sequence thenxn ∈(x1, . . . , xn−1).
Proof. By (3.2.4) we have (x1, . . . , xn−1) ⊆ AnnRK(xn−1;Y ), hence (x1, . . . , xn−1) ⊆ zRK(xn−1;Y ) as K(xn−1;Y ) 0, and it follows that xn∈(x1, . . . , xn−1)as desired.
Corollary4.3.LetY 0belong toD−(R)and letᑾ be an ideal inR. AnyY-sequence, respectively, strong Y-sequence inᑾcan be extended to a maximalY-sequence, respectively, a maximal strongY-sequence inᑾ.
Proof. The assertions follow immediately by 4.2 asRis Noetherian.
Depth4.4. Letᑾbe an ideal inR and leta = a1, . . . , at be a finite set of generators forᑾ. By definition, cf. [10, Sec. 2], theᑾ-depthofY ∈ D(R)is the number
depthR(ᑾ, Y )=t−sup K(a;Y );
it is, of course, independent of the choice of generating seta.
We note that depthR(ᑾ, Y ) < ∞ if and only if K(a;Y ) 0 for some, equivalently any, finite set of generators forᑾ. Thus, by (2.2.1) and (3.2.5) we have
(4.4.1) depthR(ᑾ, Y ) <∞ ⇐⇒ suppRY ∩V(ᑾ)= ∅, forY ∈Db(R).
Proposition4.5.LetY ∈D−(R), letᑾbe a proper ideal inR, and letM belong toD0f(R)withSuppRM =V(ᑾ). The following equalities hold:
depthR(ᑾ, Y )= −supRHomR(R/ᑾ, Y )=inf{depthRᒍYᒍ|ᒍ∈V(ᑾ)}
= −supRHomR(M, Y ).
Proof. The first equality is [10, 6.1], the second and third both follow by (1.4.2).
Remark4.6. It follows from the first equality in 4.5 that theᑾ-depth for complexes extends the usual concept ofᑾ-depth for modules, cf. [11, 16.7];
furthermore, it generalizes the concept of depth over local rings, that is, depthRY =depthR(ᒊ, Y )forY ∈D−(R), whenRis local with maximal ideal ᒊ. By the second equality in 4.5 the next inequality holds for allY ∈D−(R) and allᒍ∈SpecR.
depthR(ᒍ, Y )≤depthRᒍYᒍ.
Part (a) of the next theorem is often referred to as the ‘depth sensitivity of the Koszul complex’.
Theorem 4.7.Let Y ∈ D(R)and let ᑾ be an ideal inR. The following hold:
(a) For any sequencex=x1, . . . , xninᑾthere is an equality:
depthR(ᑾ,K(x;Y ))=depthR(ᑾ, Y )−n.
(b) For any idealᑿ⊆ᑾthere is an inequality:
depthR(ᑿ, Y )≤depthR(ᑾ, Y ).
Proof. Let a = a1, . . . , at be a set of generators for ᑾ and let x = x1, . . . , xn be a sequence inᑾ. Alsox,a = x1, . . . , xn, a1, . . . , at is a gen- erating set forᑾ, and by (3.2.1) we have K(x,a;Y )=K(a;K(x;Y )). Hence,
depthR(ᑾ, Y )=n+t−sup K(x,a;Y )
=n+t−sup K(a;K(x;Y ))
=n+depthR(ᑾ,K(x;Y ));
and this proves (a).
To prove (b), letb = b1, . . . , bu be a generating set for ᑿ, then b,a = b1, . . . , bu, a1, . . . , at is a generating set forᑾ. If sup K(b;Y ) = ∞the in- equality is trivial, so we assume that K(b;Y ) ∈ D−(R). As above we have K(b,a;Y ) = K(a;K(b;Y )), so it follows by 3.6 (a) that sup K(b,a;Y ) ≤ sup K(b;Y )+t, whence
depthR(ᑾ, Y )=u+t −sup K(b,a;Y )≥u+t−(sup K(b;Y )+t)
=depthR(ᑿ, Y ), as desired.
