Homological and combinatorial aspects of virtually Cohen--Macaulay sheaves
Christine Berkesch, Patricia Klein, Michael C. Loper, Jay Yang
aa r X i v : . [ m a t h . A C ] D ec HOMOLOGICAL AND COMBINATORIAL ASPECTS OFVIRTUALLY COHEN–MACAULAY SHEAVES
CHRISTINE BERKESCH, PATRICIA KLEIN, MICHAEL C. LOPER, AND JAY YANGA
BSTRACT . When studying a graded module M over the Cox ring of a smooth projective toric variety X , thereare two standard types of resolutions commonly used to glean information: free resolutions of M and vectorbundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometricinformation that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraicand combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, whichcapture desirable geometric information and are also amenable to algebraic and combinatorial study. The theoryof virtual resolutions includes a notion of a virtually Cohen–Macaulay property, though tools for assessingwhich modules are virtually Cohen–Macaulay have only recently started to be developed.In this paper, we continue this research program in two related ways. The first is that, when X is a productof projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings, whichwe show to be virtually Cohen–Macaulay via explicit constructions of appropriate virtual resolutions reflectingthe underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety X ,we develop homological tools for assessing the virtual Cohen–Macaulay property. Some of these tools giveexclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. Wealso use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen–Macaulay properties. I NTRODUCTION
Let X be a smooth projective toric variety over an algebraically closed field k with Cox ring S and ir-relevant ideal B (see [CLS11, §5.2]). The Cox ring S is a positively Pic( X ) -graded polynomial ring,and there is a correspondence between Pic( X ) -graded B -saturated modules M over S and sheaves f M on X [Aud91, Mus94, Cox95] (see [Mus02] when X is not smooth). Unfortunately, the numerics of the mini-mal Pic( X ) -graded free resolutions for such S -modules do not obviously provide many geometric insightsfor M when X is not projective space. For example, a minimal Pic( X ) -graded free resolution of M may besignificantly longer than the dimension of X . However, this failure appears to be a consequence of imposingtoo much algebraic structure on the resolution. Approaching the problem from the geometric perspective,vector bundle resolutions of f M are bounded in length by the dimension of X , but vector bundles on X aresignificantly more complicated than line bundles on X . A proposed solution comes from [BES20], in whichthe authors introduce a type of resolution of M by free S -modules, which they call a virtual resolution , thatbetter captures geometrically meaningful properties of Pic( X ) -graded S -modules, such as unmixedness,well-behavedness of deformation theory, and regularity of tensor products. Because virtual resolutions aredefined up to the sheafification of M , the object of study is intrinsically geometric. Because the resolutionsthemselves is in the category of S -modules, they are naturally amenable to algebraic techniques.Although virtual resolutions are desirable for their ability to encode geometric information, they do not yethave the wealth of tools that exists for graded free resolutions. In particular, there are not yet many methodsfor constructing short virtual resolutions or for establishing the minimum possible length among the virtualresolutions of a chosen Pic( X ) -graded S -module M . We provide some of each in this paper.Our broad goal in this article is to work towards a rich understanding of virtually Cohen–Macaulay modules(or virtually Cohen–Macaulay coherent sheaves of modules) as an analogue to Cohen–Macaulay modulesover the coordinate rings of affine or projective space. We provide two methods to construct short virtual res-olutions either from longer virtual resolutions or from short resolutions of closely-related modules. Thesemethods can be helpful in establishing that modules are virtually Cohen–Macaulay (see Propositions 4.1and 4.5). We also obtain homological obstructions to being virtually Cohen–Macaulay (see Section 5). CB was partially supported by NSF Grants DMS 1661962 and 2001101.JY was supported by NSF RTG Grant 1745638.
Guiding these structural developments is our production of a large class of virtually Cohen–MacaulayStanley–Reisner rings in Section 3. The results on this class are hard won through the careful applica-tion of Hochster’s formula, interpreted in a virtual setting, together with an analysis of the spectral sequenceassociated to a certain nerve complex. This not only provides us with a new source of examples of virtuallyCohen–Macaulay modules as we work to develop the theory, but also, given the difficulty of studying evenStanley–Reisner rings in this context, highlights the need for the advent of more virtual homological tools.
Acknowledgements.
We would like to thank Daniel Erman and Gregory G. Smith for helpful conversationsrelated to this work. 2. B
ACKGROUND AND S TATEMENTS OF M AIN R ESULTS
Throughout this article, let X be a smooth projective toric variety over the algebraically closed field k , and let S = Cox( X ) . All S -modules are assumed to be finitely generated and Pic( X ) -graded, and all sheaves areassumed to be coherent. Let M be an S -module. As in [BES20, Definition 1.1], a free graded S -complex F • = [ F ←− F ←− · · · ] is a virtual resolution of M (or of f M ) if the corresponding complex e F • ofvector bundles is a locally-free resolution of the sheaf f M . Next, define the virtual dimension of M , denoted vdim M , to be the minimal length of a virtual resolution of M . As noted in [BES20, Proposition 2.5], forproducts of projective spaces, there is an inequality vdim M ≥ codim M ; in light of this and an analogue tothe affine case, we say that M is virtually Cohen–Macaulay if vdim M = codim M , the minimum possible.We say that a subscheme V ⊂ X is virtually Cohen–Macaulay if its Cox ring is virtually Cohen–Macaulayas an S -module. Although there is a precise description in the literature for when complexes are virtualresolutions (see [Lop]), little is known about how to assess the virtual dimension of a module or how toconstruct virtual resolutions of minimal length, even when that minimal length is known.In this paper, we construct virtual resolutions of minimal length for a family of Stanley–Reisner rings inorder to show that they are virtually Cohen–Macaulay. Before stating that result, we review the Stanley–Reisner correspondence between simplicial complexes and squarefree monomial ideals. For a detailedintroduction, we refer the reader to [MS05].
Definition 2.1.
Let ∆ be a simplicial complex on { , , . . . , n } and R = k [ x , . . . , x n ] . Define the Stanley–Reisner ideal of ∆ to be I ∆ = h x i i · · · x i k | { i i , . . . , i k } / ∈ ∆ i and the Stanley–Reisner ring of ∆ to be R/I ∆ .We now state our main result on the existence of a new family of virtually Cohen–Macaulay rings (seeTheorem 3.1). Theorem.
Let S be the Cox ring of X = P n × P n × · · · × P n r . If ∆ is an r -dimensional simplicial complexand the variety V ( I ∆ ) ⊆ X is equidimensional, then S/I ∆ is virtually Cohen–Macaulay. Relationships between vdim M and dim X have been of interest since the introduction of virtual resolutions.In [BES20, Proposition 1.2, Theorem 5.1] a Hilbert Syzygy Theorem-type bound, vdim M ≤ dim X ,was given for an arbitrary Pic( X ) -graded S -module M when X is a product of projective spaces andfor an arbitrary punctual scheme in any smooth projective toric variety X . Further, [Yan21] shows that vdim S/I ≤ dim X when I is a relevant monomial ideal of S and X is a smooth projective toric variety.Our new result most directly compares with a similar theorem in the case of pure and balanced simplicialcomplexes, which are necessarily of dimension r − (see [KLM + , Theorem 1.3]). Our proof is constructive,and we illustrate its use in building explicit resolutions in Examples 3.10 and 3.11.Our second construction of short virtual resolutions for the purpose of realizing the virtual Cohen–Macaulayproperty comes by way of a mapping cone. It is precisely stated and proved as Proposition 4.1 and summa-rized below. OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 3
Proposition.
