Dimension of finite free complexes over commutative Noetherian rings
aa r X i v : . [ m a t h . A C ] S e p DIMENSION OF FINITE FREE COMPLEXES OVERCOMMUTATIVE NOETHERIAN RINGS
LARS WINTHER CHRISTENSEN AND SRIKANTH B. IYENGAR
To Roger and Sylvia Wiegand on the occasion of their aggregate th birthday. Abstract.
Foxby defined the (Krull) dimension of a complex of modules overa commutative Noetherian ring in terms of the dimension of its homologymodules. In this note it is proved that the dimension of a bounded complexof free modules of finite rank can be computed directly from the matricesrepresenting the differentials of the complex.
Introduction
This short note concerns certain homological invariants—specifically, dimensionand depth—of complexes of modules over commutative Noetherian local rings. Theconcepts of depth and dimension for modules, introduced by Krull and by Auslanderand Buchsbaum, respectively, need no recollection. Both concepts were extended tocomplexes of modules by Foxby [5], and also by Iversen [9]. Their extensions agreeup to a normalization; in what follows we work with Foxby’s definitions, recalledfurther below, for they are better suited to computations in the derived category.The depth and dimension of a complex depend only on the quasi-isomorphism classof the complex; said differently, they are defined on the derived category of the ring.To compute these invariants one can usually reduce to the case where the com-plex is finite free, for they are independent of the domain. Indeed, if Q → R is asurjective map of rings with Q a regular local ring, then the depth and dimension ofan R -complex M coincide with the corresponding invariants of M viewed as a com-plex over Q . And, at least when M is homologically finite, it is quasi-isomorphic,over Q , to a finite free complex. Thus, in what follows we consider a complex overa local ring R of the form: F := 0 −→ F b ∂ −−→ · · · ∂ −−→ F a −→ F i is a free R -module of finite rank. We assume that F is minimal, inthat ∂ ( F ) ⊆ m F , where m is the maximal ideal of R .For such a complex F , the depth can be read off easily: The equality of Auslan-der and Buchsbaum for modules of finite projective dimension applies equally tocomplexes—this was proved by Foxby [4]—and yields that the depth of F equalsdepth R − b , provided that F b = 0. In this note we establish a formula that expressesthe dimension of F in terms of the ranks of the modules F i and the Fitting ideals Date : 9 September 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Codimension, dimension, perfect complex.L.W.C. was partly supported by Simons Foundation collaboration grant 428308; S.B.I. waspartly supported by NSF grant DMS-1700985. of the differentials; see Theorem 1. We were lead to it in an attempt to relate thecodimension, in the sense of Bruns and Herzog [3], to other homological invariants.It turns out that the codimension of F equals dim R − dim R Hom R ( F, R ); see Re-mark 3. This observation gives a different perspective on, and different proofs of,certain results in [3] related to the homological conjectures; see Proposition 6 andTheorem 8.
Dimension
Let R be a ring. By an R -complex we mean a complex of R -modules, with lowergrading: X := · · · −→ X n ∂ n −−→ X n − −→ · · · A graded R -module, such as the homology H( X ) of X , is viewed as an R -complexwith zero differentials; in particular, we use the same grading convention for suchobjects. Dimension.
Let R be a commutative Noetherian ring. In [5, Section 3] Foxbyintroduced the dimension of an R -complex X to be(1) dim R X := sup { dim( R/ p ) − inf H( X ) p | p ∈ Spec R } . By [5, Proposition 3.5] this invariant can be computed in terms of the homology:(2) dim R X = sup { dim R H n ( X ) − n | n ∈ Z } . The convention is that the dimension of the zero module is −∞ . Finite free complexes.
By a finite free R -complex we mean a bounded R -complex(3) F := 0 −→ F b ∂ b −−→ F b − −→ · · · −→ F a +1 ∂ a +1 −−→ F a −→ F i is a free R -module of finite rank. For such a complex F we set(4) s n = X i n ( − n − i rank R F i for each n ∈ Z . Given a map ϕ between finite free modules we write I s ( ϕ ) for the ideal generatedby the s × s minors of a matrix representing ϕ ; see, for example, [3, p. 21]. Itis convenient to adopt the convention that the determinant and all minors of theempty matrix is 1; in particular, for s s × s minor of any matrix is 1. For s > s × s minor of a non-empty matrix is 0 if s exceeds the number of rowsor columns. For the differentials of the complex (3) this means that I s n ( ∂ n +1 ) = R holds for integers n outside [ a, b ]. Theorem 1.
With F and s n as above, there is an equality dim R F = sup { dim( R/I s n ( ∂ n +1 )) − n | n ∈ Z } . Proof.
