2.1 Some resolutions of toric subvarieties in $\mathbb A^2$

We work over an algebraically closed field ${\mathbb K}$ with characteristic $p\geq 0$ . We start with some motivating examples and nonexamples. We denote by $\mathbb T^k$ the (split) algebraic torus of dimension k. The examples are chosen to highlight the strategy of Theorem 3.5 of reducing resolutions of toric subvarieties to resolutions of points in a quotient stack.

Example 2.1 (Resolving $z_0=z_1$ ).

Take the line $\phi \colon \mathbb A^1\to \mathbb A^2 $ parameterized by $z\mapsto (z, z)$ . Let $i \colon \mathbb A^2\setminus \{(0,0)\}\to \mathbb A^2$ be the inclusion and let $\mathbb T^1= \mathbb A^1 \setminus \{0\}$ . The relevant maps of fans are drawn on the left-hand side of Figure 3. We have a projection $\pi \colon \mathbb A^2\setminus \{(0,0)\}\to \mathbb P^1 $ . We give $\mathbb P^1$ the coordinates $[x_0: x_1]$ . Let e denote the toric variety which is a single point, and let $\phi _e \colon e\to \mathbb P^1 $ be the inclusion whose image is $[1:1]\in \mathbb P^1$ – that is, the origin of the algebraic torus. This inclusion is presented on the right-hand side of Figure 3. We have that $\pi ^{-1}(e)=\mathbb T^1$ .

Figure 3 Reducing the resolution of a line in $\mathbb A^2$ to the resolution of a point in $\mathbb P^1$ . Compare to Figure 2.

Now, something a bit unusual happens. Let $C_{\bullet }(S^{\phi _e}, {\mathcal O}^{\phi _e})$ be the resolution of $(\phi _*){\mathcal O}_e$ on $\mathbb P^2$ given by the complex ${\mathcal O}_{\mathbb P}(-1)\xrightarrow {x_0-x_1}{\mathcal O}_{\mathbb P}\dashrightarrow \mathcal \phi ^{\prime }_*O_e$ . This resolution can be naturally ‘drawn’ on the real torus $M_{\mathbb R}/M$ by

(3)

This resolution of the point comes from Theorem 3.1. To see that this is indeed the desired resolution, we can restrict to smooth charts by specializing to $x_0=1$ or $x_1=1$ , where it becomes the Koszul resolution of the point $1 \in \mathbb A^1$ . See also Example 3.11.

Now pull Equation (3) along $\pi $ , and push forward along i. This need not be an exact resolution (as $i_*$ is not exact) of ${\mathcal O}_{\mathbb A_1}$ (as $i_*(\phi _{\mathbb T^1}){\mathcal O}_{\mathbb T^1}$ is not $\phi _*{\mathcal O}_{\mathbb A^1}$ ) by line bundles (as pushforward, in general, does not send line bundles to line bundles). A priori, we do not even know if $i_*\pi ^{*}C_{\bullet }(S^{\phi _e}, {\mathcal O}^{\phi _e})$ is a complex of coherent sheaves (as i is not proper). Disregarding all these red flags, one can compute that $i_*\pi ^{*}C_{\bullet }(S^{\phi _e}, {\mathcal O}^{\phi _e})$ is the following complex:

which matches the usual resolution of $\phi _*{\mathcal O}_{\mathbb A}$ as the structure sheaf of a hypersurface in $\mathbb A^2$ . We would like to highlight that two unusual things are happening here. Namely, the pushforward $i_*$ is not exact, but it happens to preserve exactness of our resolution and resolve the desired sheaf. Additionally, since the inclusion i has complement of codimension two, $i_*$ sends line bundles to line bundles (Hartog’s principle). In general, $i_*$ will not satisfy the second property, and so we will have to address this issue.

We now provide an example that looks similar to the one before in that we reduce resolving a subvariety to resolving a point on a quotient, but it is not a special case of our main theorem.

Example 2.2 (Resolving $z_0^2=z_1$ ).

We now describe a resolution of the parabola, which does not quite fit into our proof strategy. Take the parabola $\phi \colon \mathbb A\hookrightarrow \mathbb A^2 $ parameterized by $z\mapsto (z, z^2)$ . To construct the map $\pi $ , we observe that $\mathbb T^1 = \mathbb A\setminus \{0\}\hookrightarrow \mathbb A^2$ is the inclusion of a subgroup of the algebraic torus. We therefore consider the inclusion $i\colon \mathbb A^2\setminus \{0\}\to \mathbb A^2$ . The maps of fans are drawn in Figure 4.

Figure 4 Reducing the resolution of a parabola to the resolution of a point.

We now mimic the previous example. There is a projection

$$ \begin{align*} \pi \colon \mathbb A^2\setminus \{0\}\to \mathbb P(1, 2) & & (z_0, z_1)\mapsto [z_0: z_1]_{(1,2)}. \end{align*} $$

Then, the identity in the algebraic torus $e=[1:1]\in \mathbb P(1, 2)$ satisfies the property that $\pi ^{-1}(e)=\mathbb T^1$ . We now must produce a resolution of $(\phi _e)_*{\mathcal O}_e$ . This is not one of the examples covered in Theorem 3.1, which cover resolving points in smooth toric varieties. For this particular example, we happen to know a resolution $C_{\bullet }$ of e given by

As before, disregard all warnings and compute $i_*\pi ^{*} C_{\bullet }$ to obtain

the usual resolution of $\phi _*{\mathcal O}_{\mathbb A}$ .

In both examples, the first step – producing the map $\pi $ – is clear. We restrict the subvariety to the open torus which is a subgroup of the larger torus. Then, we study a point in the quotient by that subgroup. However, the next step – producing a resolution of that point – can be difficult in general as both the quotient space and morphism $\pi $ may not be particularly nice. In particular, the strategy of restricting to smooth charts in Figure 3 cannot be immediately applied to Example 2.2.

