Virtual Resolutions for a Product of Projective Spaces

Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry. The geometric and algebraic sources of locally-free resolutions have complementary advantages. To see the differences, consider a smooth projective toric variety X together with its Pic(X)-graded Cox ring S. The local version of the Hilbert Syzygy Theorem implies that any coherent OX -module admits a locally-free resolution of length at most dimX ; see [Har77, Exercise III.6.9]. The global version of the Hilbert Syzygy Theorem implies that every saturated module over the polynomial ring S has a minimal free resolution of length at most dimS−1, so any coherent OX -module has a locallyfree resolution of the same length; see [Cox95, Proposition 3.1]. Unlike the geometric approach, this algebraic method only involves vector bundles that are a direct sum of line bundles. When X is projective space, these geometric and algebraic constructions usually coincide. However, when the Picard number of X is greater than 1, the locally-free resolutions arising from the minimal free resolution of an S-module are longer, and typically much longer, than their geometric counterparts. To enjoy the best of both worlds, we focus on a more flexible algebraic source for locally-free resolutions. The following definition, beyond providing concise terminology, highlights this source. Definition 1.1. A free complex F := [F0←− F1←− F2←− ·· · ] of Pic(X)-graded S-modules is called a virtual resolution of a Pic(X)-graded S-module M if the corresponding complex F̃ of vector bundles on X is a locally-free resolution of the sheaf M̃. In other words, a virtual resolution is a free complex of S-modules whose higher homology groups are supported on the irrelevant ideal of X . The benefits of allowing a limited amount of homology are already present in other parts of commutative algebra including almost ring theory [GR03], where one accepts homology annihilated by a given idempotent ideal, and phantom homology [HH93], where one admits cycles that are in the tight closure of the boundaries. For projective space, minimal free resolutions are important in the study of points [GGP95,EP99], curves [Voi02, EL15], surfaces [GP99, DS00], and moduli spaces [Far09, DFS16]. Our overarching goal is to demonstrate that the right analogues for subschemes in a smooth complete toric variety use virtual resolutions rather than minimal free resolutions. This distinction is not apparent on 2010 Mathematics Subject Classification. 13D02; 14M25, 14F05. CBZ was partially supported by the NSF Grant DMS-1440537, DE was partially supported by the NSF Grants DMS-1302057 and DMS-1601619, and GGS was partially supported by the NSERC. 1 ar X iv :1 70 3. 07 63 1v 1 [ m at h. A C ] 2 2 M ar 2 01 7 2 C. BERKESCH ZAMAERE, D. ERMAN, AND G.G. SMITH projective space because the New Intersection Theorem [Rob87] establishes that a free complex with finite-length higher homology groups has to be at least as long as the minimal free resolution. In contrast, for other toric varieties such as products of projective spaces, allowing irrelevant homology may yield simpler complexes; see Example 1.4. Throughout this paper, we write Pn := Pn1×Pn2×·· ·×Pnr for the product of projective spaces with dimension vector n :=(n1,n2, . . . ,nr)∈N over a field k. Let S :=k[xi, j : 16 i6 r, 06 j 6 ni] be the Cox ring of Pn and let B := ⋂r i=1 〈 xi,0,xi,1, . . . ,xi,ni 〉 be its irrelevant ideal. We identify the Picard group of Pn with Zr and partially order the elements via their components. If e1,e2, . . . ,er is the standard basis of Zr, then the polynomial ring S has the Zr-grading induced by deg(xi, j) := ei. We first reprove the existence of short virtual resolutions; compare with Corollary 1.14 in [EES15]. Proposition 1.2. Every finitely-generated Zr-graded B-saturated S-module has a virtual resolution of length at most |n| := n1 +n2 + · · ·+nr = dimPn. Since dimS− dimPn = r, we see that a minimal free resolution can be arbitrarily long when compared with a virtual resolution. A proof of Proposition 1.2 appears in Section 2. Besides having shorter representatives, virtual resolutions also exhibit a closer relationship with Castelnuovo–Mumford regularity than minimal free resolutions. On projective space, Castelnuovo– Mumford regularity has two equivalent descriptions: one arising from the vanishing of sheaf cohomology and another arising from the