Corollary 4.8. Let Y ∈ D−(R) and let ᑾ be a proper ideal in R. If depthR(ᑾ, Y ) <∞then the following hold for a sequencex =x1, . . . , xn in ᑾ.
(a) Ifxis a weakY-sequence thenxis aY-sequence.
(b) Ifxis aY-sequence thenxis maximal inᑾif and only ifᑾ⊆zRK(x;Y ). (c) Ifx is a strongY-sequence thenxis maximal in ᑾif and only ifᑾ ⊆
ZRK(x;Y ).
Proof. Denote byᑿ the ideal generated by x. It follows by 4.7 (b) that depthR(ᑿ, Y ) < ∞, in particular, K(x;Y ) 0, cf. 4.4. The three assertions are now immediate by the definitions in 3.3.
Proposition4.9.LetY 0belong toD−(R)and letx=x1, . . . , xnbe a weakY-sequence. The next inequality holds for any idealᑾcontainingx.
n≤depthR(ᑾ, Y )+supY.
Proof. Letᑿbe the ideal generated by the sequencex=x1, . . . , xninᑾ. By 4.4, 4.7 (b), and 3.7 we have
n=depthR(ᑿ, Y )+sup K(x;Y )≤depthR(ᑾ, Y )+supY.
Corollary4.10.LetY 0belong toDb(R)and letx=x1, . . . , xnbe a strongY-sequence. The following inequality holds:
(a) n≤inf{depthRᒍYᒍ+supYᒍ|ᒍ∈suppRY ∩V(x)};
and ifY ∈Dbf(R), also the next inequality holds.
(b) n≤inf{dimRᒍ|ᒍ∈SuppRY ∩V(x)}.
Proof. Let Y ∈ Db(R)and assume that x = x1, . . . , xn is a strong Y- sequence inᒍ∈suppRY. By 3.11 the sequencex1/1, . . . , xn/1 in the maximal ideal ofRᒍis a strongYᒍ-sequence, so by 4.9 we haven≤depthRᒍYᒍ+supYᒍ, and this proves (a). IfY ∈ Dbf(R)then suppRY = SuppRY, cf. (2.2.2), and Yᒍ ∈ Dbf(Rᒍ), so (b) follows from (a) as depthRᒍYᒍ+supYᒍ ≤ dimRᒍ by (1.4.4).
Theorem 4.11. Let Y ∈ D−f(R) and let ᑾ be a proper ideal in R. If depthR(ᑾ, Y ) < ∞ then the following conditions are equivalent for a Y- sequencex=x1, . . . , xninᑾ:
(i) xis a maximalY-sequence inᑾ.
(ii) ᑾ⊆zRK(x;Y ).
(iii) depthR(ᑾ,K(x;Y ))+sup K(x;Y )=0.
(iv) depthR(ᑾ, Y )+sup K(x;Y )=n.
Proof. We assume thatY ∈D−f(R)with depthR(ᑾ, Y ) <∞; the equival- ence of (i) and (ii) is 4.8 (b). From 4.7 (a) it follows that(iii) ⇔ (iv); this leaves us with one equivalence to prove:
SetK =K(x;Y )ands=supK (∈Z); by 4.5 and (1.3.1) we have
−depthR(ᑾ, K)=supRHomR(R/ᑾ, K)≤s,
and equality holds if and only if HomR(R/ᑾ,Hs(K))=0, cf. (1.3.2). Since Hs(K)is a finite module, cf. (3.2.6), it is well-known that HomR(R/ᑾ,Hs(K))
=0 if and only ifᑾ⊆zRK, and this proves the equivalence of (ii) and (iii).
Remarks 4.12. LetY, ᑾ, andx be as in 4.11. Since K(x;Y ) ∈ D−f(R), cf. (3.2.6), it follows that
ᑾ⊆zRK(x;Y ) ⇐⇒ ᑾ⊆ᒍ for some ᒍ∈assRK(x;Y ).
This should be compared to (ii) and (iii) in 4.15.