Our approach is to instead take the stack quotient. The stack $[\mathbb A^2\setminus \{0\}/\mathbb T^1]$ , where $t \in \mathbb T^1$ acts by $t\cdot (z_1, z_2)=(tz_1, t^2z_2)$ , is a smooth stack and can be covered with smooth stacky charts. The resolution of a point on this toric stack is covered by Theorem 3.1', which we work out in Example 2.42. When taking this approach, we obtain the following resolution for the parabola:

which is easily seen to be homotopy equivalent to the one produced in Example 2.2. This new resolution also has the nice property that all the structure coefficients of the differential are given by primitive monomials, which was not the case with the previous resolution.

2.2 Line bundles on toric stacks

Before going further with resolutions, we will discuss aspects of the theory of toric stacks that we will use. There are several notions of toric stacks in the literature. We will use toric stacks in the sense of [Reference Geraschenko and SatrianoGS15] on which the following exposition is based.

Suppose we have a short exact sequence of groups $0\to M'\xrightarrow {\underline \phi } M\to C_\phi \to 0$ . Applying Cartier duality gives us $0\to G_\phi \to \mathbb T_{N'}\to \mathbb T_{N}\to 0$ , so that $G_\phi = \widehat {C_\phi }$ . Since Cartier duality is a duality, we also have $C_{\phi }=\widehat {G_{\phi }}$ . When $G_{\phi }$ is a finite group, its Cartier dual is isomorphic to its group of characters giving us the identification

(4) $$ \begin{align} \operatorname{\mathrm{coker}}(\underline\phi^{*}\colon M'\to M) =\hom_{gp}(G_{\phi}, \mathbb G_m). \end{align} $$

A stacky fan Footnote ³ is a pair $(\Sigma , \underline {\beta } \colon L\to N)$ , where $\Sigma $ is a fan on L, and $\underline {\beta }$ is a morphism of lattices with finite cokernel. The fan $\Sigma $ gives us a toric variety $X_\Sigma $ . As $\underline \beta $ has finite cokernel, the induced map $\mathbb T_{\beta } \colon \mathbb T_L\to \mathbb T_N$ is surjective. The toric stack associated with a stacky fan is the quotient stack ${\mathcal X}_{\Sigma , \underline {\beta }}:=[X_\Sigma /G_\beta ]$ . We will frequently drop the subscripts and write ${\mathcal X}$ or $\mathcal Y$ for a toric stack.

Example 2.5 (Projective line).

The projective line can be presented as a toric stack as in Figure 5. That is, we consider the fan $\Sigma $ on $L_{\mathbb R}=\mathbb R^2$ whose cones are $\{0, \mathbb R\cdot e_1, \mathbb R\cdot e_2\}$ so that $X_\Sigma = \mathbb A^2\setminus \{0\}$ . Let $\underline \beta \colon L \to N=\mathbb Z $ be the projection $\begin {pmatrix} 1 & -1 \end {pmatrix}$ . $G_\beta = \mathbb T^1$ , and it acts by $g\cdot (z_1, z_2)= (gz_1, gz_2)$ . This defines a toric stack. Since the action is free, we can identify ${\mathcal X}_{\Sigma , \beta }=X_\Sigma /G_\beta = \mathbb P^1$ .

Figure 5 A presentation of $\mathbb P^1$ as a toric stack.

Let $(\Sigma ,\underline {\beta }\colon L\to N)$ and $(\Sigma ', \underline {\beta }^\circ \colon L'\to N')$ be two stacky fans. A morphism of stacky fans $(\underline \phi ,\underline \Phi )\colon (\Sigma ', \underline {\beta }^\circ )\to (\Sigma , \underline {\beta })$ is a pair of lattice morphisms making the following diagram commute:

Additionally, we ask that $\underline \Phi $ be a morphism of fans (so that for all $\sigma '\in \Sigma '$ , there exists $\sigma \in \Sigma $ so that $\underline \Phi (\sigma ')\subset \sigma $ ). A morphism of stacky fans induces a toric morphism of the corresponding toric stacks which we denote by $\phi \colon {\mathcal X}_{\Sigma ', \underline {\beta }'}\to {\mathcal X}_{\Sigma , \underline {\beta }}$ . More descriptively, we obtain from this data a group homomorphism $f_\phi \colon G_\beta \to G_\beta '$ such that the map of toric varieties $\Phi \colon X_\Sigma \to X_{\Sigma ,'}$ is an $f_\phi $ -map (see Appendix B).

We will be interested in line bundles on ${\mathcal X}_{\Sigma , \underline {\beta }}$ that can be described by a line bundle on $X_{\Sigma }$ equipped with a $G_\beta $ action. The sections of a line bundle on ${\mathcal X}_{\Sigma , \underline {\beta }}$ are the $G_\beta $ -equivariant sections of the respective bundle on $X_\Sigma $ .

We first give a short and standard description of line bundles on toric varieties in preparation for discussing line bundles on toric stacks. A torus invariant Cartier divisor on $X_\Sigma $ is uniquely determined by a support function $F \colon |\Sigma | \to \mathbb Z $ , which is a continuous function on the support of the fan $|\Sigma | \subset L_{\mathbb R}$ whose restriction to every $\sigma \in \Sigma $ is an integral linear function. Namely, the divisor associated to F is $D_F =\sum _{\rho \in \Sigma (1)} -F(u_{\rho }) D_{\rho }$ , where $u_{\rho }$ is the primitive generator of $\rho $ and $D_{\rho }$ the orbit closure associated to $\rho $ . Thus, a support function determines a line bundle $\mathcal {O}_{X_\Sigma }(F) := \mathcal {O}_{X_\Sigma }(D_F)$ , and, in fact, every line bundle on $X_\Sigma $ is isomorphic to a line bundle associated to a torus invariant divisor. Two support functions determine isomorphic line bundles when their difference is a linear function.

