1. Main result

We give a short new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties and their multigraded Cox rings. Throughout, we let X be a simplicial, projective toric variety over an algebraically closed field k with $\operatorname {Cl}(X)$ -graded Cox ring S. Our main result (Theorem 1.2) was first proven in [Reference Hanlon, Hicks and LazarevHHL], but our proof is independent from their methods. Our approach is more algebraic and simpler, while their approach is more explicit and connects to a wider range of topics, including symplectic geometry and homological mirror symmetry. See also the work of Favero-Huang [Reference Favero and HuangFH], which was completed simultaneously with [Reference Hanlon, Hicks and LazarevHHL] and whose main results coincide with some of Hanlon-Hicks-Lazarev’s.

Our interest in these topics begins with a program to extend results on syzygies to multigraded or toric settings. The basic perspective, introduced by Berkesch-Erman-Smith in [Reference Berkesch, Erman and SmithBES20], is that many classical results about minimal free resolutions will have strong analogues in the toric setting, as long as one replaces minimal free resolutions with the more flexible notion of a virtual resolution.

The following is a consequence of Hanlon-Hicks-Lazarev’s main result [Reference Hanlon, Hicks and LazarevHHL, Theorem A].

And here is our short proof of Theorem 1.2. The proof relies on some elementary facts about toric varieties that we recall in Lemma 2.2 below.

We now describe applications of Theorem 1.2 and their history. For a fuller discussion, see [Reference Hanlon, Hicks and LazarevHHL, §1]. We start with a special case, first proven by Hanlon-Hicks-Lazarev:

Special cases of Theorem 1.3 were studied in [Reference Brown and ErmanBE21, Reference Brown and SayrafiBS22, Reference CanonacoCan03], and [Reference Bayer, Popescu and SturmfelsBPS01, Reference AndersonAnd] study closely related questions. It was known that this result would immediately yield proofs of two conjectures that also had received independent interest. The first conjecture is due to Berkesch-Erman-Smith [Reference Berkesch, Erman and SmithBES20, Question 1.3] and was proven by Hanlon-Hicks-Lazarev:

Hilbert’s Syzygy Theorem gives a bound of $\dim S=\dim X + \operatorname {rank} \operatorname {Cl}(X)$ ; Theorem 1.4 implies that the added flexibility of virtual resolutions allows for significantly shorter resolutions, especially when $\operatorname {rank} \operatorname {Cl}(X)$ is large. See [Reference Berkesch, Erman and SmithBES20, Reference Harada, Nowroozi and Van TuylHNVT22, Reference Berkesch, Klein, Loper and YangBKLY21] and elsewhere for many examples of this phenomenon. Prior to [Reference Hanlon, Hicks and LazarevHHL], Theorem 1.4 had been proven in several special cases: when ${\operatorname {rank} \operatorname {Pic}(X)=1}$ , it essentially follows from Hilbert’s Syzygy Theorem; for products of projective spaces, it was shown in [Reference Berkesch, Erman and SmithBES20, Theorem 1.2] (see also [Reference Eisenbud, Erman and SchreyerEES15, Corollary 1.14]); Yang proved it for any monomial ideal in the Cox ring of a smooth toric variety [Reference YangYan21]; and Brown-Sayrafi proved it for smooth projective toric varieties of Picard rank 2 [Reference Brown and SayrafiBS22].

The second conjecture, due to Orlov, is the special case of [Reference OrlovOrl09, Conjecture 10] for toric varieties. This was first proven by Favero-Huang in [Reference Favero and HuangFH, Theorem 1.2], and independently and essentially simultaneously in [Reference Hanlon, Hicks and LazarevHHL, Corollary E].

Special cases of Theorem 1.5 had been established in [Reference Bai and CôtéBC23, Reference Ballard and FaveroBF12, Reference Ballard, Duncan and McFaddinBDM19, Reference Ballard, Favero and KatzarkovBFK19] before Favero-Huang and Hanlon-Hicks-Lazarev proved it in general. The full version of Orlov’s Conjecture states that Theorem 1.5 extends to any smooth quasi-projective variety; see [Reference Bai and CôtéBC23, §1.2] for a list of known cases of this conjecture.

Theorem 1.2 easily implies Theorems 1.3, 1.4 and 1.5. To prove Theorem 1.3, observe that the diagonal $X \subseteq X \times X$ satisfies the conditions of Theorem 1.2. To prove Theorem 1.4, one can simply follow the method of [Reference Berkesch, Erman and SmithBES20, Proof of Theorem 1.2]. For Theorem 1.5, one can use standard techniques on derived categories; see, for example, the proof of [Reference Hanlon, Hicks and LazarevHHL, Corollary E].

Our proof of Theorem 1.2 is quite simple, perhaps embarrassingly so given the prior partial results on these questions cited above. It is not yet clear how to compare our resolutions to those obtained in [Reference Hanlon, Hicks and LazarevHHL], but we believe that the two constructions agree in the case of Theorem 1.3. Their work gives a creative perspective on building these resolutions, drawing motivation from the symplectic side of the mirror symmetry functor and involving a wide array of ideas.Footnote ² The resolutions they obtain are quite explicit; indeed, their resolution of the diagonal yields a canonical generating set for the derived category of any normal toric variety, proving a claim of Bondal [Reference Hanlon, Hicks and LazarevHHL, Corollary D]. However, some algebraic aspects of their constructions are harder to determine. For instance, if $F_\bullet $ is the free complex of S-modules corresponding to one of their resolutions, their work implies that the modules $H_i(F_\bullet )$ correspond to the zero sheaf on X for all $i>0$ , but it is not clear whether $H_i(F_\bullet )$ equals the zero module on the nose (i.e., it is not clear if $F_\bullet $ is acyclic as a complex of S-modules). The S-module that arises as $H_0(F_\bullet )$ is also unclear. By comparison, the complexes that arise in our construction are always acyclic, and they resolve normalizations of coordinate rings. However, we are not able to give as explicit of a description of the terms. It would be very interesting to better compare these complexes, and to compare them with those in [Reference Brown and ErmanBE21, Reference Brown and SayrafiBS22]. Favero-Huang’s approach [Reference Favero and HuangFH] can almost certainly yield all of the above results as well, and it would be interesting to compare to those resolutions too.