For a finiteR-moduleMand an idealᑾinRit follows by (4.4.1) and (2.2.2) that depthR(ᑾ, M) <∞ ⇐⇒ M/ᑾM =0.
Spelling out 4.11 for modules – as done in 4.14 – we recover the property 1◦ advertised in the introduction. Thus, in a sense, 4.11 describes the correspond- ing property for complexesY ∈ D−f(R); but unlessR is local (see 5.4) the length of a maximal sequence need not be a well-determined integer:
LetYandᑾbe as in 4.11. If depthR(ᑾ, Y )+supY =0 thenᑾ⊆zRY, so the empty sequence is the onlyY-sequence inᑾ. If depthR(ᑾ, Y )+supY =1 then all maximalY-sequences inᑾare of length 1, but if depthR(ᑾ, Y )+supY >1 there can be maximalY-sequences inᑾof different length. This is illustrated by the example below.
Example 4.13. Let k be a field, set R = k[U, V], and consider the R- complexY =0→R/(U−1)→0→k →0 concentrated in degrees 2, 1, and 0. Letᑾbe the maximal idealᑾ=(U, V), then depthR(ᑾ, Y )=2−2=0 and it is straightforward to check that U as well as V , U is a maximal Y- sequence inᑾ.
Corollary4.14.LetM be a finiteR-module and letᑾbe a proper ideal inR. IfdepthR(ᑾ, M) < ∞ then the next four conditions are equivalent for anM-sequencex =x1, . . . , xninᑾ.
(i) xis a maximalM-sequence inᑾ. (ii) ᑾ⊆zRM/(x1, . . . , xn)M.
(iii) depthR(ᑾ, M/(x1, . . . , xn)M)=0.
(iv) depthR(ᑾ, M)=n.
In particular, the maximal length of anM-sequence inᑾis a well-determined integer:depthR(ᑾ, M)= inf{ ∈Z |ExtR(R/ᑾ, M) = 0}, and all maximal M-sequences inᑾhave this length.
Proof. By 3.8 we have K(x;M) M/(x1, . . . , xn)M =0, in particular, sup K(x;M)= 0. The equivalence of the four conditions now follows from 4.11, and the last assertions are immediate, cf. 4.5.
Other well-known characterizations of maximal sequences for finite mod- ules are recovered by readingM/(x1, . . . , xn)M for K(x;M)in the next the- orem.
Theorem 4.15. Let Y ∈ Dbf(R) and let ᑾ be a proper ideal in R. If depthR(ᑾ, Y ) < ∞then the next four conditions are equivalent for a strong Y-sequencex=x1, . . . , xninᑾ.
(i) xis a maximal strongY-sequence inᑾ. (ii) ᑾ⊆ZRK(x;Y ).
(iii) ᑾ⊆ᒍfor someᒍ∈AssRK(x;Y ).
(iv) There is a prime idealᒍ ∈ SuppRY containingᑾ such that the strong Yᒍ-sequencex1/1, . . . , xn/1inRᒍis a maximalYᒍ-sequence.
Proof. The equivalence(i) ⇔ (ii)is immediate as depthR(ᑾ, Y ) < ∞, cf. 4.8 (c).
(ii)⇔(iii): Clearly, (iii) implies (ii). On the other hand, K(x;Y )∈Dbf(R) by (3.2.7), so ZRK(x;Y ) = ∪ᒍ∈AssRK(x;Y )ᒍ is a finite union, cf. 2.4. Thus, ifᑾ ⊆ ZRK(x;Y )thenᑾmust be contained in one of the prime idealsᒍ ∈ AssRK(x;Y ).
(iii)⇔(iv): Letᒍbe a prime ideal in SuppRYcontainingᑾ, then depthRᒍYᒍ
< ∞, cf. (2.2.2) and (2.2.4), and by 3.11 the sequence of fractionsx/1 = x1/1, . . . , xn/1 inRᒍis a strongYᒍ-sequence. By (3.2.2) there is an equality:
depthRᒍK(x;Y )ᒍ+sup K(x;Y )ᒍ=depthRᒍK(x/1;Yᒍ)+sup K(x/1;Yᒍ).