We can also construct a fan on $L\oplus \mathbb Z$ from the data of a support function, whose cones are

$$\begin{align*}\Sigma_F:=\{(\sigma, F(\sigma))\;\operatorname{\mbox{ | }}\; \sigma\in \Sigma\}\cup \{\langle (0, 1) , (\sigma, F(\sigma))\rangle\;\operatorname{\mbox{ | }}\; \sigma\in \Sigma\}.\end{align*}$$

There is a map of fans $\underline {\Pi } \colon \Sigma _{F}\to \Sigma $ from projection onto the first factor. The toric variety $X_{\Sigma _F}$ is the total space of the corresponding line bundle over $X_{\Sigma }$ .

The sections of $\Sigma _F\to \Sigma $ are generated by the toric monomial sections. Each of these can be expressed as a map of fans $\underline S \colon \Sigma \to \Sigma _F $ so that $\underline {\Pi }\circ \underline S=\underline{\mathrm {id}}_\Sigma $ (i.e., linear maps $k\colon L\to \mathbb Z$ with the property that $k(u_{\rho })\geq F(u_{\rho })$ for every $\rho \in \Sigma (1)$ ). Typically, the monomial sections are identified with the points of the lattice polytope

$$\begin{align*}\Delta(F):=\{k\in K \;\operatorname{\mbox{ | }}\; \forall \rho\in \Sigma(1), k(u_{\rho})\geq F(u_{\rho})\}. \end{align*}$$

A linear map inducing an isomorphism of line bundles can also be seen as an isomorphism of fans intertwining with the projection $\underline {\Pi }$ .

Example 2.7 (Line bundles on $\mathbb P^1$ ).

Let $X_\Sigma = \mathbb P^1$ with complete fan generated by $u_{\rho _0}= -1$ and $u_{\rho _1}= 1$ . Consider the support function F determined by

$$ \begin{align*} F(u_{\rho_0})= 0, F(u_{\rho_1})=-3 \end{align*} $$

so that $\mathcal {O}_{\mathbb P^1}(F)\simeq \mathcal {O}_{\mathbb P^1}(3)$ . We have drawn the fans $\Sigma $ and $\Sigma _{F}$ in Figure 6.

Figure 6 The total space of the line bundle ${\mathcal O}_{\mathbb P^1}(3)$ . The three equivariant sections of this map are labeled by dashed lines, which are also described by the polygon on the right-hand side.

The dashed lines are the images of the sections $\underline S\colon \Sigma \to \Sigma _{F}$ . These are in bijection with the black diamonds on the right-hand side of the figure, which are the lattice points in the polytope $\Delta (F)=\{k\in K \operatorname {\mbox { | }} k(u_{\rho _0})\geq 0, k(u_{\rho _1}) \geq -3 \}$ .

We now apply this construction to toric stacks. As before, to each support function $F \colon |\Sigma | \to \mathbb R,$ we can associate the fan $\Sigma _F$ on $L\oplus \mathbb Z$ . We define $\underline {\beta }_F:=\underline {\beta }\oplus \underline{\mathrm {id}} \colon L\oplus \mathbb Z\to N\oplus \mathbb Z $ .

The projection $\underline {\Pi } \colon \Sigma _F\to \Sigma $ extends to a morphism of stacky fans $(\underline \pi , \underline {\Pi })\colon (\Sigma _F, \underline {\beta }_F)\to (\Sigma , \underline {\beta })$ in the obvious way. The total space ${\mathcal X}_{\Sigma _F, \underline {\beta }_{F}}$ now comes with a $G_\beta $ action, making the sheaf of sections of $\pi \colon {\mathcal X}_{\Sigma _F, \underline {\beta }_{F}}\to {\mathcal X}_{\Sigma }$ a $G_\beta $ -equivariant sheaf. We write ${\mathcal O}_{{\mathcal X}_{\Sigma , \beta }}(F)$ for this sheaf.

A different way to understand this sheaf is to describe the $G_\beta $ action on fibers (see Appendix B for notation). Let $(\Sigma , \underline \beta )$ be a stacky fan containing a saturated (not necessarily top dimensional) smooth cone $\sigma $ of $\dim (N)$ with the property that $\underline \beta |_{\sigma }$ is injective. Let $\Sigma _{\sigma }$ be the fan of cones subordinate to $\sigma $ , so we have an open immersion of stacky fans $(\underline i, \underline I)\colon (\Sigma _{\sigma }, \underline \beta )\to (\Sigma , \underline {\beta })$ .

Let $L'= L/\ker (\underline \beta ), N'=N$ and $\underline \beta ' \colon L'\to N' $ , and let $\Sigma ^{\prime }_\sigma $ be the image of $\Sigma _\sigma $ in the quotient. This gives us a stacky fan $(\Sigma _\sigma ', \underline {\beta }')$ . The map $(\underline \pi , \underline {\Pi }):(\Sigma _\sigma , \underline \beta )\to (\Sigma _\sigma ', \underline \beta ')$ induces an equivalence of stacks ${\mathcal X}_{\Sigma _\sigma , \beta }=[\mathbb A^{|\sigma |}/G_{\beta '}]$ .