2. Some elementary facts about toric varieties

Proof. Since $Y \hookrightarrow X$ is a toric morphism, it induces an embedding $T_Y \hookrightarrow T_X$ on tori and hence a surjection $p \colon M_X \twoheadrightarrow M_Y$ of lattices. Taking the pushout of the surjection p and the canonical map $M_X \to \mathbb {Z}^{\dim {S}}$ yields the morphism

(2.3)

of exact sequences. The abelian group $M'$ is isomorphic to $\mathbb {Z}^r \oplus A$ , where r is defined to be $\dim (S) - \dim (X) + \dim (Y)$ , and A is some finite abelian group. We observe that I coincides with the radical of ; note that $k[M']$ need not be reduced when $\operatorname {char}(k) \ne 0$ , since $M'$ may have torsion, and so J need not be radical. Let us verify that $I = \operatorname {rad}(J)$ : since p is surjective, the Snake Lemma implies that q is surjective, and so J is the defining ideal of the closure of $\operatorname {Spec}(k[M'])$ in $\operatorname {Spec}(S)$ . Diagram (2.3) induces the following morphism of short exact sequences of algebraic groups:

It follows that $\alpha ^{-1}(T_Y)$ is equal to the image of $\operatorname {Spec}(k[M'])$ in $\operatorname {Spec}(k[\mathbb {Z}^{\dim S}])$ . Since Z is equal to the closure of $\alpha ^{-1}(T_Y)$ in $\operatorname {Spec}(S)$ , we conclude that $I = \operatorname {rad}(J)$ .

Writing $R = k[\mathbb {Z}^r]$ and $A = \bigoplus _{i = 1}^t \mathbb {Z} / (n_i)$ , we have

$$ \begin{align*} k[M'] \cong R[z_1, \dots, z_t] /(z_1^{n_1} -1, \dots, z_t^{n_t} - 1). \end{align*} $$

The quotient of $k[M']$ by its nilradical is therefore a product of copies of R, and so I is a finite intersection of prime ideals arising as kernels of ring homomorphisms $S \to R$ . It therefore follows from [Reference Cox, Little and SchenckCLS, Proposition 1.1.8] that the irreducible components of Z are affine toric varieties of dimension r. If $\operatorname {Cl}(X)$ is torsion-free, then the bottom row of Diagram (2.3) splits, and so $A = 0$ , which means I is prime. This proves (1). As for (2), we have shown that $\dim (Z) = r$ , which is precisely $\dim (S) - \operatorname {codim}(Y \subseteq X)$ .

3. Examples

Example 3.2. Let X be the weighted projective space $\mathbb P(1,1,2)$ and T the Cox ring $k[x_0,x_1,x_2,y_0,y_1,y_2]$ of $X\times X$ . By a calculation similar to Example 3.1, the ring $T/I_\Delta $ is isomorphic to the semigroup ring $ k[x_0,x_1,x_2,tx_0,tx_1,t^2x_2], $ which is not normal because $tx_2$ lies in the fraction field and satisfies the integral equation $(tx_2)^2 - x_2 \cdot (t^2x_2)=0$ . Let R be the normalization of $T/I_\Delta $ . A presentation matrix for R as a T-module is given as follows, where the rows correspond to the generators $1$ and $tx_2$ :

The free resolution of R as a T-module is given by

(3.3) $$ \begin{align} \begin{matrix} T \\ \oplus\\ T(-1,-1)\end{matrix} \xleftarrow{\left[\begin{smallmatrix} x_{1}y_{0}-x_{0}y_{1}&x_{2}y_{0}&x_{2}y_{1}&x_{0}y_{2}&x_{1}y_{2}\\ 0&-x_{0}&-x_{1}&-y_{0}&-y_{1} \end{smallmatrix}\right]} \begin{matrix} T(-1,-1) \\ \oplus\\ T(-2,-1)^2 \\ \oplus \\ T(-1,-2)^2 \end{matrix} \xleftarrow{\left[\begin{smallmatrix} -x_{2}&0&-y_{2}\\ x_{1}&-y_{1}&0\\ -x_{0}&y_{0}&0\\ 0&-x_{1}&-y_{1}\\ 0&x_{0}&y_{0} \end{smallmatrix}\right]} \begin{matrix} T(-3,-1) \\ \oplus\\ T(-2,-2)\\ \oplus \\ T(-1,-3) \end{matrix} \gets 0. \end{align} $$

Additionally, we have the short exact sequence $ 0\to T/I_\Delta \to R \to Q \to 0, $ and $Q = tx_2\cdot k[x_2,y_2]$ . One can directly compute that the sheaf $\widetilde {Q}$ corresponding to Q is the zero sheaf on $X \times X$ . In fact, since Q is annihilated by $x_0,x_1,y_0$ and $y_1$ , we can reduce to checking that $\widetilde {Q}$ is also zero on the affine patch $D(x_2y_2)$ . The global sections of $\widetilde {Q}$ on this patch are $Q[x_2^{-1},y_2^{-1}]_{(0,0)}=0$ , and thus $\widetilde {Q}=0$ , as desired.