By 4.11 and the definition of associated prime ideals it now follows thatx/1 is a maximalYᒍ-sequence if and only ifᒍ∈AssRK(x;Y ).
5. Local Rings
In this sectionR is local with maximal idealᒊand residue fieldk = R/ᒊ. We focus on (strong) sequences for complexes inD−f(R)and strengthen some of the results from the previous section. The results established here are essen- tially those lined out by H. -B. Foxby in [8, Sec. 12], exceptions are 5.7 and 5.9.
Proposition5.1.Let Y 0 belong toD−f(R); the following hold for a sequencex=x1, . . . , xninᒊ:
(a) There are inequalities
sup K(x;Y )≥ · · · ≥sup K(xj;Y )≥sup K(xj−1;Y )≥ · · · ≥supY; in particular,K(x;Y )0.
(b) The next three conditions are equivalent.
(i) xis a weakY-sequence.
(ii) xis aY-sequence.
(iii) sup K(x;Y )=supY.
(c) Ifxis aY-sequence then so is any permutation ofx.
Proof. (a): The inequalities hold by Nakayama’s lemma and 3.6(c); in particular we have sup K(x;Y )≥supY >−∞, so K(x;Y )0.
(b): It follows by 3.7 thatx is a weakY-sequence if and only if equality holds in each of the inequalities in (a). This proves the equivalence of (i) and (iii); also(i)⇔(ii)is immediate by (a).
(c): By commutativity of the tensor product the number sup K(x;Y ) is unaffected by permutations ofx, so the last assertion follows by (b).
The next corollary is an immediate consequence of 5.1(a). The example below shows that the equality sup K(x;Y ) = supY need not hold, not even for strongY-sequence, ifY does not have finite homology modules.
Corollary5.2.Letᑾbe a proper ideal inR. IfY ∈D−f(R)then depthR(ᑾ, Y ) <∞ ⇐⇒ Y 0.
Example5.3. Let R be a local integral domain, not a field, and letB = R(0) = Rbe the field of fractions. Consider the complex Y = 0 → B −→0 R→0. For anyᒍ∈SpecRwe have supYᒍ=supY, so AssRY =assRY = AssRB= {0}. Letx=0 be an element in the maximal ideal ofR, it follows thatx ∈ZRY and K(x;Y ) R/(x) = 0, sox is a strongY-sequence, but sup K(x;Y ) <supY.
Theorem5.4.LetY 0belong toD−f(R)and letᑾbe a proper ideal in R. The next four conditions are equivalent for aY-sequencex = x1, . . . , xn
inᑾ.
(i) xis a maximal sequence inᑾ. (ii) ᑾ⊆zRK(x;Y ).
(iii) depthR(ᑾ,K(x;Y ))+supY =0.
(iv) depthR(ᑾ, Y )+supY =n.
In particular, the maximal length of aY-sequence inᑾis a well-determined integer:depthR(ᑾ, Y )+supY, and all maximalY-sequences inᑾ have this length.
Proof. By 5.2 and 5.1 we have depthR(ᑾ, Y ) < ∞ and sup K(x;Y ) = supY, so the equivalence is a special case of 4.11, and the last assertions follow.
Corollary5.5.LetY 0belong toD−f(R); the integer depthRY +supY
is the maximal length of aY-sequence, and any maximalY-sequence is of this length. Furthermore, the following inequalities hold:
depthRY +supY ≤dimRHsupY(Y )≤dimR.
Proof. A Y-sequence must be contained in ᒊ, and the first part is 5.4 applied toᑾ=ᒊ. The inequalities are (1.4.4).
Corollary5.6.LetY 0belong toD−f(R)andM ∈D0f(R). The maximal length of aY-sequence inAnnRM is a well-determined integern:
n= −supRHomR(M, Y )+supY
=inf{depthRᒍYᒍ|ᒍ∈V(AnnRM)} +supY; and any maximalY-sequence inAnnRM is of this length.