Betti numbers in a minimal free resolutions. However, on more general toric varieties, the multigraded Castelnuovo–Mumford regularity is not determined by a minimal free resolution; see Theorem 1.5 in [MS04] or Theorem 4.7 in [BC17]. From this perspective, we demonstrate that virtual resolutions improve on minimal free resolutions in two ways. First, Theorem 3.1 proves that the set of virtual resolutions of a module determines its multigraded Castelnuovo–Mumford regularity. Second, the next theorem, from Section 4, demonstrates how to use regularity to extract a virtual resolution from a minimal free resolution. Theorem 1.3. Let M be a finitely-generated Zr-graded B-saturated S-module that is d-regular. If G is the free subcomplex of a minimal free resolution of M consisting of all summands generated in degree at most d+n, then G is a virtual resolution of M. For convenience, we refer to the free complex G as the virtual resolution of the pair (M,d). Algorithm 4.4 shows that it can be constructed without computing the entire minimal free resolution. The following example illustrates that a virtual resolution can be much shorter and much thinner than the minimal free resolution. Surprisingly, it follows that a majority of the summands in the minimal free resolution are often irrelevant for geometric purposes. Example 1.4. A hyperelliptic curve C of genus 4 can be embedded as a curve of bidegree (2,8) in P1×P2; see [Har77, Theorem IV.5.4]. For instance, the B-saturated S-ideal I := 〈 x3 1,1x2,0− x3 1,1x2,1 + x3 1,0x2,2, x2 1,0x 2,0 + x2 1,1x 2,1 + x1,0x1,1x 2,2, x2 1,1x 2,0− x2 1,1x 2,0x2,1− x1,0x1,1x 2,1x2,2− x2 1,0x 2,2, x1,0x1,1x 2,0− x1,0x1,1x 2 2,0x2,1−x 1,0x 2,1x2,2 +x2 1,1x2,0x 2,2−x 1,1x2,1x 2,2, x1,1x 2,0x 2,1−x1,1x 2,0x 2,1−x1,0x 2,1x2,2−x1,0x 2,0x 2,2 +x1,0x 2,0x2,1x 2,2− x1,1x2,0x 4 2,2 + x1,1x2,1x 4 2,2, x1,1x 5 2,0− x1,1x 2,0x2,1− x1,0x 2,0x 2,1x2,2 + x1,1x 2,1x 2,2 + x1,0x 2,2, x1,0x 2,0− x1,0x 2,0x2,1 + x1,1x 2,1x2,2+ x1,1x 3 2,0x 2 2,2−x1,1x 2,0x2,1x 2,2 +x1,0x 2,1x 2,2, x8 2,0−2x 2,0x2,1 +x6 2,0x 2,1 +x6 2,1x 2,2 +3x3 2,0x 2,1x 2,2−3x 2,0x 2,1x 2,2−x2,0x 2,2 +x2,1x 2,2 〉 VIRTUAL RESOLUTIONS 3 defines such a curve. Macaulay2 [M2] shows that the minimal free resolution of S/I has the form S1←− S(−3,−1)1 ⊕ S(−2,−2)1 ⊕ S(−2,−3)2 ⊕ S(−1,−5)3 ⊕ S(0,−8)1 ←−− S(−3,−3)3 ⊕ S(−2,−5)6 ⊕ S(−1,−7)1 ⊕ S(−1,−8)2 ←− S(−3,−5)3 ⊕ S(−2,−7)2 ⊕ S(−2,−8)1 ←− S(−3,−7)1←− 0 . (1.4.1) Using the Riemann–Roch Theorem [Har77, Theorem IV.1.3], one verifies that the module S/I is (2,1)-regular, so the virtual resolution of the pair ( S/I,(2,1) ) has the much simpler form S1←− S(−3,−1)1 ⊕ S(−2,−2)1 ⊕ S(−2,−3)2 φ ←−− S(−3,−3)3←− 0 . (1.4.2) If the ideal J ⊂ S is the image of the first map in (1.4.2), then we have J = I∩Q for some ideal Q whose radical contains the irrelevant ideal. Using Proposition 2.4, we can even conclude that S/J is Cohen–Macaulay and J is the ideal of maximal minors of the 4×3 matrix (1.4.3) φ :=  x2 2,1 x 2 2,2 −x2 2,0 −x1,1(x2,0− x2,1) 0 x1,0x2,2 x1,0 −x1,1 0 0 x1,0 x1,1  . As an initial step towards our larger goal, we formulate a novel analogue for properties of points in projective space. Although any punctual subscheme of projective space is arithmetically Cohen–Macaulay, this almost always fails for a zero-dimensional subscheme of Pn; see [GVT15]. However, by considering virtual resolutions, we obtain the following variant. Theorem 1.5. Let Z ⊂ Pn be a zero-dimensional scheme and let I be its corresponding B-saturated S-ideal. There exists an S-ideal Q, whose radical contains B, such that the minimal free resolution of S/(I ∩Q) has length |n|. In particular, the minimal free resolution of S/(I ∩Q) is a virtual resolution of S/I. This theorem, proven in Section 5, does not imply that S/(I∩Q) is itself Cohen–Macaulay, as the components of Q will often have codimension less than |n|. However, when the ambient variety is P1×P1, the ring S/(I∩Q) will be Cohen–Macaulay of codimension 2. In this case, Corollary 5.2 shows that there is a matrix whose maximal minors cut out Z scheme-theoretically. Proposition 5.8 extends this to general points on any smooth toric surface. As a second and perhaps more substantial step, we apply virtual resolutions to deformation theory. On projective space, there are three classic situations in which the particular structure of the minimal free resolution allows one to show that all deformations have the same structure: arithmetically 