Let F • be a virtual resolution of an S -module M of length t such that Ext t ( M, S ) ∼ = 0 . If Ext t ( M, S ) admits a free resolution of length at most t + 1 , then we can construct a virtual resolution of M of length t − . Additionally, we propose a notion of a virtually regular element (see Definition 4.4) and show (as Proposi-tion 4.5) that it can be used to produce virtual resolutions.
Proposition. If M is a virtually Cohen–Macaulay S -module and f is a virtually regular element on M ,then M/f M is virtually Cohen–Macaulay.
In Example 4.8, we use Proposition 4.1 to show that a particular squarefree monomial ideal defines a virtu-ally Cohen–Macaulay quotient ring. We then quotient by a sequence of virtually regular elements to arriveat a virtually Cohen–Macaulay quotient ring outside of the squarefree monomial setting. This virtuallyCohen–Macaulay quotient ring has an embedded associated prime, which we notice is irrelevant.These constructions allow us to give a variety of examples of virtually Cohen–Macaulay modules that aregeometrically but not arithmetically Cohen–Macaulay. Motivated by these examples and in light of a previ-ous unmixedness result for virtually Cohen–Macaulay modules over the Cox rings of products of projectivespace ([BES20, Proposition 5.1]), we show the following (see Theorem 6.1 and Corollary 6.4).
Theorem. If M is an arithmetically Cohen–Macaulay S -module, then M is virtually Cohen–Macaulay.If M is virtually Cohen–Macaulay, then M is geometrically Cohen–Macaulay. Moreover, if M is virtu-ally Cohen–Macaulay, then, if it has any associated primes of height other than codim M , they must beirrelevant. Moreover, we show in Examples 3.10 and 6.2 that both of the implications of the first two sentences of thetheorem above are strict.In addition to the tools above, we also provide some homological tools that provide exclusionary criteriafor a module to have the virtually Cohen–Macaulay property. We also show that virtual resolutions can beused to compute
Ext and
Tor modules up to sheafification (as Theorems 5.1 and 5.7), which implies thefollowing (as Corollaries 5.2 and 5.8).
Theorem.
If the S -module M has a virtual resolution of length ℓ , then Ext iS ( M, N ) ∼ = 0 = Tor Si ( M, N ) ∼ for all S -modules N and all i > ℓ . We use this result to give an example of a module that cannot be virtually Cohen–Macaulay (see Exam-ple 5.3), a result that there were not pre-existing tools in the literature to establish. We also show that theconverses to Proposition 4.5 and Corollaries 5.2 and 5.8 are false. In particular, we see that the virtualdimension of a module is bounded below by the homological dimension of the associated coherent sheaf.Further, the homological dimension may be strictly greater than the virtual dimension, as demonstrated inExample 6.2. 3. V
IRTUALLY C OHEN –M ACAULAY S TANLEY –R EISNER RINGS
The purpose of this section is to prove the following theorem.
Theorem 3.1.
Let S be the Cox ring of X = P n × P n × · · · × P n r . If ∆ is an r -dimensional simplicialcomplex and its associated variety V ( I ∆ ) ⊆ X is equidimensional, then S/I ∆ is virtually Cohen–Macaulay. Let S = k [ x i,j | ≤ i ≤ r, ≤ j ≤ n i ] be the Cox ring of X (of Theorem 3.1) and B the irrelevant ideal of S . Throughout this section, we will consider simplicial complexes on the vertex set X corresponding to thevariables ( x i,j ) ≤ i ≤ r, ≤ j ≤ n i of S . The vertices in X corresponding to x i, • are said to have color i . Let ∆ bea simplicial complex with vertices in X . Define the color set of a face σ ∈ ∆ to be the set of the colors of the CHRISTINE BERKESCH, PATRICIA KLEIN, MICHAEL C. LOPER, AND JAY YANG vertices of σ , denoted by color( σ ) . We say that a face σ ∈ ∆ is relevant if color( σ ) = [ r ] = { , , . . . , r } and irrelevant otherwise. A simplicial complex ∆ is relevant if it contains at least one relevant face, andit is irrelevant otherwise. Note that if ∆ is an irrelevant simplicial complex on X , then S/I ∆ is irrelevant,meaning that the support of S/I ∆ is contained in V ( B ) = { P ∈ Spec( S ) | B ⊆ P } . If ∆ is a relevantsimplicial complex on X , then ∆ is said to be virtually Cohen–Macaulay if S/I ∆ is virtually Cohen–Macaulay.Our proof of Theorem 3.1 begins with a lemma treating irrelevant faces of a fixed dimension. We aim tounderstand reduced simplicial homology of complexes associated with Stanley–Reisner rings in order toapply Reisner’s criterion, which we will use to detect the virtually Cohen–Macaulay property. We will needto recall two pieces of standard terminology and introduce one new piece of notation. Recall that if σ is aface of the simplicial complex ∆ , then we define the link of σ in ∆ to be lk σ (∆) = { σ ′ ∈ ∆ | σ ∪ σ ′ ∈ ∆ , σ ∩ σ ′ = ∅ } . Recall also that e H i (∆; k ) denotes the i th reduced simplicial homology of the simplicial complex ∆ withcoefficients in k . Finally, let B r = { σ ⊆ X | dim σ ≤ r, σ is irrelevant } , the simplicial complex of all atmost r -dimensional irrelevant simplices. Lemma 3.2.
The ring
S/I B r is Cohen–Macaulay on the punctured spectrum. Also, e H r − ( B r ; k ) = k and e H i ( B r ; k ) = 0 for i < r with i = r − .Proof. We will show that for all σ ∈ B r \ ∅ , e H i (lk σ ( B r ); k ) = 0 for i < dim(lk σ ( B r )) = r − − dim( σ ) , e H r − ( B r ; k ) = k , and e H i ( B r ; k ) = 0 for i = r − and i < r .Let σ ∈ B r be arbitrary. Let ∆ = lk σ ( B r ) . Now for C ⊂ [ r ] with C c ∪ color( σ ) = [ r ] , considerthe subcomplex ∆ C given by the faces of ∆ that do not include the colors in C . Note in particular that ∆ C ∩ ∆ D = ∆ C ∪ D .For every face γ ∈ ∆ , since γ ∪ σ ∈ B r , it is irrelevant. Thus, there exists an i such that i / ∈ color( γ ∪ σ ) .In particular, i satisfies both i / ∈ color( σ ) and γ ∈ ∆ { i } . Putting these together, (cid:8) ∆ { i } (cid:9) i/ ∈ color( σ ) provides acovering of ∆ , which induces the Mayer–Vietoris spectral sequence: E p,q = M | C | = p +1 > C c ∪ color( σ ) =[ r ] H q (∆ C ; k ) ⇒ H p + q (∆; k ) . We claim that ∆ C is the ( r − − dim( σ )) -skeleton of the simplex on all vertices with color in C c , excludingthose vertices in σ . To see this, recall that we are restricting to C with C c ∪ color( σ ) = [ r ] . Thus for everysimplicial complex γ on X , with dim γ ≤ dim(lk σ ( B r )) = r − − dim( σ ) and color( γ ) ⊂ C c , it must bethat γ ∪ σ is irrelevant and belongs to B r , so γ ∈ ∆ C .Now for σ = ∅ , we must show that e H i (∆; k ) = 0 for i < dim(∆) = r − − dim( σ ) for σ = ∅ . Since ∆ C is the ( r − − dim( σ )) -skeleton of a simplex, it cannot have reduced homology in degrees lower than r − − dim( σ ) , and therefore H q (∆ C ; k ) = 0 for < q < r − − dim( σ ) . Thus for p + q < r − − dim( σ ) with q = 0 , we have that E p,q = 0 . In light of this, it suffices to show that the maps on E p +1 , → E p, give homology. But this can be observed in the total complex, since H (∆ C ; k ) = k for C = [ r ] , and ∆ C = ∅ if and only if C = [ r ] , the complex given by these maps is simply the simplicial chain complex for the nerveof the covering of ∆ by (cid:8) ∆ { i } (cid:9) i/ ∈ color( σ ) , where the nerve is the simplicial complex given by N (cid:16)(cid:8) ∆ { i } (cid:9) i/ ∈ color( σ ) (cid:17) = ( F ⊂ [ r ] \ color( σ ) (cid:12)(cid:12)(cid:12)(cid:12) \ i ∈ F ∆ { i } = ∅ ) . OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 5
In this case, this nerve is the simplex on [ r ] \ color( σ ) . Thus E , = k and E p,q = 0 for < p + q
Note that the intersection of an exterior and an interior face is an exterior face.