Let F be a support function $\Sigma $ . Then, ${\mathcal O}_{\mathbb A^{|\sigma |}}(i^{*}F)$ has a canonical trivialization given by the linear map $k':L' \to \mathbb Z$ satisfying $k'|_{\underline {\Pi }(\sigma )}=F|_{\underline {\Pi }(\sigma )},$ whose section $s_k\in \Gamma ({\mathcal O}_{\mathbb A^{|\sigma |}}(i^{*}F))$ we declare to be the constant e-section. In coordinates on $\mathbb T_L$ , the section is parameterized by

$$ \begin{align*} s_{k'}:\mathbb T_{L'}\mapsto \mathbb T_{L'}\times \mathbb G_m\\ x \mapsto (x, \chi^{k'}(x)). \end{align*} $$

We now prove that with respect to this trivialization, the $G_{\beta '}$ action on the sheaf ${\mathcal O}_{\mathbb A^{|\sigma |}}(i^{*}F)$ is given by multiplication on the fibers by the character $\chi ^{k'}\in \hom (G_{\beta '}, \mathbb G_m)$ .

Let $x_0$ denote the origin in $\mathbb A^{|\sigma |}$ , which is fixed on the $G_{\beta '}$ -action on $\mathbb A^{|\sigma |}$ . For all $g\in G_{\beta }$ , we compare

(5) $$ \begin{align} s_{k'}(g\cdot x_0) = (x_0, \chi^{k'}(x_0)) && g\cdot s_{k'} (x_0 )= (x_0,\chi^{k'}(g)\cdot \chi^{k'}(x_0)) \end{align} $$

It follows that the $G_\beta $ action on the sections of $\mathbb T_{L'}\times \mathbb G_m\to \mathbb T_{L'}$ in this trivialization is given multiplication by the character $\chi ^{k'}(g)$ .

In this fashion, F defines not only a line bundle ${\mathcal O}_{X_\Sigma }(F)$ but also a $G_\beta $ action on the sheaf, making it a $G_\beta $ -equivariant sheaf. We define ${\mathcal O}_{{\mathcal X}_{\Sigma , \underline {\beta }}}(F)$ to be the sheaf of $G_{\beta }$ -equivariant sections.

Among these $G_\beta $ -equivariant sections are those which are additionally monomial sections. Each monomial section corresponds to a map of fans $\underline S \colon \Sigma \to \Sigma _F $ that is a section and extends to a map of stacky fans $(\underline s, \underline S)\colon (\Sigma , \underline {\beta })\to (\Sigma _F, \underline {\beta }_F)$ . The set of such sections is

$$\begin{align*}\Delta_\beta(F):=\{k\in K \;\operatorname{\mbox{ | }} \;\forall \rho\in \Sigma, k(u_{\rho}) \geq F(u_{\rho}) \text{ and there exists } m\in M \text{ with } k=\underline{\beta}^{*} m)\} \end{align*}$$

and can also be written as

$$\begin{align*}\Delta_\beta(F)=\{m\in M \;\operatorname{\mbox{ | }}\; \forall \rho\in \Sigma, \underline{\beta}^{*}m( u_{\rho} )\geq F(u_{\rho})\}\end{align*}$$

since $\underline \beta ^{*}$ is injective. This is a subset of $\Delta (F)$ (reflecting that all $G_\beta $ -equivariant monomial sections are monomial sections, but not all monomial sections are $G_\beta $ -equivariant).

We base our notation for toric line bundles on support functions (as opposed to toric divisors) for several expositional reasons. One reason for using the support function notation is that it allows us to compare different $G_\beta $ actions on ${\mathcal O}_{X_\Sigma }(F)$ . Since this is a line bundle, the set of $G_\beta $ -structures is a torsor of over the character group of $G_\beta $ . Moreover, the action of $\hom (G_\beta , \mathbb G_m)$ can be understood via its identification with $\operatorname {\mathrm {coker}}(\underline {\beta })$ and Equation (5). In summary,

Proposition 2.9. Let F be a support function for $\Sigma $ , and $\underline {\beta } \colon L\to N$ . Let $\chi ^{(-)}: K\to \hom (G_\beta , \mathbb G_m)$ be the identification from Equation (4). For any $k\in K$ , we have a canonical isomorphism

$$\begin{align*}{\mathcal O}_{{\mathcal X}_{\Sigma, \beta}}(F+k) = {\mathcal O}_{{\mathcal X}_{\Sigma, \beta}}(F)_{\chi^{k}},\end{align*}$$

where the latter bundle is the bundle ${\mathcal O}_{{\mathcal X}_{\Sigma , \beta }}(F)$ where the $G_{\beta }$ action has been twisted by multiplication by $\chi ^k(g)$ on the fibers.

Secondly, some functoriality properties of line bundles are easier to state in the language of support functions. For example,

Proof. Recall that the line bundle ${\mathcal O}_{{\mathcal X}}(F)$ is a line bundle ${\mathcal O}_{X}(F)$ on X with a $G_\beta $ action, and the map $\Phi \colon X'\to X$ is $f_\phi \colon G_{\beta '}\to G_{\beta }$ equivariant. Given a $G_{\beta }$ -sheaf on ${\mathcal X}$ , the pullback sheaf naturally inherits a $G_{\beta '}$ action via pushforward of the $G_{\beta }$ action via the homomorphism $f_\phi :G_\beta \to G_{\beta '}$ . It is a classical fact on toric varieties that $\Phi ^{*}{\mathcal O}_X(F)\cong {\mathcal O}_{X'}(\underline \Phi ^{*} F)$ .

We need to also check that the $G_{\beta }$ action on $ {\mathcal O}_{X'}(\underline \Phi ^{*} F)$ arises from the pushforward $f_\phi $ . From Proposition 2.9, we have the constant e section over a top dimensional cone of $X'$ which is parameterized by the section $s^{\prime }_{k'}(x')=(x', \chi ^{k'}(x'))$ . We have that the $G_{\beta '}$ action on this section is given by multiplication with character $\chi ^{k'}(g')$ .