To prove the inequality “ > ” we verify thatdim( R/I s n ( ∂ n +1 )) dim R F + n holds for every integer n in [ a, b ]. Fix such an n . The inequality above holds if andonly if one has I s n ( ∂ n +1 ) p = R p for every p ∈ Spec R with dim( R/ p ) > dim R F + n .For such a prime ideal and any integer i n one getsdim R H i ( F ) dim R F + i < dim( R/ p ) IMENSION OF FINITE FREE COMPLEXES 3 where the first inequality holds by (2). This yieldsH i ( F ) p = 0 for all i n , which implies that the homology of the complex(5) ( F n +1 ) p ( ∂ n +1 ) p −−−−−→ ( F n ) p −→ · · · −→ ( F a +1 ) p ( ∂ a +1 ) p −−−−−→ ( F a ) p −→ n . It follows that the image of ( ∂ n +1 ) p is a free R p -module ofrank s n . Hence one has I s n ( ∂ n +1 ) p = R p .To prove the opposite inequality, “ ”, we show thatdim R H n ( F ) − n ≤ sup { dim( R/I s i ( ∂ i +1 )) − i | a i b } holds for each integer n in [ a, b ]. Let t be the supremum above. One needs to verifythat H n ( F ) p = 0 holds for primes p with dim( R/ p ) > t + n . Fix such a p ; for every i n one has dim( R/I s i ( ∂ i +1 )) t + i t + n < dim( R/ p )so that I s i ( ∂ i +1 ) p = R p . We now argue by induction on i that the homology of thecomplex (5) is zero in degrees n ; in particular, one has H n ( F ) p = 0, as desired.With f i = rank R F i and K i = ker ∂ i the argument goes as follows: In the basecase i = a one applies [3, Lemma 1.4.9] to the presentation of the image of ∂ a +1 afforded by (5), and one concludes that it is a free submodule of F a of rank f a , i.e.the whole thing. One also notices that a free module contained in K a +1 has rankat most s a +1 . In the induction step one applies op.cit to the presentation of theimage of ∂ i +1 and concludes that it is a free module of rank f i +1 − s i +1 = s i . Bythe induction hypothesis a free module contained in K i has rank at most s i , so thecomplex is exact at ( F i ) p . (cid:3) Codimension.
Let R be a commutative Noetherian ring and F a finite free R -complex as in (3). For each integer n set r n := X i > n ( − i − n rank R ( F i ) . For n in [ a + 1 , b ] this is the expected rank of the map ∂ n ; see [3, p. 24]. Corollary 2.
With F and r n as above there is an equality dim R Hom R ( F, R ) = sup { dim( R/I r n ( ∂ Fn )) + n | n ∈ Z } . Proof.
Set G := Hom R ( F, R ). This too is a finite free complex, concentrated indegrees [ − b, − a ], with differentials ∂ Gn = Hom R ( ∂ F − n , R ) for each n . It is now easyto check that the expected ranks r n of F and the invariants s n of G , from (4),determine each other: s n ( G ) = r − n ( F ) for each n. Whence one gets equalitiesdim R G = sup { dim( R/I s n ( ∂ Gn +1 )) − n | n ∈ Z } = sup { dim( R/I r − n ( ∂ F − n ) − n | n ∈ Z } = sup { dim( R/I r n ( ∂ Fn )) + n | n ∈ Z } . (cid:3) L. W. CHRISTENSEN AND S. B. IYENGAR
Remark . Bruns and Herzog [3, Section 9.1] have introduced a notion of “codimen-sion” for finite free complexes. This is perhaps a misnomer: Applied to the minimalfree resolution of a module, the codimension does not equal the usual codimensionof the module. In fact, Corollary 2 yields that the codimension, in their sense, ofany finite free R -complex F , is precisely dim R − dim R Hom R ( F, R ).Foxby also has a notion of codimension for an R -complex X , namely the invariantcodim R X := inf { dim R p + inf H( X ) p | p ∈ Spec R } = inf { codim R H n ( X ) + n | n ∈ Z } ;see [5, Lemma 5.1] and the definition preceding it. From the definitions one im-mediately gets codim R X + dim R X dim R ; equality holds if R is local, catenary,and equidimensional. For a finite free complex F over such a ring one thus hascodim R Hom R ( F, R ) = dim R − dim R Hom R ( F, R ) . In particular, the codimension of F in the sense of [3] is the codimension of thedual complex, Hom R ( F, R ), in the sense of [5].In Bruns and Herzog’s [3] treatment of the homological conjectures—most ofwhich are now theorems thanks to Andr´e [2]—their notion of codimension of a finitefree complex is key. Per Remark 3 this suggests that estimates on the dimensionof Hom R ( F, R ) are useful, and that motivates the development below.
Support.
Let R be a commutative Noetherian ring. The large support of an R -complex X is the support of the graded module H( X ), i.e.Supp R X := { p ∈ Spec R | H n ( X ) p = 0 for some n } . Foxby [5, Section 2] also introduced the (small) support of X to be the setsupp R X := { p ∈ Spec R | H( κ ( p ) ⊗ L R X ) = 0 } ;as usual, κ ( p ) denotes the residue field of the local ring R p . Support is connectedto the finiteness of the depth of X :supp R X = { p ∈ Spec R | depth R p X p < ∞} . We recall that the depth of a complex X over local ring R with residue field k isdepth R X := inf { n ∈ Z | Ext nR ( k, X ) = 0 } . This invariant can also be computed in terms of the Koszul homology, and thelocal cohomology, of X ; see [6]. Proposition 4.