Example 3.5.
Consider the following example in P × P , where in Figure 1 the first copy of P is coloredred and the second copy of P is colored blue. Consider the link of the red square vertex, whose facesconsist of the red triangle vertices, blue pentagon vertices and dashed lines. The exterior faces are the redtriangle vertices and the interior faces are the blue pentagon vertices and the dashed lines. Notice that theblue hexagons and dashed line on the right hand edge of the diagram are irrelevant, but are still interiorfaces. F IGURE
1. Each column of vertices corresponds to a copy of P . •• • •• The idea of interior and exterior faces can become considerably more complex. Consider the followingillustration of the link of a cell in an example ∆ on P n × P m × P ℓ . In Figure 2, the vertices corresponding toeach of the parts of the product are colored red, blue, and green. Only the link is illustrated, and it is the linkof a vertex that would be colored blue. Then the bold faces are the exterior faces, and the others are interiorfaces. Lemma 3.6.
Let ∆ be a pure, relevant r -dimensional simplicial complex. If σ = ∅ is a simplex of ∆ , thenevery facet of lk σ (∆) has at most two codimension faces that are interior faces of lk σ (∆) . Moreover, CHRISTINE BERKESCH, PATRICIA KLEIN, MICHAEL C. LOPER, AND JAY YANG F IGURE
2. The link in some ∆ of a certain blue vertex on P n × P m × P ℓ . • • • ••••• •• • ••• (1) A facet of lk σ (∆) has no codimension faces that are interior if and only if σ uses some color atleast twice. (2) A facet τ of lk σ (∆) has exactly one codimension face that is interior if and only if τ shares a colorwith σ . (3) A facet τ of lk σ (∆) has exactly two codimension faces that are interior if and only if τ uses somecolor at least twice.Proof. Let τ be a facet of lk σ (∆) . By assumption, τ ∪ σ is relevant and dim( τ ∪ σ ) = r . Since τ ∪ σ is relevant, it has color [ r ] ; since τ ∪ σ has dimension r , it has exactly r + 1 vertices. Putting these twofacts together, it follows that in τ ∪ σ exactly one color is used twice. The rest of the argument proceeds bycarefully considering the locations of that twice-used color.Let the vertices of the twice-used color in τ ∪ σ be labeled v and v . Now, a codimension 1 face γ of τ isan interior face if and only if τ \ γ ⊂ { v , v } , since both of these conditions are the same as requiring that γ ∪ σ be colored by [ r ] . But this immediately implies that τ contains at most two codimension faces thatare interior.Moving now to the exact number of codimension faces of τ that are interior faces, we consider which of τ or σ contains each of the vertices v and v .(1) There are no codimension faces of τ that are interior faces if and only if v , v ∈ σ , which isequivalent to σ using some color twice.(2) There is one codimension face of τ that is an interior face if and only if v i ∈ σ for precisely one i ∈ { , } . In this case, for j = i , we have v j ∈ τ . Therefore, color( σ ) ∩ color( τ ) = ∅ .(3) There are two codimension faces of τ that are interior faces if and only if both v , v ∈ τ . This isequivalent to τ using some color at least twice. (cid:3) We are now prepared to prove Theorem 3.3. The proof makes heavy use of Reisner’s criterion, which werecord below for convenience. Reisner showed in his thesis that
S/I ∆ is Cohen–Macaulay if and only if ∆ is Cohen–Macaulay as a simplicial complex. It is for this reason, combined with the statement of Reisner’scriterion, that the proof of Theorem 3.3 centers on the computation of reduced simplicial homology. Theorem 3.7 (Reisner’s Criterion) . A simplicial complex ∆ is Cohen–Macaulay if and only if e H i (lk σ (∆); k ) =0 for all i < dim lk σ (∆) .Proof of Theorem 3.3. First note that we may assume that all facets of ∆ are relevant. We claim that all r -dimensional relevant-connected simplicial complexes having no irrelevant facets are pure. Note that an r -dimensional simplicial complex corresponds to a -dimensional subvariety of P n and that all 1-dimensional OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 7 subvarieties of P n are equidimensional. Thus, there cannot be any lower dimensional relevant facets of ∆ .Then since ∆ contains only relevant facets, ∆ is pure.We will produce a simplicial complex ∆ ′ that is Cohen–Macaulay and differs from ∆ in only irrelevantfaces. Note the theorem is trivially true when r = 1 , since this is the case of a single projective space, and,in this case, ∆ must be a 1-dimensional pure and relevant-connected simplicial complex. It now followseasily from Reisner’s criterion that ∆ is Cohen–Macaulay.When r > , there are two cases. First, consider the case that ∆ is of the form ∆ = join( τ, Ω) for a face τ = ∅ and a simplicial subcomplex Ω with color( τ ) ∩ color(Ω) = ∅ . Since τ is a simplex, there is abijection between the top-dimensional cells of ∆ and those of Ω . Thus, since ∆ is relevant-connected, Ω is,too. Further, since dim ∆ = r , dim Ω = r − dim τ − . Moreover, since any face of ∆ uses at most onecolor twice, we know that either dim τ = | color( τ ) | or dim τ = | color( τ ) | − . In the first case, we find that dim Ω = | color(Ω) | − . Now restricting to the colors in color(Ω) and applying [KLM + , Theorem 1.3] to Ω , we can construct a Cohen–Macaulay simplicial complex Ω ′ that differs from Ω only on irrelevant faces.On the other hand, if dim σ = | color( σ ) | − , then dim Ω = | color(Ω) | − and dim Ω < r so by replacing ∆ with Ω and by using induction on r , we will construct a Cohen–Macaulay simplicial complex Ω ′ differingfrom Ω only on irrelevant faces.Now we take the simplicial complex Ω ′ and let ∆ ′ = join( σ, Ω ′ ) . Then ∆ ′ differs from ∆ only in irrelevantfaces. Since σ is a simplex, join( σ, Ω ′ ) can be constructed by iteratively taking the cone over Ω ′ by thevertices in σ . Since the cone over a Cohen–Macaulay simplicial complex is Cohen–Macaulay, ∆ ′ is Cohen–Macaulay, and so ∆ is virtually Cohen–Macaulay.For the second and final case, suppose that ∆ is not of the form ∆ = join( τ, Ω) , where τ = ∅ is a face of ∆ and color( τ ) ∩ color(Ω) = ∅ . Then, define ∆ ′ = ∆ ∪ B r . We claim that ∆ ′ is Cohen–Macaulay, and wewill show this using Reisner’s criterion, i.e., we will show that e H i (lk σ (∆ ′ ); k ) = 0 for each face σ of ∆ ′ and all i < d = dim lk σ (∆ ′ ) . (3.1)Since ∆ ′ = ∆ ∪ B r , it follows that lk σ (∆ ′ ) = lk σ (∆) ∪ lk σ ( B r ) . Then the long exact sequence of a pairyields the exact sequence e H i (lk σ ( B r ); k ) → e H i (lk σ (∆ ′ ); k ) → H i (lk σ (∆ ′ ) , lk σ ( B r ); k ) → e H i − (lk σ ( B r ); k ) . (3.2)Then for any i , so long as e H i (lk σ ( B r ); k ) = e H i − (lk σ ( B r ); k ) = 0 , it suffices to show H i (lk σ (∆ ′ ) , lk σ ( B r ); k ) = 0 . (3.3)We will first