What makes a complex a virtual resolution?

Virtual resolutions are homological representations of finitely generated $\text{Pic}(X)$-graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.


Introduction
In a famous paper, Buchsbaum and Eisenbud present two criteria that completely determine whether or not a chain complex is exact over a Noetherian ring [BE73]. This is done without examining the homology of the complex. These criteria are useful in investigating a module by examining the minimal free resolution.
In turn, these criteria can be used to study the geometry of projective space. Coherent sheaves over projective space correspond to finitely generated graded modules over a standard-graded polynomial ring. Properties of this module and the sheaf associated to the graded module such as degree, dimension, and Hilbert polynomial, are encoded in the minimal free resolution of the graded module.
Before stating the main theorem from [BE73], we must introduce some notation. A map of free S-modules ϕ : F → G can be expressed as a matrix with entries in S by choosing bases of F and G. Denote by I r (ϕ) the ideal generated by the r × r minors of ϕ. Then rank(ϕ) will be the largest r such that I r (ϕ) = 0. The ideal I rank(ϕ) (ϕ) will be the most important of these ideals of minors, and we set I(ϕ) := I rank(ϕ) (ϕ). By convention, we define I k (ϕ) = S for every integer k ≤ 0.
In fact, these ideals of minors can be extended to projective modules (that may not be finitely generated). Indeed ϕ gives rise to a map r ϕ : r F → r G, and the rank of ϕ is the largest r such that r ϕ = 0. In this context, I r (ϕ) is the image of the map r F ⊗ ( r G) * → S. The main theorem from [BE73] can now be stated.
When the toric variety is projective space, the locally free resolutions of coherent sheaves over P N and the free resolutions of Cox(P N )-modules coincide. Unfortunately when studying coherent sheaves over more general smooth projective toric varieties, the situation is not as well-behaved. Locally free resolutions of a coherent sheaf are often shorter and thinner than the corresponding minimal free resolutions of the modules. Tying these concepts more closely together, Berkesch, Erman, and Smith introduced the notion of virtual resolutions in [BES20]. The main theorem in the present paper (Theorem 1.3) is the virtual analogue to the main theorem of [BE73] (Theorem 1.1).
Notation. Throughout this paper, X will be a smooth projective toric variety and S will denote the Cox ring of X over an algebraically closed field k. The Cox ring S is graded by the Picard group of X, which we denote by Pic(X) [Cox95,§1]. In particular, S is a polynomial ring with a multigrading by Z r for r = rank Pic(X). Let B denote the irrelevant ideal of S, which is radical; M will be a finitely generated Pic(X)-graded module over S and M denotes the sheaf of M over X, as constructed in [Cox95,§3]. Given an ideal I of S, we denote the set of all homogenous prime ideals containing I by V (I).
Definition 1.2. A graded free complex of S-modules is called a virtual resolution of M if the corresponding complex of vector bundles F • is a locally free resolution of the sheaf M .
The above definition uses the geometric language, but virtual resolutions can be equivalently defined algebraically. The Pic(X)-graded S-module associated to a sheaf F over X is defined to be and for every i ≥ 1, there is an such that B H i (F ) = 0, where again B is the irrelevant ideal of S.
The main result of this paper is the following theorem.
Theorem 1.3. Let X be a smooth projective toric variety with S = Cox(X). Suppose is a Pic(X)-graded complex of free S-modules. Then F • is a virtual resolution if and only if both of the following conditions are satisfied: (a) rank(ϕ i ) + rank(ϕ i+1 ) = rank(F i ) (with ϕ n+1 = 0), for each i = 1, 2, ..., n.
As in [BE73], we assign the unit ideal infinite depth, so that condition (b) holds if I(ϕ i ) : B ∞ = S, i.e., I(ϕ i ) is irrelevant. The difference between Theorem 1.1 and Theorem 1.3 is the replacement of exactness with virtuality and the addition of the saturation of ideals of minors by the irrelevant ideal B. Of course, any complex of graded free S-modules that is exact will also be a virtual resolution. Further, if a complex is exact, then the conditions of Theorem 1.3 will be satisfied by Theorem 1.1. On the other hand, below is an example of a complex that is a virtual resolution but is not exact.