Let R be a commutative noetherian ring. For every finite free R -complex F one has dim R Hom R ( F, R ) dim R + sup H( F ) . Proof.
For every prime ideal p the complex F p has finite projective dimension, sothe Auslander–Buchsbaum Formula combines with standard (in)equalities betweeninvariants to yield − inf H(Hom R ( F, R )) p = sup { m ∈ Z | Ext mR p ( F p , R p ) = 0 } = proj . dim R p F p = depth R p − depth R p F p dim R p + sup H( F ) p . IMENSION OF FINITE FREE COMPLEXES 5
From the definition (1) one now getsdim R Hom R ( F, R ) sup { dim( R/ p ) + dim R p + sup H( F ) p | p ∈ Spec R } dim R + sup H( F ) . (cid:3) Remark . For the minimal free resolution of a finitely generated module of fi-nite projective dimension, the codimension considered in [3] is non-negative by theBuchsbaum–Eisenbud acyclicity criterion; see the comment before [3, Lemma 9.1.8].This compares to the inequality in Proposition 4, rewritten asdim R − dim R Hom R ( F, R ) > − sup H( F ) . Balanced big Cohen–Macaulay modules.
Let ( R, m ) be local and M a bigCohen–Macaulay module; that is, a module with depth R M = dim R and m M = M .Hochster [7, 8] proved that such a module exists for every equicharacteristic localring, and Andr´e [1] proved their existence over local rings of mixed characteristic. Abig Cohen–Macaulay R -module M is called balanced if every system of parametersfor R is an M -regular sequence. The m -adic completion of any big Cohen–Macaulaymodule is balanced; see [3, Theorem 8.5.3]. Sharp [10] demonstrated that thesemodules behave much like maximal Cohen–Macaulay modules. Of interest here isthe fact that for a balanced big Cohen–Macaulay module M one has(6) depth R p M p = dim R − dim( R/ p ) for each p ∈ supp R M ;this is part ( iii ) in [10, Theorem 3.2]. Note that what Sharp calls the supersupportof M is the support of M , in the sense above; this follows from comparison of [5,Remark 2.9] and part ( v ) in op. cit. Proposition 6.
Let R be a local ring, F a finite free R -complex, and M a balancedbig Cohen–Macaulay module. One has sup H( F ⊗ R M ) = dim R Hom R ( F, R ) − dim R .
Proof.
Set G := Hom R ( F, R ). There is an isomorphism F ⊗ M ∼ = Hom R ( G, M ). Inthe computation below, the first equality holds by [5, Proposition 3.4]. The secondequality follows from (6) and the fourth one follows from (1).sup H(Hom R ( G, M )) = − inf { depth R p M p + inf H( G ) p | p ∈ Spec R } = − inf { dim R − dim( R/ p ) + inf H( G ) p | p ∈ Spec R } = sup { dim( R/ p ) − inf H( G ) p | p ∈ Spec R } − dim R = dim R G − dim R . (cid:3)
Given Remark 3, the theorem above recovers [3, Lemma 9.1.8]:
Corollary 7.
Let R be a local ring and F := 0 → F b → · · · → F → a finitefree R -complex. If dim R − dim R Hom R ( F, R ) > holds, then for any balanced bigCohen–Macaulay module M one has H i ( F ⊗ R M ) = 0 for all i > . In [3, Section 9.4] it is shown how to derive various intersections theorems, in-cluding the New Intersection Theorem, from Corollary 7. The latter sheds furtherlight on the invariant dim R − dim R Hom R ( F, R ). Theorem 8.
Let R be a local ring and F a finite free R -complex. One has dim R + inf H( F ) dim R Hom R ( F, R ) dim R + sup H( F ) . L. W. CHRISTENSEN AND S. B. IYENGAR
Proof.
The right-hand inequality holds by Proposition 4. Since R is local, one canapply the version of the New Intersection Theorem recorded by Foxby [5, Lemma4.1] to the complex Hom R ( F, R ) to getdim R − dim R Hom R ( F, R ) proj . dim R Hom R ( F, R ) . As Hom R ( F, R ) is also a finite free complex one hasproj . dim R Hom R ( F, R ) = − inf H(Hom R (Hom R ( F, R ) , R )) = − inf H( F ) . (cid:3) Remark . Let R be a local ring and M a nonzero finitely generated R -module offinite projective dimension. Applying Theorem 8 to a finite free resolution of M yields the equalitymax { dim R Ext nR ( M, R ) + n | n ∈ Z } = dim R .
Notice that with p := proj . dim R M one gets inequalitiesdim R − dim R M proj . dim R M dim R − dim R Ext pR ( M, R ) ;the inequality on the left is the version of the New Intersection Theorem that wentinto the proof of Theorem 8.
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