treat the case of σ = ∅ and then separately handle the case of σ = ∅ .For σ = ∅ , since H i (lk σ ( B r ); k ) = 0 for i < d , by Lemma 3.2 and thanks to (3.3), it suffices to show that H i (lk σ (∆ ′ ) , lk σ ( B r ); k ) = 0 for all i < d . Notice that H i (lk σ (∆ ′ ) , lk σ ( B r ); k ) = H i (lk σ (∆) , Ex ( σ, ∆); k ) , where Ex ( σ, ∆) = lk σ (∆) ∩ lk σ ( B r ) . To complete the proof for the case σ = ∅ , we will show that H i (lk σ (∆) , Ex ( σ, ∆); k ) = 0 , starting with i < d − . We will then treat separately the cases i = d − and i = d − .When i < d − , let τ be an i -face of lk σ (∆) of codimension at least . Then we claim that τ is an exteriorface. To see this, let ˜ τ be a facet of lk σ (∆) that contains τ . Because τ is of codimension at least in ˜ τ , it iscontained in at least codimension faces of ˜ τ , and therefore, by Lemma 3.6, it is contained in at least oneexterior facet of ˜ τ . Hence τ ∈ Ex ( σ, ∆) , so C i (lk σ (∆) , Ex ( σ, ∆); k ) = 0 and H i (lk σ (∆) , Ex ( σ, ∆); k ) = 0 for i < d − , as desired.When i = d − , we must show that H d − (lk σ (∆) , Ex ( σ, ∆); k ) = 0 . To do so, we will again show thatevery ( d − -face τ in lk σ (∆) is a boundary relative to Ex ( σ, ∆) . Without loss of generality, assume that τ is not in Ex ( σ, ∆) . Let ˜ τ be a facet of lk σ (∆) containing τ . Since τ is of codimension in lk σ (∆) , τ CHRISTINE BERKESCH, PATRICIA KLEIN, MICHAEL C. LOPER, AND JAY YANG is contained in exactly two codimension faces of ˜ τ . Further, since τ is not in Ex ( σ, ∆) , it must be thatboth of these codimension faces of ˜ τ are interior faces; call one of them ξ . By Remark 3.4, the othercodimension faces of ξ besides τ must be in Ex ( σ, ∆) . Therefore, up to sign, the relative boundary withrespect to Ex ( σ, ∆) of ξ is τ . Since τ was arbitrary, H d − (lk σ (∆) , Ex ( σ, ∆); k ) = 0 , as desired.Finally, when i = d − , we must show that H d − (lk σ (∆) , Ex ( σ, ∆); k ) = 0 . To do so, we will useLemma 3.6 to construct a graph (with loops). In the graph G , there is a distinguished vertex ∗ , while theother vertices correspond to the codimension faces of lk σ (∆) that are interior. Edges are placed to connectvertices corresponding to interior faces that are both contained in a common facet of lk σ (∆) . When a facetof lk σ (∆) has one codimension face that is interior, then an edge is placed between the vertex for thatfacet and ∗ . Finally, if a facet of lk σ (∆) has no interior faces, a loop is placed at ∗ .Recall that we are currently in the case that ∆ is not of the form join( τ, Ω) , where τ is a face of ∆ and color( τ ) ∩ color(Ω) = ∅ . We claim that, in this case, lk σ (∆) contains at least one facet that has at most onecodimension face that is interior. To see this, by way of contradiction, suppose that in lk σ (∆) , all facetscontain exactly two codimension 1 faces that are interior. Let τ ∈ lk σ (∆) be such a facet, in which case τ ∪ σ is a facet of ∆ . Since τ has exactly two codimension faces that are interior, by Lemma 3.6, τ usessome color at least twice. Then, since τ ∪ σ contains r + 1 vertices and there are only r possible colors,it must be that the colors used in σ are present only in σ and not in τ . But since a relevant simplex mustcontain all colors, the relevant facets of τ ∪ σ must all contain σ . Further, since ∆ is relevant-connected,repeating this for the successive neighbors of τ ∪ σ in ∆ , we find that all facets of ∆ contain σ , and thus ∆ = join( σ, Ω) for some Ω , a contradiction. Therefore, it must be that lk σ (∆) contains at least one facetfor which at most one of its codimension faces is interior.By the previous paragraph, the graph G is connected, and there is a commutative diagram C d (lk σ (∆) , Ex ( σ, ∆); k ) C d − (lk σ (∆) , Ex ( σ, ∆); k ) C ( G, ∗ ; k ) C ( G, ∗ ; k ) . ∼ = ∼ = (3.4)The surjectivity of the bottom map in (3.4) is a consequence of the fact that G is connected, so H ( G, ∗ , k ) =0 . Since the vertical maps in the diagram are isomorphisms, the top map in (3.4) is also surjective. Therefore, H d − (lk σ (∆) , Ex ( σ, ∆); k ) = 0 , which concludes the proof of (3.1) for any face σ = ∅ in ∆ ′ .It now remains to show that condition (3.1) holds for σ = ∅ . Before beginning this portion of the proof,note that the argument in the cases that σ = ∅ and i ≤ d − case above apply here as well to show that H i (∆ ′ , B r ; k ) = 0 for i ≤ r − . (3.5)Now consider the case that σ = ∅ and i < r − , where r = dim(∆ ′ ) . By Lemma 3.2, H i ( B r ; k ) = 0 for i < r − . Putting this together with (3.5), it now follows from (3.2) that e H i (∆ ′ ; k ) = 0 for all i < r − .It remains to show that condition (3.1) holds for σ = ∅ in the cases i = r − and i = r − . The long exactsequence of a pair together with (3.5) yield the exact sequence: H r − ( B r ; k ) → H r − (∆ ′ ; k ) → H r − (∆ ′ , B r ; k ) → H r − ( B r ; k ) → H r − (∆ ′ ; k ) → . Applying Lemma 3.2, this simplifies to → H r − (∆ ′ ; k ) → H r − (∆ ′ , B r ; k ) → k → H r − (∆ ′ ; k ) → . (3.6)Thus, it suffices to show that H r − (∆ ′ , B r ; k ) = k and that the map H r − (∆ ′ ; k ) → H r − (∆ ′ , B r ; k ) is the zero map, since this would imply that the map H r − (∆ ′ , B r ; k ) → k is an isomorphism, so that H r − (∆ ′ ; k ) = H r − (∆ ′ ; k ) = 0 , as desired.To see that H r − (∆ ′ , B r ; k ) = k , note first that the codimension faces of any ( r − -simplex are irrelevantfor dimension reasons. Thus, the boundary of any ( r − -face in ∆ ′ belongs to B r , so the ( r − -faces of OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 9 ∆ provide a generating set for H r − (∆ ′ , B r ; k ) . By Lemma 3.6, every r -dimensional relevant face of ∆ hasprecisely two relevant ( r − -faces in its boundary. Further, since ∆ is relevant-connected, every relevant ( r − -face of ∆ ′ is nonzero and homologically equivalent up to sign. Therefore, H r − (∆ ′ , B r ; k ) = k andany relevant ( r − -face of ∆ ′ gives a generator of this homology group.Finally, to see that H r − (∆ ′ ; k ) → H r − (∆ ′ , lk σ ( B r )) is the zero map, let Γ be a simplex on the color set [ r ] . There is a projection ∆ ′ → Γ , given by mapping all vertices of a given color i onto the vertex i . Sinceany relevant ( r − -face of ∆ ′ gives a generator of H r − (∆ ′ , B r ; k ) = k and the induced map sends thisgenerator to a nonzero element of H r − (Γ , ∂ Γ) = k , this induced map H r − (∆ ′ , B r ) → H r − (Γ , ∂ Γ) is anisomorphism.Now consider the following diagram induced by the map ∆ ′ → Γ H r − (∆ ′ ) H r − (Γ) = 0 H r − (∆ ′ , B r ) H r − (Γ , ∂ Γ) . ∼ = This diagram commutes, and, thus, the map H r − (∆ ′ ) → H r − (∆ ′ , B r ) is the zero map. Applying this factto the exact sequence (3.6), we get H r − (∆ ′ ; k ) = H r − (∆ ′ ; k ) = 0 . (cid:3) Example 3.8.