[BES20, Theorem 1.3] states that if X = P n 1 × P n 2 × · · · × P nr , the S = Cox(X)-ideal I is B-saturated (as it is in this example), and d is an an element in the multigraded Castelnuovo-Mumford regularity of S/I (see [MS04] for a detailed introduction to multigraded regularity), then the chain complex consisting of all twists less than or equal to d+(n 1 , n 2 , ..., n r ) is a virtual resolution. They call this the virtual resolution of the pair (S/I, d ). In this example, (1, 1) is in the multigraded regularity of S/I and the virtual resolution of the pair (S/I, (1, 1)) is shown below.
Notice that this virtual resolution is both shorter and thinner than the minimal free resolution. This virtual resolution has nonzero first homology module S/ x 0 , x 1 , which is annihilated by B. Consequently, this virtual resolution is not exact. Calculating the depth of the (non-B-saturated) ideal of minors of each differential yields depth(I(ϕ 1 )) = 3, depth(I(ϕ 2 )) = 2, depth(I(ϕ 3 )) = 2, indicating again that this virtual resolution is not exact, because the depth of I(ϕ 3 ) is less than three. Though the complex is not exact, depth(I(ϕ 3 ) : B ∞ ) = 3, so it is indeed a virtual resolution.
Remark 1.5. In the special case of the smooth projective toric variety P N k , the Cox ring is S = k[x 0 , x 1 , ..., x N ] with the standard grading, and the irrelevant ideal is the maximal homogeneous ideal B = m = x 0 , x 1 , ..., x N . If the length of the complex F is not longer than N +1 (the bound from the Hilbert Syzygy Theorem), then Theorem 1.3 recovers Theorem 1.1. This is because each I(ϕ i ) is homogeneous (in fact, every minor of ϕ i is homogeneous), and for any homogeneous ideal I m, where the second to last equality follows from the fact that any homogeneous radical ideal other than m is m-saturated. Therefore, condition (b) in Theorem 1.3 becomes which exactly matches condition (b) in Theorem 1.1.
As condition (a) is the same in the two theorems, we have thus proved the following proposition. A slightly different proof is offered below.
Since F • is a virtual resolution, all of its homology modules are annihilated by a power of B. Assume F • has a nonzero homology module and let H : The Peskine-Szpiro Acyclicity Lemma [PS73, Lemma 1.8] guarantees depth(H) ≥ 1 and hence B has a nonzerodivisor on H, a contradiction to the assumption that F • is a virtual resolution. Therefore, in this setting, if F • is a virtual resolution, it must be exact.
If the length of F • is larger than N + 1, then virtual resolutions do not need to be exact. An example is illustrated below.
That is, Then the length of F • is 6 > 3 and F • is a virtual resolution as the only nonzero higher homology module is H 3 (F • ) = k which is annihilated by the irrelevant ideal x, y, z .
Outline. Section 2 lays the groundwork for the rest of the paper. This includes notation, relevant definitions, and some preliminary facts about B-saturated homogeneous prime ideals. These primes are important, because the homogeneous localization of a module at a B-saturated homogeneous prime corresponds to taking the stalk of a sheaf over the toric variety X. Section 3 contains the proof of Theorem 1.3. In Section 4, the invariance of saturated Fitting ideals of virtual presentations is presented along with an obstruction to the number of generators of a module up to saturation. Finally, Section 5 contains a connection between saturated Fitting ideals and locally free sheaves. It ends with a useful result concerning unbounded virtual resolutions. The results in Sections 4 and 5 can be used to prove the reverse direction of Theorem 1.3 in the same way that results of Fitting ideals can be used to prove Theorem 1.1.
Acknowledgements. The author is incredibly grateful to Christine Berkesch for her guidance while this work was conducted. He would also like to thank Daniel Erman and Jorin Schug for helpful conversations and Patricia Klein, Gennady Lyubeznik, and Robert Walker for suggestions that improved the readability of this note. Finally, he also thanks an anonymous referee for a suggestion of a strengthening of Lemma 3.2 that resulted in a simpler proof of Theorem 1.3. The author was partially supported by the National Science Foundation RTG grant DMS-1745638.

B-Saturated Prime Ideals
In this section the structure of B-saturated prime ideals in the Cox ring S is discussed, which will aid in later proofs. Indeed, in order to show that a complex is a virtual resolution, we will show that after sheafifying, the complex of vector bundles is acyclic by showing it is exact in each place. The latter will be true if and only if the stalk at each B-saturated homogeneous prime is exact.
Recall the saturation of an ideal I by B is There is a correspondence between B-saturated ideals of S and closed subschemes of X [CLS11, Proposition 6.A.7].
Throughout the paper we will often be concerned with the structure of the homogeneous B-saturated prime ideals of S, which form a proper subset of the homogeneous prime ideals of S. It will be important to see which prime ideals are B-saturated.