Proof. It follows by 5.4 that aY-sequence x = x1, . . . , xn in AnnRM is maximal if and only ifn = depthR(AnnRM, Y )+supY. AsM is finite SuppRM =V(AnnRM), and the desired equalities follow by 4.5.
It follows from the last remark in 3.13 that 5.6 has no counterpart for strong sequences, but 5.5 does have one:
Corollary5.7 (to 4.15). LetY 0belong toDbf(R). A maximal strong Y-sequence is a maximalY-sequence; in particular, the maximal length of a strongY-sequence is a well-determined integern:
n=depthRY +supY ≤dimRHsupY(Y )≤dimR;
and any maximal strongY-sequence is of this length.
Proof. Letx=x1, . . . , xnbe a maximal strongY-sequence, that is, max- imal inᒊ. Since depthRY <∞by 5.2 it follows by 4.15 thatxis a maximal Y-sequence, and the desired equality and inequalities follow from 5.5.
The number depthRY+supY provides an upper bound for the length of a Y-sequence, even ifY does not have finite homology modules, cf. 4.9. In view of 5.5 it is natural to ask if also dimRis a bound. If dimR=0 it obviously is, cf. (1.4.1), and so it is if dimR= 1 and depthRHsupY(Y ) <∞(this follows by [10, 2.3]); but the next example shows that the answer is negative. For bounded complexes, however, a bound involving dimRis available, see 5.9.
Example5.8. Letkbe a field and consider the local ringR=k[[U,V]]/(UV) with dimR=1. The residue classesuandvof, respectively,UandVgenerate prime ideals inR; we setY =0→R(v)−→0 R/(u)→0. Multiplication byu onR(v)is an isomorphism,vis aR/(u)-sequence, and it follows thatu, vis a Y-sequence.
Corollary5.9 (to 4.9). LetY ∈Db(R)and letx=x1, . . . , xnbe a weak Y-sequence inᒊ. Ifᒊ∈suppRY thenxis a sequence, and
n≤depthRY +supY ≤dimRY +supY ≤dimR+ampY.
Proof. It follows by (2.2.4) that depthRY <∞, soxis aY-sequence by 4.8 (a). The first inequality is a special case of 4.9. The inequality depthRY ≤ dimRY holds by [7, 3.9]; this gives the second inequality, and the third one follows as dimRY ≤dimR−infY by the definition of dimension.
We close with an example, illustrating an application of sequences for com- plexes.
Example5.10 (Parameter Sequences). In the following we assume thatR admits a dualizing complexD, cf. [9], and letx=x1, . . . , xnbe a sequence in R. ForY 0 inDbf(R)it follows by (3.2.7), 3.6, and well-known properties of dualizing complexes that
dimRK(x;Y )=dimRY−n ⇐⇒ xis aRHomR(Y, D)-sequence;
and by [7, 3.12] there is an equality:
dimRK(x;Y )=sup{dimR(Y⊗LRH(K(x)))−|∈Z}.
LetM be a finiteR-module; we say thatxis anM-parameter sequenceif and only if dimRM/(x1, . . . , xn)M = dimRM−n, that is, if and only ifx is part of a system of parameters forM. It follows by the definition of Krull dimension, Nakayama’s lemma, and (3.2.3) that
dimRK(x;M)=sup{dimR(M⊗LRH(K(x)))−|∈Z}
=sup{dimR(M⊗RH(K(x)))−|∈Z}
=dimRM/(x1, . . . , xn)M.
Thus,x is anM-parameter sequence if and only ifx is aRHomR(M, D)- sequence. In particular, anyM-sequence is aRHomR(M, D)-sequence. Only ifMis Cohen-Macaulay willRHomR(M, D)have homology concentrated in one degree, that is, be equivalent to a module up to a shift.
Acknowledgments.The work presented here was sparked off by a ques- tion from my supervisor prof. Hans-Bjørn Foxby. I thank him for this, and for the numerous discussions over which this work evolved.
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MATEMATISK AFDELING UNIVERSITETSPARKEN 5 DK-2100 COPENHAGEN Ø DENMARK
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