Continuing with Example 3.5, one of the critical steps in the proof of Theorem 3.3 is thereduction of the some of the more troublesome homology groups (in the case that σ = ∅ and i = d − )to the homology of a graph by the construction of the graph given by the interior faces of the link. It isLemma 3.6 that allows such a graph to be constructed. In Figure 3 that graph is shown with the vertices givenby × symbols, the edges given by dashed lines, and the half edges are illustrated with an edge terminatedwith a ◦ symbol.F IGURE
3. The graph associated to the complex of interior faces of the link from Figure 2. • • • ••••• •• • •••× × ×××× × ×× × × × ××× × ×
In light of Theorem 3.3, to complete the proof of Theorem 3.1 it remains to show that it is enough to showthat
S/I ∆ is virtually Cohen–Macaulay on each of the components of its support. Proposition 3.9.
Let S be the Cox ring of a smooth projective toric variety X , and let M be a finitelygenerated Pic( X ) -graded S -module. If M is module with equidimensional support X = F X i with disjointcomponents X i , then M is virtually Cohen–Macaulay if each M | X i is virtually Cohen–Macaulay.Proof. Let N = L M | X i . Then we claim that f M ∼ = e N . Since N is a direct sum, we can decompose e N as e N = M ] M | X i . And since X = F X i , we have f M = M ] M | X i . Thus a virtual resolution of N is a virtual resolution of M . Since M | X i is virtually Cohen–Macaulay, vdim M | X i = codim M | X i . Since X is equidimensional, we have that vdim M | X i = codim M . Finally,a direct sum of virtual resolutions is a virtual resolution of the direct sum, so vdim M = vdim N =codim M . (cid:3) Proof of Theorem 3.1.
This result is now an immediate consequence of Theorem 3.3 and Proposition 3.9,where M in the proposition is S/I ∆ and the X i correspond to the relevant-connected components of ∆ . (cid:3) Example 3.10.
Let k [ x , . . . , x ] be the Cox ring of X = P and consider the ideal J = h x x , x x , x x , x x i , for which S/J has free resolution S h x x x x x x x x i ←−−−−−−−−−−−−−−−−−−−−− S − x − x x − x − x x x x ←−−−−−−−−−−−−−−−−−−− S x − x − x x ←−−−−− S ←− . Note that J corresponds to a 1-dimensional simplicial complex with a single color, so Theorem 3.1 impliesthat S/J is virtually Cohen–Macaulay, with a short virtual resolution of the form S " x x x x ←−−−−−−−−−−−−−− S − x x − x x ←−−−−−−−−−− S ←− . See Example 5.3 for a discussion of the subscheme of P d cut out by J when d > . Example 3.11.
Let X = P × P , and consider the simplicial complex ∆ that is homeomorphic to a acylinder, as shown in Figure 4. F IGURE
4. A cylindrical ∆ on P × P . y y y x x x The Stanley–Reisner ideal corresponding to ∆ is I ∆ = h x y , x y , x y , x x x , y y y i . Since e H (∆; k ) =0 and dim ∆ = 2 , Reisner’s criterion implies that S/I ∆ is not Cohen–Macaulay. On the other hand, if weconsider the simplicial complex given by B ∪ ∆ , which is illustrated in Figure 5 and corresponds to theideal J = h x y , x y , x y i , then one can check that Reisner’s criterion is satisfied in this case. Since f I ∆ = e J , we conclude that S/I ∆ is virtually Cohen–Macaulay. OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 11 F IGURE
5. Adding irrelevant faces to ∆ in Figure 4 yields a Cohen–Macaulay complex. y y y x x x We will show in Example 4.9 that S/ √ I being virtually Cohen–Macaulay does not imply the same for S/I ,even when I is a monomial ideal. This example shows that the general monomial case cannot be reducedto the squarefree monomial case; moreover, it highlights that being virtually Cohen–Macaulay is a scheme-theoretic property rather than a set-theoretic one. For this reason, it is essential to develop tools to checkthe intuition developed through Theorem 3.1 in the not-necessarily-radical case and to use the examples itdelivers us to scaffold new ones as we build towards a theory of virtual depth. The remainder of this paperis directed at that transition. 4. N EW VIRTUAL RESOLUTIONS FROM OLD
We will now consider homological aspects of the virtually Cohen–Macaulay property. In particular, we willnow introduce two homological constructions that allow us to build new virtually Cohen–Macaulay modulesfrom those we have shown to be virtually Cohen–Macaulay. Then, in the next section of the article, we givehomological obstructions to being virtually Cohen–Macaulay.For the remainder of the article, let X be an arbitrary smooth projective toric variety with Cox ring S , andlet M be a finitely generated Pic( X ) -graded S -module.4.1. A mapping cone construction.