This proves the second part of the proposition.
In order to show the first part, it suffices to show that if P and Q are prime ideals so that P does not contain Q, then P : Q ∞ = P . Suppose s ∈ P : Q ∞ and let ∈ Z ≥0 be such that sQ ⊂ P . Since Q ⊂ P , there is a q ∈ Q − P . Thus q ∈ Q − P , but sq ∈ P so s ∈ P .
This first proposition says that every homogenous prime ideal of S is either B-saturated or irrelevant. The lemma below implies we need only consider prime ideals of small enough height.
Lemma 2.2. If I is a homogeneous ideal of codimension greater than dim X, then Proof. Homogeneous ideals that saturate to all of S are ideals that correspond to the empty subvariety of X. It is enough to show that if I corresponds to a nonempty subvariety of X, then the height of I is less than or equal to the dimension of X. Let d := dim X, c := codim I, and N be the number of variables of the polynomial ring S. Suppose x is in the subvariety of X corresponding to I. Considering the quotient construction of X, there is a torus G that acts on N -dimensional affine space A N . As G acts freely on A N \V (B) [CLS11, Exercise 5.1.11], the dimension of both G and the orbit Gx in Therefore, when considering homogeneous B-saturated primes of the Cox ring S of X, we need only consider the homogeneous primes of codimension at most the dimension of X that do not contain any prime components of the irrelevant ideal B.

When Complexes are Virtual Resolutions
In this section Theorem 1.3 is proved. The proof relies on a lemma relating the minimum of the depths of homogeneous localization of an ideal with depth of the ideal after saturation by the irrelevant ideal B. There is some care that must be taken in examining degree zero component of the localization of the ideal. Fortunately, Proposition 3.1 guarantees that every element of the localization can be written as the product of a unit and a degree zero element so S P and S (P ) may be viewed as "the same up to units." In order to prove the proposition, we use the combinatorial structure of the smooth toric variety and exploit a fact about determinants of two maps in a short exact sequence.
Let Σ be the complete fan of the smooth projective toric variety X, and σ denote a cone in Σ. Then as in [Cox95], let where x ρ is the variable of the Cox ring corresponding to the ray ρ ∈ Σ (see [CLS11, Section 5.2] for a more detailed exposition). The irrelevant ideal B is generated by the monomials xσ as σ ranges over the maximal cones of Σ.
Proposition 3.1. Let X be a smooth projective toric variety. Let σ be a maximal cone of the fan of X. The elements of Pic(X), {deg(x ρ )|ρ / ∈ σ} generate the Picard group.
Proof. Consider the short exact sequence where L is the lattice of characters of X and Div T N (X) is the group of torus invariant Weil divisors of X. Each of L, Div T N , and Pic(X) are free abelian groups. Here the matrix A contains the rays of Σ as rows. Since σ is a smooth cone, the determinant of the rows corresponding to the rays of σ is ±1. Number these rows i 1 , i 2 , ..., i r . Since the image of B is the cokernel of A, the determinant of B obtained by omitting columns i 1 , i 2 , ..., i r also has determinant ±1. The columns we have omitted correspond to rays in σ so omitting these columns corresponds to rays not in σ. Therefore, the degrees of the {x ρ } ρ / ∈σ generate Pic(X). Proof. First we claim depth(I P ) = depth(I (P ) ) for B-saturated homogeneous primes P . As P is B-saturated, it does not contain xσ for some maximal cone σ. Letting xσ = ρ / ∈σ x ρ , by Proposition 3.1, {deg(x ρ )} ρ / ∈σ generate Pic(X). So if f is a homogeneous element of S P , then there is a unit u ∈ S P so that f u ∈ S (P ) . As multiplying by a unit does not change whether or not an element is a zero divisor, this proves that depth(I P ) = depth(I (P ) ). Then as S is Cohen-Macaulay, Proof of Theorem 1.3. Suppose first that F • is a virtual resolution. Then as F • is exact, it is also true that (F • ) (P ) is exact for every B-saturated prime ideal P . In particular, it is true for the zero ideal, so (F • ) ( 0 ) is an exact sequence of vector spaces and condition (a) is satisfied. Further, by Lemma 3.2 depth(I(ϕ i ) : B ∞ ) = min{depth(I(ϕ i ) (P ) )|P is a B-saturated prime ideal of S}, and by Theorem 1.1 min{depth(I(ϕ) (P ) )|P is a B-saturated prime ideal of S} ≥ i.