Let $k=\underline \Phi ^{*}(k')$ ; then $s_k$ is the constant e section. The $G_\beta $ action on $s_k$ is multiplication by the pullback character

$$ \begin{align*} \chi^{k'}(\mathbb T_\Phi g)=\mathbb T_\Phi^{*}\chi^{k'}(g)=\chi^{\underline \Phi^{*} k'}(g)=\chi^k(g), \end{align*} $$

where the second equality uses that Cartier duality is a contravariant functor. This matches the $G_\beta $ action on ${\mathcal O}_{{\mathcal X}_{\Sigma ,'}}(\underline \Phi ^{*} F)$ .

Finally, some of the intuition for our constructions come from homological mirror symmetry (Section 1.2), where support functions make an appearance in the form of a certain Hamiltonian function on the mirror space.

Example 2.11 (Orbifold line).

We describe $[\mathbb A^1/(\mathbb Z/2\mathbb Z)]$ as a toric stack, its line bundles, and a resolution of the point e in this stack. We take $L=N=\mathbb Z$ , $\underline {\beta }:L\xrightarrow {(2)} N$ , and $\Sigma =\{0, \langle e_1\rangle \}$ . Then, $X_{\Sigma }=\mathbb A^1$ , and the toric stack ${\mathcal X}_{\Sigma , \underline {\beta }}$ is the orbifold line $[\mathbb A^1/(\mathbb Z/2\mathbb Z)]$ .

A support function F on L is determined by its value on $e_1$ , so let $F_a$ be the support function such that $F_{a}(e_1)=a$ . On $X_{\Sigma }$ , the line bundles ${\mathcal O}(F_a)$ are all isomorphic. However, up to isomorphism, there are two line bundles on ${\mathcal X}_{\Sigma , \underline {\beta }}$ . In particular, observe that when $G: N\to \mathbb Z$ is a linear function, $\underline {\beta }^{*} \colon L\to \mathbb Z $ takes values in $2\mathbb Z$ ; therefore, $F_1$ is not linearly equivalent to $F_0$ . More concretely,

• • The sheaf ${\mathcal O}_{{\mathcal X}_{\Sigma , \underline {\beta }}}$ corresponds to the $\mathbb Z/2\mathbb Z$ -invariant sections of ${\mathcal O}_{\mathbb A^1}$ . Under identification of ${\mathcal O}_{\mathbb A^1}$ with ${\mathbb K}[x]$ , these sections correspond to the even degree polynomials.

• • The second sheaf ${\mathcal O}_{{\mathcal X}_{\Sigma , \underline {\beta }}}(F_1)$ has sections corresponding to the $\mathbb Z/2\mathbb Z$ anti-invariant sections of ${\mathcal O}_{\mathbb A^1}$ (i.e., the odd degree polynomials (where the action of $G_\beta $ on sections is multiplication by $-1$ )).

A specific example with sections given by maps of fans, and their corresponding points in the lattice polytope is drawn in Figure 7.

Figure 7 Sections of a line bundle on $[\mathbb A/(\mathbb Z/2\mathbb Z)]$ where the support function takes the value $F(u_{\rho _1 })=-1$ . On the left-hand side, the dashed lines represent the (not necessarily $\mathbb Z/2\mathbb Z$ equivariant) sections of the bundle ${\mathcal O}_{\mathbb A}(F)$ . The green dashed lines represent the $\mathbb Z/2\mathbb Z$ equivariant sections.

2.3 Thomsen collection

In this section, we describe the set of line bundles from which our resolutions will be built. The construction of these line bundles is based on [Reference Bondal, Altmann, Batyrev and TeissierBon06]. We can assign to any $m \in M_{\mathbb R}$ a $\mathbb T_N$ -invariant divisor on $X_\Sigma $ by

(6) $$ \begin{align} m \mapsto \sum_{\rho \in \Sigma(1)} \left\lfloor - \underline \beta^{*} m(u_{\rho}) \right\rfloor D_{\rho}, \end{align} $$

where $u_{\rho }\in L$ is the primitive generator of the corresponding ray $\rho \in \Sigma (1)$ and $\lfloor \cdot \rfloor $ is the floor function.

For the remainder of this section, we will assume that $\Sigma $ is smooth. Then, we can identify the set of $\mathbb T_N$ invariant divisors on $X_\Sigma $ with the set of support functions SF $(\Sigma )$ on $\Sigma $ . Under that identification, Equation (6) becomes a map

$$ \begin{align*}\textbf{F} \colon M_{\mathbb R} \to \text{ SF}(\Sigma)\end{align*} $$

given by

$$ \begin{align*}\textbf{F}(m)(u_{\rho}) = \left\lceil \underline{\beta}^{*}m(u_{\rho}) \right\rceil\end{align*} $$

for all $\rho \in \Sigma (1)$ and where $\lceil \cdot \rceil $ is the ceiling function. Observe that $\textbf {F}(m_{\mathbb R})$ and $\textbf {F}(m_{\mathbb R}+m_{\mathbb Z})$ are linearly equivalent support functions for any $m_{\mathbb Z}\in M$ . As a result, we obtain a map

(7) $$ \begin{align} {\mathcal O}(\textbf{F}(-))\colon M_{\mathbb R}/M \to \operatorname{\mathrm{Pic}}({\mathcal X}_{\Sigma, \underline{\beta}}), \end{align} $$

where $\operatorname {\mathrm {Pic}}({\mathcal X}_{\Sigma , \underline \beta })$ is the Picard group. We call the image of this map the Thomsen collection. This nomenclature is explained by Section 5.