In this subsection, we introduce a mapping cone construction that,under certain conditions, will allow us to use a virtual resolution of the module M to construct a shortervirtual resolution of M , given the right conditions.To begin, let F • be a virtual resolution of M of length t , and assume that Ext t ( M, S ) ∼ = 0 . Let G ∗• bea free resolution of Ext t ( M, S ) , shifted and with indexing reversed as in (4.1). By [Eis95, Prop. A3.13],there is an induced map, which we denote by α ∗ , from F ∗• = Hom • S ( F, S ) to G ∗• : · · · F ∗ F ∗ · · · F ∗ t − F ∗ t − F ∗ t · · · G ∗− G ∗ G ∗ · · · G ∗ t − G ∗ t − G ∗ t . α ∗ ϕ ∗ α ∗ ϕ ∗ α ∗ ϕ ∗ t − ϕ ∗ t − α ∗ t − ϕ ∗ t α ∗ t − α ∗ t ψ ∗ ψ ∗ ψ ∗ ψ ∗ t − ψ ∗ t − ψ ∗ t = ϕ ∗ t (4.1)Dualizing yields the diagram · · · F F · · · F t − F t − F t · · · G − G G · · · G t − G t − G t . ϕ ϕ ϕ t − ϕ t − ϕ t ψ − α − ψ α ψ α ψ ψ t − α t − ψ t − α t − ψ t = ϕ t α t (4.2) Then, the mapping cone of α : G → F , denoted cone( α ) , is the complex · · · ←− G − ∂ ←−− F ⊕ G − ∂ ←−− F ⊕ G ∂ ←−− F ⊕ G ∂ ←−− · · · ←− F t − ⊕ G t − ∂ t ←−− F t ⊕ G t − ∂ t +1 ←−−−− ⊕ G t ←− , where the maps have the form ∂ i = (cid:20) ϕ i α i − ψ i − (cid:21) . Now, because ψ t = ϕ t , this reduces to the complex · · · ←− G − ∂ ←−− F ⊕ G − ∂ ←−− F ⊕ G ∂ ←−− F ⊕ G ∂ ←−− · · · ←− F t − ⊕ G t − ←− G t − ←− . (4.3) Proposition 4.1.
Let S be the Cox ring of a smooth projective toric variety X , and let M be a finitely gener-ated Pic( X ) -graded S -module. Let F • be a virtual resolution of M of length t such that Ext t ( M, S ) ∼ = 0 ,and let α be as in (4.2) . If G − = 0 in (4.3) , then (the minimization of) cone( α ) is a virtual resolution of M .Proof. There is an exact triangle G • α −−→ F • → cone( α ) → G • [1] , which induces the long exact sequencein homology · · · → H i +1 (cone( α )) → H i ( G ) → H i ( F ) → H i (cone( α )) → · · · . Since H i ( G ) ∼ = 0 for all i , it follows that the homology modules of cone( α ) is isomorphic to those for F • ,and thus cone( α ) and its minimization are virtual resolutions of M . (cid:3) The mapping cone construction of Proposition 4.1 can be iterated as long as the hypotheses hold.
Example 4.2.
Referring again to Example 3.10, the mapping cone construction of Proposition 4.1 alsoyields a short resolution for
S/J . Since the variety V ( J ) ⊂ X is simply the disjoint union of two lines, S/J is not arithmetically Cohen–Macaulay even though it is Cohen–Macaulay at every relevant
Pic( X ) -gradedprime ideal. The minimal free resolution of S/J is ← S ← S ( − ← S ( − ← S ( − ← . We will take the mapping cone of the following map of chain complexes, where the bottom chain complexis the dual of the free resolution of
Ext ( S/J, S ) ∼ = k : S S ( − S ( − S ( − S ( −
4) 0 . S S ( − S ( − S ( −
4) 0 . α − ψ α ψ α ψ α ψ α ϕ ϕ ϕ The mapping cone yields S ← S ( − ⊕ S ( − ← S ( − ⊕ S ( − ← S ( − ⊕ S ( − ← S ( − ← , which after minimizing provides a virtual resolution of S/J of length codim( J ) : S ← S ( − ← S ( − ← . Note that this resolution can also be constructed using the techniques of sheaves over simplicial complexesof [Yan03].
OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 13
Example 4.3.
Consider the hyperelliptic curve C of genus which can be embedded as a curve of bidegree (2 , in P × P found in [BES20, Example 1.4]. If I denotes the B -saturated ideal for C , then I = * x , x , − x , x , + x , x , , x , x , + x , x , + x , x , x , , x , x , − x , x , x , − x , x , x , x , − x , x , , x , x , x , + x , x , x , x , − x , x , x , + x , x , x , + x , x , x , , x , x , x , + x , x , x , − x , x , x , − x , x , x , + x , x , x , x , − x , x , x , − x , x , x , , x , x , + x , x , x , − x , x , x , x , + x , x , x , + x , x , , x , x , + x , x , x , + x , x , x , + x , x , x , + x , x , x , x , + x , x , x , , x , + 2 x , x , + x , x , + x , x , + 3 x , x , x , + 3 x , x , x , − x , x , − x , x , + . Now
S/I has minimal free resolution S ←− S ( − , − ⊕ S ( − , − ⊕ S ( − , − ⊕ S ( − , − ⊕ S (0 , − ←− S ( − , − ⊕ S ( − , − ⊕ S ( − , − ⊕ S ( − , − ←− S ( − , − ⊕ S ( − , − ⊕ S ( − , − ←− S ( − , − ←− . Applying Proposition 4.1 yields a virtual resolution for
S/I of the form S ⊕ S ( − , − ←− S ( − , − ⊕ S ( − , − ⊕ S ( − , − ⊕ S ( − , − ⊕ S ( − , − ←− S ( − , − ⊕ S ( − , − ⊕ S ( − , − ←− S ( − , − ←− . (4.4)However, this procedure can be repeated in this case, so applying Proposition 4.1 to (4.4) yields the followingvirtual resolution for S/I : S ⊕ S (0 , − ⊕ S (0 , − ⊕ S ( − , − ϕ ←−− S ( − , − ⊕ S ( − , − ⊕ S (0 , − ⊕ S ( − , − ⊕ S (0 , − ⊕ S ( − , − ⊕ S (0 , − ϕ ←−− S ( − , − ←− , where ϕ = − x y − x y x y y y − x y − x y y y + y y x y y + y y − x − x y − y − y
00 0 − x − x y + y − y x y − x − y y and ϕ = − y y − y y y − y y + y − y x x
00 0 y y + y x x y y − y − x x x − x . The quotient by a virtually regular element.
The purpose of this subsection is to introduce thenotion of a virtually regular element and to show that the quotient of a virtually Cohen–Macaulay moduleby a virtually regular element is again virtually Cohen–Macaulay. We do this by the explicit construction ofa virtual resolution of the appropriate length for the quotient module arising from a virtual resolution of theoriginal module.
Definition 4.4.
Let f ∈ S be homogeneous, and let M be an S -module. If Ann M f is irrelevant and dim M = 1 + dim M/f M , then we say that f is virtually regular on M or that f is a virtually regularelement on M .It is immediate that any regular element on M is virtually regular and that no element of a minimal prime of M can be virtually regular. The additional flexibility gained in considering virtually regular elements overregular elements alone is that an element of an embedded associated prime of M can be virtually regularif its annihilator is sufficiently well controlled. Notice also that if M ′ is an S -module satisfying f M ′ = f M ,then f is virtually regular on M if and only if f is virtually regular on M ′ . Proposition 4.5.
Let S be the Cox ring of a smooth projective toric variety X , and let M be a finitelygenerated Pic( X ) -graded S -module. If M has a virtual resolution of length ℓ and f is a virtually regularelement on M , then M/f M has a virtual resolution of length ℓ + 1 . In particular, if M is virtually Cohen–Macaulay, then M/f M is virtually Cohen–Macaulay.Proof.
Because dim M = 1 + dim M/f M , it suffices to prove the first claim. Let F • be a virtual resolutionof M of length ℓ . Consider the complex G • = 0 → S f −→ S → S/ h f i → . We claim that the totalcomplex, E • , of the double complex of F • ⊗ S G • gives a virtual resolution of M/f M . It is clear that H ( E ) = M ′ ⊗ S S/ h f i for some module M ′ satisfying f M ′ = f M . Because Ann M f is irrelevant, it followsthat H ( E ) ∼ = ( M/f M ) ∼ . A standard diagram chase shows that the higher homology of the total complexis irrelevant. Because E • has length ℓ + 1 , we have found a virtual resolution of M/f M of length ℓ + 1 , asdesired. (cid:3) Definition 4.6.