Conversely, suppose conditions (a) and (b) are satisfied. We use a similar strategy and show that F • is exact by showing that (F • ) (P ) is exact for every B-saturated prime ideal P . Proposition 3.1 implies that after homogeneous localization, (F • ) (P ) has the same ranks as F • in every homological degree so condition (a) of Theorem 1.1 is satisfied. By Lemma 3.2, depth(I(ϕ i ) (P ) ) ≥ depth(I(ϕ) : B ∞ ) ≥ i, so condition (b) is also satisfied, proving the (F • ) (P ) is exact. Therefore F • is a virtual resolution.
Remark 3.3. The same proof as above with "homogenous" omitted everywhere shows that in the setup of a Cohen-Macaulay ring R, a radical ideal J, and a bounded free chain complex of R modules

Invariance of Saturated Fitting Ideals
Between this section and Section 5, we record the groundwork of tools for an alternative proof of the reverse direction of Theorem 1.3, using B-saturated Fitting ideals in the Cox ring S of a smooth projective toric variety X. Here, we present some facts about the B-saturation of Fitting ideals of an S-module. The Fitting ideals of a finitelygenerated module M over a Noetherian ring can be calculated by a free presentation Let r denote the rank of G. The ith Fitting ideal of M , Fitt i (M ), is defined to be I r−i (ϕ). This ideal is independent of the free presentation [Fit36].
We now adapt this idea of Fitting ideals to virtual presentations of a Pic(X)-graded S-module M , beginning with the invariance of the B-saturated Fitting ideals. We then produce further facts about saturated Fitting ideals that mirror the classical theory of Fitting ideals.   Proof. We may harmlessly assume that M is already B-saturated and that

Let a chain complex
is the truncation of the minimal free resolution of M . By replacing M with coker ϕ if necessary, we may also assume that The open cover will be by the generators {xσ} σ∈Σ of the irrelevant ideal. To see that this forms an open cover of X, begin by fixing a B-saturated prime P . We shall show that P ∈ D(xσ) for some σ. Notice P ∈ D(xσ) exactly when xσ / ∈ P . If xσ ∈ P for every σ ∈ Σ, then P contains the irrelevant ideal B and is therefore not B-saturated.
By Proposition 3.1, homogeneous localization by xσ preserves free modules and so where the last equality holds because taking Fitting ideals commutes with base change. Because localization is exact, and Proposition 3.1 implies that every element of F xσ is equal to a product of a unit and an element of F (xσ) , Fitting's Lemma [Fit36] implies I r−i (ϕ (xσ) ) = I r −i (ϕ (xσ) ).
Thus I r−i (ϕ) and I r −i (ϕ ) agree on an open cover of X, and glue in the same way so the sheaves are equal.
We will call I r−i (ϕ) : B ∞ the ith saturated Fitting ideal of M . Lemma 4.1 allows us to prove that the jth saturated Fitting ideal not equaling S is an obstruction to generating a module up to saturation by j elements. It is analogous to the fact that the jth Fitting ideal not equaling S is an obstruction to generating a module by j elements in the classical theory of Fitting ideals of Noetherian rings (see [Eis95,Proposition 20.6]).
Define V(I) to be the set of homogeneous B-saturated prime ideals P of S such that P ⊃ I. In particular,  As mentioned above, the previous proposition can be rephrased as saying that Fitt j (M ) is an obstruction to generating the sheaf M by j elements. One may hope that the converse is true as well, that is, if Fitt j (M ) : B ∞ = S, then there is a module N generated by j elements such that M ∼ = N . Unfortunately, this fails to be true as shown by the following example.
Then S/I has minimal resolution The ideal Fitt 2 (S/I) is generated by the entries of the matrix above, so However, there cannot be two homogenous polynomials of S whose intersection vanishes exactly at the above three points. For if f and g are homogeneous forms of degree (a, b) and (c, d) respectively, the multigraded version of Bézout's Theorem (see example 4.9 in [Sha77]) says that the intersection multiplicity of f and g is ad + bc. For concreteness, the leftmost vertical line is V (x 0 ), the middle vertical line is V (x 1 ), the topmost horizontal line is V (y 0 ) and the middle horizontal line is V (y 1 ). Then there are two cases for which ad + bc = 3.