We can describe the regions $\textbf {F}^{-1}(F)\subset M_{\mathbb R}$ via the hyperplane arrangement $\{\langle m, \beta (u_{\rho })\rangle \in \mathbb Z\}_{\rho \in \Sigma (1)}$ . When drawing the stratification, we label each hyperplane with small ‘hairs’ indicating the direction of $\beta (u_{\rho })$ . The hyperplane arrangement induces a stratification $S_{\Sigma , \underline {\beta }}$ of $M_{\mathbb R}$ , and each stratum $\sigma \in S_{\Sigma , \underline \beta }$ is labeled by a line bundle on ${\mathcal X}_{\Sigma , \underline {\beta }}$ via $\textbf {F}$ . We call the associated line bundle ${\mathcal O}_{{\mathcal X}_{\Sigma , \underline {\beta }}}(\sigma )$ . Note that if $\tau $ is codimension 1 boundary component of $\sigma $ , and there are no ‘hairs’ pointing from $\tau $ into $\sigma $ , that ${\mathcal O}_{{\mathcal X}_{\Sigma , \underline {\beta }}}(\sigma )={\mathcal O}_{{\mathcal X}_{\Sigma , \underline {\beta }}}(\tau )$ .

The stratification is M-periodic, so we obtain an induced stratification on $M_{\mathbb R}/M$ . This stratification was originally studied in [Reference Bondal, Altmann, Batyrev and TeissierBon06].

Example 2.13 (Beilinson collection).

On $\mathbb P^n$ , the Thomsen collection coincides with the Beilinson collection. In Figure 8, we draw the stratification of $M_{\mathbb R}$ given by $\textbf {F}$ for $n = 2$ .

Figure 8 The stratification $S_\Sigma $ and Thomsen collection for $\mathbb P^2$ . The fundamental domain for the torus $M_{\mathbb R}/M_{\mathbb Z}$ is outlined by the dotted line.

A fundamental domain for the torus $M_{\mathbb R}/M_{\mathbb Z}$ is marked out by the dashed lines. Observe that every line bundle in the figure is isomorphic to one inside the fundamental domain as claimed. In particular, the line bundles which appear in the Thomsen collection are ${\mathcal O}, {\mathcal O}(-1)$ and ${\mathcal O}(-2)$ .

We can also produce a natural set of morphisms of line bundles in the Thomsen collection using the stratification $S_{\Sigma , \underline {\beta }}$ . A version of the following statement appears in [Reference Favero and HuangFH22, Proposition 5.5].

Proof. The claim is equivalent to showing that $\textbf {F}(m_1)(u_{\rho }) \geq \textbf {F}(m_2)(u_{\rho })$ for any $m_1 \in \sigma $ , $m_2 \in \tau $ and all $\rho \in \Sigma (1)$ (since $0$ is clearly in the image of $\underline \beta ^{*} \colon M\to K $ ). Observe that for all $\rho \in \Sigma (1)$ and $m_1 \in \sigma $ , we have

$$\begin{align*}\textbf{F}(m_1)(u_{\rho}) - 1 < m_1(u_{\rho}) \leq \textbf{F}(m_1)(u_{\rho}). \end{align*}$$

Thus, if $\tau < \sigma $ and $m_2 \in \tau $ , we have

$$\begin{align*}\textbf{F}(m_1)(u_{\rho}) - 1 \leq m_2(u_{\rho}) \leq \textbf{F}(m_1)(u_{\rho}), \end{align*}$$

and, therefore,

$$\begin{align*}\textbf{F}(m_2) (u_{\rho}) = \lceil m_2(u_{\rho}) \rceil \leq \textbf{F}(m_1)(u_{\rho}), \end{align*}$$

as required.

Definition 2.15. For all $\sigma , \tau \in S_{\Sigma , \underline \beta }$ with $\tau < \sigma $ , the boundary morphism

$$ \begin{align*}\delta_{\sigma,\tau}\in \hom({\mathcal O}_{\mathcal X}(\textbf{F}(\sigma), {\mathcal O}_{\mathcal X}(\textbf{F}(\tau))\end{align*} $$

is the morphism corresponding to $0 \in \Delta _\beta (\textbf {F}(\tau )-\textbf {F}(\sigma ))$ guaranteed to exist by Proposition 2.14.

The morphisms appearing in Example 2.1 and Equation (9) are all boundary morphisms. Additional examples appear in Example 3.11 and Figure 10. Our resolutions will be built from boundary morphisms. We first observe that boundary morphisms compose to a boundary morphism.

The Thomsen collection behaves well under open inclusions.

Lemma 2.17. Let $(\underline i, \underline I):(\Sigma , \underline {\beta })\to (\Sigma ', \underline {\beta }')$ be an open inclusion. Let $m'\in M^{\prime }_{\mathbb R}$ be any point. Then

$$\begin{align*}{\mathcal O}_{{\mathcal X}_{\Sigma, \beta}}(\textbf{F}(\underline i^{*} m'))= i^{*}{\mathcal O}_{{\mathcal X}_{\Sigma', \beta'}}(\textbf{F}'(m'))\end{align*}$$

and

$$\begin{align*}i^{*}\delta_{\sigma,\tau}= \delta_{\underline i ^{*}\sigma,\underline i ^{*}\tau}.\end{align*}$$

Proof. By Proposition 2.10, we have $i^{*}{\mathcal O}_{{\mathcal X}_{\Sigma ', \beta '}}(\textbf {F}'(m'))={\mathcal O}_{{\mathcal X}_{\Sigma , \beta }}(\underline I^{*}\textbf {F}'(m'))$ . We check that this support function agrees with $\textbf {F}(\underline i^{*}m')$ by evaluation against primitive vectors $u_{\rho }$ for the one-dimensional cones $\rho \in \Sigma (1)$ .

$$ \begin{align*} \textbf{F}(\underline i^{*} m')(u_{\rho})&=\lceil\underline{\beta}^{*}(\underline i^{*} m')(u_{\rho})\rceil =\lceil \underline I^{*}(\underline{\beta}')^{*}m'(u_{\rho})\rceil \end{align*} $$

Because $(\underline i, \underline I)$ is an open inclusion, $I(u_{\rho })\in \Sigma '(1)$ , allowing us to substitute the definition of $\textbf {F}'$ ,

$$ \begin{align*} &=\textbf{F}'(m')(\underline I u_{\rho})=\underline I^{*}\textbf{F}'(m')(u_{\rho}) \end{align*} $$

A similar computation proves that pullback sends boundary morphisms to boundary morphisms.