4 C. BERKESCH ZAMAERE, D. ERMAN, AND G.G. SMITH Cohen–Macaulay subschemes of codimension 2, arithmetically Gorenstein subschemes in codimension 3, and complete intersections; see [Har10, Sections 2.8–2.9]. We generalize these results about unobstructed deformations in projective space as follows. Theorem 1.6. Consider Y ⊂ Pn and let I be the corresponding B-saturated S-ideal. Assume that the generators of I have degrees d1,d2, . . . ,ds and that the natural map (S/I)di → H0 ( Y,OY (di) ) is an isomorphism for all 16 i6 s. If any one of the following conditions hold (i) the subscheme Y has codimension 2 and there is d ∈ reg(S/I) such that the virtual resolution of the pair (S/I,d) has length 2; (ii) each factor in Pn has dimension at least 2, the subscheme Y has codimension 3, and there is d ∈ reg(S/I) such that the virtual resolution of the pair (S/I,d) is a self-dual complex (up to a twist) of length 3; or (iii) there is d ∈ reg(S/I) such that virtual resolution of the pair (S/I,d) is a Koszul complex of length codimY ; then the embedded deformations of Y in Pn are unobstructed and the component of the multigraded Hilbert scheme of Pn containing the point corresponding to Y is unirational. To illustrate this theorem, we can reuse the hyperelliptic curve in Example 1.4. Example 1.7. By reinterpreting Example 1.4, we see that the hyperelliptic curve C ⊂ P1×P2 satisfies condition (i) in Theorem 1.6. It follows that the embedded deformations of C are unobstructed and the corresponding component of the multigraded Hilbert scheme of P1×P2 can be given an explicit unirational parametrization by varying the entries in the 4×3 matrix φ from (1.4.3). Three other geometric applications for virtual resolutions are collected in Section 6. The first, Proposition 6.1, provides an unmixedness result for subschemes of Pn that have a virtual resolution whose length equals its codimension. The second, Proposition 6.5, gives sharp bounds on the Castelnuovo–Mumford regularity of a tensor products of coherent OPn-modules; compare with [Laz04, Prop. 1.8.8]. Lastly, Proposition 6.8 describes new vanishing results for the higher-direct images of sheaves, which are optimal in many cases. The final section presents some promising directions for future research. Conventions. In this article, we work in the product Pn :=Pn1×Pn2×·· ·×Pnr of projective spaces with dimension vector n := (n1,n2, . . . ,nr) ∈ Nr over a field k. Its Cox ring is the polynomial ring S := k[xi, j : 16 i6 r, 06 j 6 ni] and its irrelevant ideal is B := ⋂r i=1 〈 xi,0,xi,1, . . . ,xi,ni 〉 . The Picard group of Pn is identified with Zr and the elements are partially ordered componentwise. If e1,e2, . . . ,er is the standard basis of Zr, then S has the Zr-grading induced by deg(xi, j) := ei. We assume that all S-modules are finitely generated and Zr-graded. Acknowledgements. Some of this research was completed during visits to the Banff International Research Station (BIRS) and the Mathematical Sciences Research Institute (MSRI), and we are very grateful for their hospitality. We also thank Lawrence Ein, David Eisenbud, Craig Huneke, Nathan Ilten, Rob Lazarsfeld, Diane Maclagan, Frank-Olaf Schreyer, and Ian Shipman for helpful conversations. VIRTUAL RESOLUTIONS 5 2. EXISTENCE OF SHORT VIRTUAL RESOLUTIONS This section, by proving Proposition 1.2, establishes the existence of virtual resolutions whose length is bounded above by the dimension of Pn. In particular, these virtual resolutions are typically shorter than a minimal free resolution. Moreover, Proposition 2.4 shows that Proposition 1.2 provides the best possible uniform bound. Our proof of Proposition 1.2 is based on a minor variation of Beilinson’s resolution of the diagonal; compare with Proposition 3.2 in [Căl05] or Lemma 8.27 in [Huy06]. Given an OX j-module F j for all 16 j 6 n, their external tensor product is F1 F2 · · · Fm := p1 F1⊗OX1×X2 p ∗ 2 F2⊗OX1×X2 · · ·⊗OX1×X2 p ∗ m Fm where p j denotes the projection map from the Cartesian product X1×X2× ·· · ×Xm to X j. In particular, for all u ∈ Zr, we have OPn(u) = OPn1 (u1) OPn2 (u2) · · · OPnr (ur). With this notation, we can describe the resolution of the diagonal Pn ↪→ Pn×Pn. Lemma 2.1. If T ei Pn := OPn1 OPn2 · · · OPni−1 TPni OPni+1 · · · OPnr for 16 i6 r, then the diagonal Pn ↪→ Pn×Pn is the zero scheme of a global section of ⊕r i=1 OPn(ei) T ei Pn(−ei). Hence, the diagonal has a locally-free resolution of the form