We say that the sequence f , . . . , f k is a virtually regular sequence on the module M if f is virtually regular on M and if f i is virtually regular on M/ h f , . . . , f i − i M for all < i ≤ k .The following corollary is a immediate from Proposition 4.5. Corollary 4.7.
If the S -module M is virtually Cohen–Macaulay, and f , . . . , f k is a virtually regular se-quence on M , then M/ h f , . . . , f k i M is virtually Cohen–Macaulay. Example 4.8.
Let S = k [ x , . . . , x ] be the Cox ring of P and consider the ideal J = h x , x , x i ∩ h x , x , x i . With M = S/J and F • the minimal free resolution of M , the construction in Proposition 4.1 yields a virtualresolution of M of length codim M = 3 , which shows that M is virtually Cohen–Macaulay. We claim that OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 15 x − x is a virtually regular element on M , that x − x is a virtually regular element on M/ h x − x i M ,and that x − x is a virtually regular element on M/ h x − x , x − x i M . Because x − x is a regularelement, it is automatically a virtually regular element. Observe that M = M h x − x i M ∼ = S h x x , x x , x x , x x , x x , x x , x x , x x , x , x i∼ = S h x , x , x , x i ∩ h x , x , x , x i ∩ h x , x , x , x , x , x i . Now x − x is not a regular element on M , but it is not in either minimal prime of M , and so dim M =1 + dim M / h x − x i M . The isomorphism presented above is given by x i x i for i = 5 and x x − x . After application of this isomorphism, it is easy to see that Ann M h x − x i = h x i M , whichis irrelevant. Hence, x − x is a virtually regular element that is not a regular element on M . A similarcomputation shows that x − x is in an embedded prime of M/ h x − x , x − x i M and has, after applyingan analogous isomorphism to the one described above, an irrelevant annihilator generated by x and x .Hence, x − x , x − x , x − x is a virtually regular sequence on M .Therefore, since M is virtually Cohen–Macaulay, so are each of the modules M/ h x − x i M , M/ h x − x , x − x i M , and M/ h x − x , x − x , x − x i M by Proposition 4.5.It is worth noting, however, that the converse to Proposition 4.5 is false, even over the Cox ring of a singleprojective space, as seen in the example below. Example 4.9. If S = k [ x , x , x ] is the Cox ring of P , and M = S/ h x , x x i , then there is no f ∈ S so that S/ h f i ∼ ∼ = M ∼ . Thus, the virtual dimension of M is at least of length while codim M = 1 , so M is not virtually Cohen–Macaulay. However, x is a (virtually) regular element on M , and M/ h x i M ∼ ∼ = S/ h x , x i ∼ , which is clearly virtually Cohen–Macaulay. Hence, we have an example of a module thatis not virtually Cohen–Macaulay and a virtually regular on it so that the quotient by that virtually regularelement yields virtually Cohen–Macaulay module. Notice also that x , x , x is a virtually sequence on M while x , x , x is not, which shows that virtually regular sequences need not be permutable. Additionally,because S/ h x i is virtually Cohen–Macaulay, this example shows that it is possible that a monomial ideal I does not define a virtually Cohen–Macaulay scheme while its radical √ I does.5. D ERIVED FUNCTORS VIA VIRTUAL RESOLUTIONS
In this section, we show that the sheafifications of
Ext and
Tor functors can be computed using virtualresolutions in place of free resolutions. This perspective gives necessary conditions on a module for it to bevirtually Cohen–Macaulay. The goals of this treatment are to relate the length of shortest virtual resolutionswith homological dimension and to describe some conditions on modules that prevent them from beingvirtually Cohen–Macaulay.As in the previous section, X will always denote an arbitrary smooth projective toric variety with Cox ring S and that all S -modules will be Pic( X ) -graded and finitely generated. Nevertheless, we will endeavor torepeat these hypotheses in the statements of theorems. Theorem 5.1.
Let M and N be finitely generated Pic( X ) -graded modules over the Cox ring S of a smoothprojective toric variety X . If F • is any virtual resolution of M , then Ext iS ( M, N ) ∼ is the sheafification ofthe i th homology module of Hom S ( F • , N ) . As usual, denote by B the irrelevant ideal of S .Proof. Fix a virtual resolution F • of M and an injective resolution I • of N . We consider the spectralsequence of the double complex whose ( i, j ) th entry on page is Hom S ( F i , I j ) . Taking homology alongeach column yields a page 1 whose only nonzero entries are Hom S ( F • , N ) . Hence, the module in position ( i, j ) on page ∞ is the i th homology module of Hom S ( F • , N ) . On the other hand, taking homology along rows first yields a page 1 whose entry in position ( i, j ) is Hom( L i , I j ) where L i is the i th homology of F • , which is supported only on B unless i = 0 , in which case L = M . Hence, the entry in position ( i, j ) on page ∞ is H i ⊕ K i for some module K i supported only on B and H i ⊆ Ext iS ( M, N ) with Ext iS ( M, N ) /H i supported only on B . Hence, Ext iS ( M, N ) ∼ ∼ = ( H i ⊕ K i ) ∼ ,as desired. (cid:3) It is a classical result that, for a module M over the Cox ring S of P d , the condition that H iB ( M ) ∨ ∼ =Ext d +1 − iS ( M, S ) be irrelevant (equivalently, finite length) for all i < dim( M ) is equivalent to the conditionthat M be equidimensional and Cohen–Macaulay at its localization at every non-irrelevant prime in itssupport. We will see in Section 6 that virtually Cohen–Macaulay implies geometrically Cohen–Macaulayand equidimensional. Putting these facts together, we have that when X is a single projective space, M beingvirtually Cohen–Macaulay implies that Ext jS ( M, S ) is irrelevant for all j > codim( M ) . The followingcorollary, which is immediate from Theorem 5.1, extends that result to the arbitrary smooth projective toricsetting. Corollary 5.2.
Let S be the Cox ring of a smooth projective toric variety X , and let M be a finitelygenerated Pic( X ) -graded S -module. If M has a virtual resolution of length ℓ , then Ext iS ( M, N ) ∼ = 0 forall Pic( X ) -graded finitely generated S -modules N and all i > ℓ . (cid:3) Example 5.3.
In Example 3.10, we saw that I = h x , x i ∩ h x , x i defined a virtually Cohen–Macaulaysubscheme of P . Corollary 5.2 implies that I does not define a virtually Cohen–Macaulay subscheme of P d whenever d > . In particular, with S = k [ x , . . . , x d ] , we have that Ext S ( S/I, S ) ∼ = S/ h x , x , x , x i ,which is not irrelevant. Remark 5.4.
With notation as above, the virtual dimension of M is greater than or equal to the homologicaldimension of f M . To see this, observe that if M has a virtual resolution of length ℓ , then, because every virtualresolution of M gives rise to a locally free resolution of f M , the homological dimension of f M is at most ℓ .However, it is not true that a module M is virtually Cohen–Macaulay if and only if its sheafification hashomological dimension equal to its codimension. For example, any module whose sheafification is a vectorbundle that does not split as a direct sum of line bundles has homological dimension while its shortestvirtual resolution must have positive length. Example 6.2 examines an explicit example of this type.We have seen in Corollary 5.2 that an S -module M cannot have a virtual resolution of length ℓ if there issome Ext iS ( M, N ) ∼ = 0 for some S -module N and some i > ℓ . The following two corollaries combine toshow that one need not consider all possible N but that it is instead sufficient to check only N = S . Corollary 5.5.