2.4 Pushforward along finite quotients

As illustrated by Example 2.23, we will need to compute the pushforward of a line bundle along a finite quotient. The goal of this section is to make such a computation.

The map is called a change of group with finite cokernel for the following reason. From the commutativity of the diagram

we obtain $\mathbb T_{\beta '}\circ \mathbb T_{L'}\circ i_{\mathbb T}=0$ and conclude that there exists a group homomorphism $f_\pi \colon G_{\beta }\to G_{\beta '}$ such that the morphism of toric stacks $[X_\Sigma /G_\beta ]\to [X_\Sigma /G_{\beta '}]$ arises from an $f_\pi $ map. We claim that the cokernel of $f_\pi $ is finite.

Proposition 2.19. Suppose that $(\underline \pi , \underline {\Pi }): (\Sigma , \underline {\beta })\to (\Sigma ', \underline {\beta }')$ is a change of group with finite cokernel. Let $\tilde {\pi }^{*} \colon M^{\prime }_{\mathbb R}/M'\to M_{\mathbb R}/M$ and $f_\pi : G_{\beta }\to G_{\beta '}$ be the induced maps. We have the following relations between groups

$$\begin{align*}\operatorname{\mathrm{coker}}(f_\pi) =\ker(\mathbb T_{\pi})\leftarrow \text{ Cartier Dual } \to \operatorname{\mathrm{coker}}(\underline\pi^{*} \colon M'\to M)=\ker(\tilde \pi^{*}).\end{align*}$$

Proof. First, we prove $\operatorname {\mathrm {coker}}(f_\pi \colon G_{\beta }\to G_{\beta '})=\ker (\mathbb T_\pi ) $ . We observe that the map $\mathbb T_\beta \colon \mathbb T_L\to \mathbb T_N $ is surjective (see the discussion following [Reference Geraschenko and SatrianoGS15, Definition 2.4]) so $\operatorname {\mathrm {coker}}(\mathbb T_\beta )=0$ . Then, we apply the zig-zag lemma to obtain the diagram:

verifying the first claim.

Now, we prove the second equality. Given $[m]\in \operatorname {\mathrm {coker}}(\underline \pi ^{*})$ , consider the element $m\otimes _{\mathbb R} 1\in M_{\mathbb R}$ . Since $\pi $ has a trivial kernel and finite cokernel, the map $\underline \pi _{\mathbb R}^{*} \colon M^{\prime }_{\mathbb R}\to M_{\mathbb R} $ is an isomorphism. Consider the lift $(\underline \pi ^{*}_{\mathbb R})^{-1}(m\otimes _{\mathbb R} 1)\in M^{\prime }_{\mathbb R}$ , and let $[(\pi ^{*}_{\mathbb R})^{-1}(m\otimes _{\mathbb R} 1)]\in M^{\prime }_{\mathbb R}/M'$ . Since $\underline \pi ^{*}([(\underline \pi ^{*}_{\mathbb R})^{-1}(m\otimes _{\mathbb R} 1)])=[m\otimes _{\mathbb R} 1]$ , we learn that $[(\underline \pi ^{*}_{\mathbb R})^{-1}(m\otimes 1)]\in \ker (\tilde {\pi }^{*})$ . This yields a map $\operatorname {\mathrm {coker}}(\underline \pi ^{*})\to \ker (\tilde {\pi }^{*})$ .

Conversely, any element $[m'\otimes _{\mathbb R}r]\in \ker (\tilde \pi ^{*})$ , we have $\underline \pi ^{*}_{\mathbb R}(m'\otimes r)\in M$ and thus represents a class in $\operatorname {\mathrm {coker}}(\underline \pi ^{*} \colon M'\to M) $ . This gives the inverse $\operatorname {\mathrm {coker}}(\underline \pi ^{*})\leftarrow \ker (\tilde {\pi }^{*})$ .

A morphism of stacky fans satisfying Definition 2.20 is called a finite quotient because the stack ${\mathcal X}_{\Sigma ', \underline {\beta }'}$ is the stacky quotient of ${\mathcal X}_{\Sigma , \underline \beta }$ by a finite group.

Lemma 2.22 (Pushforward along finite quotients).

Assume that $|\operatorname {\mathrm {coker}}(\underline \pi _*)|\in {\mathbb K}^\times $ . Let F be a support function on $\Sigma $ and let $(\underline \pi , \underline {\Pi }) \colon (\Sigma , \underline {\beta }) \to (\Sigma ', \underline {\beta }')$ be a finite quotient. Define the support function $\underline {\Pi }_*F$ to be the pushforward of F along $\underline {\Pi }$ . Then,

(8) $$ \begin{align} \pi_*{\mathcal O}(F)=\bigoplus_{[m]\in \operatorname{\mathrm{coker}}(\underline\pi^{*}:M'\to M)}{\mathcal O}_{{\mathcal X}'}(\underline{\Pi}_*(F-\underline{\beta}^{*}m)). \end{align} $$

Proof. The pushforward is given by the coinduced representation, which can be understood in this setting via the finite Fourier transform (Appendix B.1.1)

$$ \begin{align*} \pi_*{\mathcal O}(F)=\phi_c{\mathcal O}(F) =& \bigoplus_{\chi\in \hom(\operatorname{\mathrm{coker}}(f_\pi), \mathbb G_m)} {\mathcal O}(\underline \Pi_* F)_\chi \end{align*} $$