Case 1: One of the terms (ad or bc) is 3 and one is 0. Without loss of generality, assume a = 3, d = 1. If c = 0, then deg(g) = (0, 1), which cannot vanish at all three points. Hence, b = 0 and deg(f ) = (3, 0). Plugging in x 0 = 0, x 1 = 1 gives a degree 0 form that must vanish at [0 : 1] and [1 : 0] which means f must vanish on all of V (x 0 ). Similarly, plugging in x 0 = 0, x 1 = 1 to g yields a degree 1 form that also vanishes at these two points, implying g vanishes on all of V (x 0 ). Therefore V (f ) ∩ V (g) = X.
Case 2: One of the terms is 2, and the other is equal to 1. It suffices to assume a = 2 and b = c = d = 1. Then both f and g restricted to V (x 0 ) are degree one polynomials in y 0 and y 1 that vanish at two points so again V (x 0 ) ⊂ V (f ) ∩ V (g). Therefore, no ideal generated by two elements will saturate to the ideal I.

Saturated Fitting Ideals and Locally Free Sheaves
In the classical theory of Fitting ideals, a module is projective if and only if its first nonzero Fitting ideal is the whole ring. Sheafifying a projective module yields a locally free sheaf. The situation is similar for modules over the Cox ring and saturated Fitting ideals. Proof. First, if M is locally free of rank r, then for any B-saturated prime P , M (P ) ∼ = M P is a free S (P ) -module of rank r. Since P is B-saturated, there is some xσ so that xσ / ∈ P , which implies M P = S P M (P ) . Therefore, M P has the free presentation  Thus ϕ P = 0, which implies N P ∼ = S r P and N (P ) ∼ = S r (P ) . Since each the stalk at each point of the sheaf N is free of rank r, this means N is locally free of rank r, as desired.
Notice that in everything up to this point, we have required that the complex be bounded. This may seem unsatisfying as Definition 1.2 does not require the length of a virtual resolution to be finite. The following result does not require this hypothesis, again mirroring the classical theory of Fitting ideals and exactness. Proof. The sequence is right exact and a free presentation of coker ϕ. Letting rank(G) = n and rank(ϕ) = r,  By Proposition 5.1, coker ϕ is a locally free sheaf of constant rank n−r. As localization is exact, homogeneous localization at a B-saturated prime P gives F (P ) → G (P ) → coker(ϕ (P ) ) → 0.
This proposition can be used in an alternate proof of the reverse direction of Theorem 1.3.
Proof. We prove the conditions (a) and (b) imply the complex is a virtual resolution.
It is enough to show for each B-saturated homogeneous prime ideal P of S, the complex (F • ) P is exact. This is because (F • ) P is a graded complex so if it is exact it will remain exact after taking the degree zero strand (F • ) (P ) . Since S is a polynomial ring, it is an integral domain, so each I(ϕ i ) contains a nonzerodivisor. Therefore, as localization commutes with taking Fitting ideals [Eis95, Corollary 20.5], the hypotheses are not weakened. We will show that ( F • ) P is exact by induction on the codimension of P . The unique minimal prime ideal of S is 0 , and localizing at this ideal gives a complex of vector spaces. The complex (F • ) 0 therefore becomes exact by assumption (a). Now suppose P is a B-saturated prime of codimension c > 0. By assumption (b), depth(I(ϕ c+1 ) : B ∞ ) ≥ c + 1, so I(ϕ c+1 ) : B ∞ is not contained in P . Therefore I(ϕ c+1 ) is also not contained in P . Indeed, if I(ϕ c+1 ) ⊂ P , then I(ϕ c+1 ) : B ∞ ⊂ P : B ∞ = P . Hence I(ϕ c+1 ) P = S P , so (F • ) P can be broken into two complexes: where ϕ c is induced from ϕ c . Now since I((ϕ i ) P ) = S P for every i ≥ c + 1, the complex (G • ) P is exact. All that is left to show is that (H • ) P is exact. Since I((ϕ c+1 ) P ) = S P , the S P -module (coker ϕ c+1 ) P is projective and hence free. Also notice that I((ϕ c ) P ) = I((ϕ c ) P ).
By induction, (H • ) P becomes exact when localizing at any homogeneous B-saturated prime ideal Q properly contained in P . So by Proposition 2.1, it becomes exact when localizing at any prime ideal contained in P (if a smaller prime contains a prime component of B, then P does as well, which means that P would not be B-saturated). Therefore the homology modules are only supported on the maximal ideal P P and thus have depth 0. The depth of each free S P -module is equal to the codimension of P (since S is Cohen-Macaulay), which is strictly positive. Therefore, applying the Peskine-Szpiro Acyclicity Lemma [PS73, Lemma 1.8] completes the proof.