Let S be the Cox ring of a smooth projective toric variety X , and let M be a finitely generated Pic( X ) -graded S -module. If Ext ℓS ( M, S ) ∼ = 0 and Ext ℓ +1 S ( M, L ) ∼ = 0 for every Pic( X ) -graded finitelygenerated S -module L , then Ext ℓS ( M, N ) ∼ = 0 for every Pic( X ) -graded finitely generated S -module N .Proof. Let N be an S -module. There is a short exact sequence for some a ≥ and some K of the form −→ K −→ S a −→ N −→ . Applying Ext S ( M, − ) to this yields Ext ℓS ( M, S a ) ∼ −→ Ext ℓS ( M, N ) ∼ −→ Ext ℓ +1 S ( M, K ) ∼ = 0 . Since
Ext ℓS ( M, S a ) ∼ = (Ext ℓS ( M, S )) a = 0 , it must be true that Ext ℓS ( M, N ) ∼ = 0 . (cid:3) Corollary 5.6.
Let S be the Cox ring of a smooth projective toric variety X . Suppose the finitely generated Pic( X ) -graded S -module M has the property that Ext iS ( M, S ) is irrelevant for all i ≥ ℓ , for some ℓ ≥ .Then Ext iS ( M, N ) is irrelevant for every Pic( X ) -graded S -module N for all i ≥ ℓ .Proof. Because free resolutions are virtual resolutions, the claim is trivial if ℓ is greater than the projectivedimension of M , denoted pdim M , so we assume that ℓ ≤ pdim M . Now, using Corollary 5.5, we proceedby induction on pdim M − ℓ . (cid:3) OMOLOGICAL AND COMBINATORIAL ASPECTS OF VIRTUALLY COHEN–MACAULAY SHEAVES 17
For completeness, we state an analogue for
Tor of Theorem 5.1, which concerned
Ext . Theorem 5.7.
Let M and N be Pic( X ) -graded finitely generated modules over the Cox ring S of a smoothprojective toric variety X . If F • is any virtual resolution of M , then Tor Si ( M, N ) ∼ is the sheafification ofthe i th homology module of F • ⊗ S N .Proof. The argument follows the proof of Theorem 5.1 but uses the spectral sequence arising from thedouble complex F • ⊗ S G • , where G • is a free resolution of N . (cid:3) Corollary 5.8.
Let S be the Cox ring of a smooth projective toric variety X . If a finitely generated Pic( X ) -graded S -module M has a virtual resolution of length ℓ , then Tor Si ( M, N ) ∼ = 0 for all finitely generated Pic( X ) -graded S -modules N and all i > ℓ . (cid:3) Connecting back to Definition 4.4, a virtually regular element has a description in terms of virtual
Tor , justas, in the affine case, a regular element can be described in terms of the vanishing of certain
Tor modules.
Proposition 5.9.
Let S be the Cox ring of a smooth projective toric variety X , M be a finitely generated Pic( X ) -graded S -module, and f ∈ S be homogeneous. Then f is virtually regular on M (as in Defini-tion 4.4) if and only if M/f M = dim M and Tor S ( M, S/ h f i ) ∼ = 0 .Proof. Tensor the short exact sequence → S f −→ S → S/ h f i → with M to see that there is anisomorphism of S -modules Tor S ( M, S/ h f i ) ∼ = Ann M f . (cid:3)
6. R
ELATIONSHIPS AMONG THE ARITHMETICALLY , VIRTUALLY , AND GEOMETRICALLY C OHEN –M ACAULAY PROPERTIES
Theorem 6.1.
Let S be the Cox ring of a smooth projective toric variety X . If M is a finitely generated Pic( X ) -graded S -module, then (1) if M is arithmetically Cohen–Macaulay, then M is virtually Cohen–Macaulay; and (2) if M is virtually Cohen–Macaulay, then M is geometrically Cohen–Macaulay.Proof. If M is arithmetically Cohen–Macaulay, then by the Auslander–Buchsbaum formula, M has a freeresolution of length codim M . Because free resolutions are virtual resolutions, M is thus virtually Cohen–Macaulay.Similarly, if M is virtually Cohen–Macaulay, then it has a virtual resolution F • of length codim M . If P is any relevant prime in the support of M , then localizing F • at P gives a free resolution ( F P ) • of M P oflength codim M . Because the codimension of Spec( S/ Ann S ( M )) in Spec( S ) is equal to the codimensionof Spec( S P / Ann S P ( M P )) in Spec( S P ) , it follows from the Auslander–Buchsbaum formula that M P isCohen–Macaulay. Hence M is geometrically Cohen–Macaulay. (cid:3) We saw several times in Section 3, for example in Example 3.10, that implication (1) of Theorem 6.1 isstrict. We now give an example showing that implication (2) is also strict.
Example 6.2.
Even over the Cox ring of projective space, a module can be geometrically Cohen–Macaulaybut not virtually Cohen–Macaulay. For example, if S is the Cox ring of P d with d > and M correspondsto the tangent bundle, i.e., M is the cokernel of the map S d +1 x ... x d ←−−− S ← , then f M is a vector bundle that does not split as a direct sum of line bundles, see [CLS11, Theorem 8.1.6].Thus M has virtual dimension but codimension . Meanwhile, for each ≤ i ≤ d , the matrix above has a unit entry after tensoring with S [1 /x i ] , whichshows that M [1 /x i ] ∼ = S [1 /x i ] , and so M is a geometrically Cohen–Macaulay. In fact, not only is M geometrically Cohen–Macaulay, but also it is a faithful module of depth d on the homogeneous maximalideal of S . These properties show that the virtual Cohen–Macaulay property is not captured by the geometricCohen–Macaulay property along with depth information coming from the affine setting.It was shown in [BES20, Proposition 5.1] that every B -saturated virtually Cohen–Macaulay module over theCox ring S of a product of projective spaces is unmixed. The argument, which we record here, generalizesto arbitrary smooth projective toric varieties. Theorem 6.3.
Let S be the Cox ring of a smooth projective toric variety X with irrelevant ideal B . If M is a finitely generated Pic( X ) -graded B -saturated S -module that is virtually Cohen–Macaulay, then dim S/P = dim M for all associated primes P of M .Proof. Suppose that M is a virtually Cohen–Macaulay module of codimension c , and suppose that there issome associated prime P of M of codimension e > c . Let F • be a virtual resolution of length c . Because M is B -saturated, every associated prime of P is relevant. Hence, ( F P ) • gives an S P -free resolution of M P oflength c . Let pdim S P M P denote the projective dimension of M P over S P , and recall that the codimensionof Spec( S/ Ann S ( M )) in Spec( S ) is equal to the codimension of Spec( S P / Ann S P ( M P )) in Spec( S P ) ,which we record as codim M = codim M P . Then we obtain a contradiction because pdim M P ≤ c < e = codim M = codim M P . (cid:3) The following corollary is immediate from Theorem 6.3.
Corollary 6.4.
Let S be the Cox ring of a smooth projective toric variety X with irrelevant ideal B , and let M and N be finitely generated Pic( X ) -graded S -modules such that M is B -saturated. If e N = f M and M is virtually Cohen–Macaulay, then any embedded prime of N is irrelevant. R EFERENCES [Aud91] Mich`ele Audin,
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