We identify $\operatorname {\mathrm {coker}}(f_\pi )$ with $\ker (\mathbb T_{\pi })$ using Proposition 2.19, and $\hom _{gp}(\ker (\mathbb T_{\pi }), \mathbb G_m)$ with $\operatorname {\mathrm {coker}}(\underline \pi ^{*} \colon M'\to M) $ using Equation (4).

$$ \begin{align*} =&\bigoplus_{[m]\in \operatorname{\mathrm{coker}}(\underline\pi^{*}:M'\to M)} {\mathcal O}(\underline \Pi_* F)_{\chi^{\underline \beta^{*}m}} \end{align*} $$

Finally, by Proposition 2.9

$$ \begin{align*}, =&\bigoplus_{[m]\in \operatorname{\mathrm{coker}}(\underline\pi^{*}:M'\to M)} {\mathcal O}(\underline \Pi_* F-\underline{\beta}^{*}m), \end{align*} $$

as claimed.

Example 2.23 (Resolving point on the orbifold line).

We pick up from Example 2.11. Observe that under the quotient map $\pi \colon \mathbb A^1\to [\mathbb A^1/(\mathbb Z/2\mathbb Z)]$ , we have

$$ \begin{align*} \pi_* {\mathcal O}_{\mathbb A^1}={\mathcal O}_{{\mathcal X}_{\Sigma, \underline{\beta}}} \oplus O_{{\mathcal X}_{\Sigma, \underline{\beta}}} (F_1) & & \pi^{*}{\mathcal O}_{{\mathcal X}_{\Sigma, \underline{\beta}}}\cong\pi^{*}{\mathcal O}_{{\mathcal X}_{\Sigma, \underline{\beta}}}(F_1)\cong {\mathcal O}_{\mathbb A^1} \end{align*} $$

Let $\phi \colon e\to [\mathbb A^1/(\mathbb Z/2\mathbb Z)] $ denote the inclusion of the identity point. We now describe a resolution of the sheaf $\phi _*{\mathcal O}_e$ . Consider the point $e'= 1 \in \mathbb A^1$ . Using the quotient map, we can push forward a resolution of ${\mathcal O}_{e'}$ to obtain a resolution for ${\mathcal O}_{e}$ . We take the standard resolution for ${\mathcal O}_{e'}$ as in Equation (12). The pushforward is

(9)

The pushforward turns out to be exact so this indeed gives us a resolution of the point e in $[\mathbb A^1/(\mathbb Z/2\mathbb Z)]$ .

We now discuss the behavior of the Thomsen collection under finite quotients.

Lemma 2.24. Let $(\underline \pi , \underline {\Pi }): (\Sigma , \underline {\beta })\to (\Sigma ', \underline {\beta }')$ be a finite quotient, where $|\operatorname {\mathrm {coker}}(f_\beta )|\in {\mathbb K}^\times $ . Let $p\in M_{\mathbb R}/M_{\mathbb Z}$ be any point. Let $\tilde \pi \colon M^{\prime }_{\mathbb R}/M^{\prime }_{\mathbb Z}\to M_{\mathbb R}/M_{\mathbb Z} $ be the associated map on tori. Then,

$$\begin{align*}\pi_* {\mathcal O}_{{\mathcal X}_{\Sigma, \underline{\beta}}}(\textbf{F}(p))= \bigoplus_{p'\in \tilde \pi^{-1}(p)} {\mathcal O}_{{\mathcal X}_{\Sigma, \underline{\beta}'}}(\textbf{F}(p')).\end{align*}$$

Furthermore, given $\sigma>\tau $ , we have

$$\begin{align*}\pi_*\delta_{\sigma,\tau}(x_1, \ldots x_k)=(\delta_{\sigma_1,\tau_1}x_1, \ldots,\delta_{\sigma_k,\tau_k}x_k ),\end{align*}$$

where $\sigma _i>\tau _i$ are the lifts of $\sigma , \tau $ which preserve adjacency.

2.5 Toric stacky charts

In this section, we introduce and discuss the notion of a chart on a toric stack. We will eventually use these charts to cover our toric stack and localize computations of our resolutions.

Every inclusion/immersion of stacky fans induces an inclusion/immersion of a toric stack. Unfortunately, since every toric stack may be represented by several different stacky fans, to represent a morphism of toric stacks, we may have to modify our (possibly preferred) choice of stacky fans.

Example 2.30 (Nonseparated line).

We return to the nonseparated line introduced in Example 2.8. We draw its stacky fan on the right-hand side in Figure 9. Observe that ${\mathcal X}_{\Sigma , \underline {\beta }}$ is covered by stabilized smooth stacky charts.

Figure 9 A smooth chart covering a portion of the nonseparated line.

We draw the map of stacky fans corresponding to one of the charts in the figure. Observe that the $\mathbb A^1$ chart does not use the usual (non-stacky) fan, but rather the stacky fan which is obtained by stabilizing the standard fan.

The doubled origin corresponds to two toric divisors $D_0, D_1$ (coming from the $G_\beta $ -invariant divisors $z_1=0, z_2=0$ ), but by the discussion in Example 2.8, we only need to consider divisors of the form $a(D_0+D_1)$ . The $G_\beta $ -equivariant sections of ${\mathcal O}_{X_\Sigma }$ are given by the polynomial ring ${\mathbb K}[z_1z_2]$ . Let $e=[1]$ in ${\mathcal X}_{\Sigma , \underline {\beta }}$ . The complex

(10)

is a resolution of ${\mathcal O}_e$ , which is seen by restricting the resolution to the $\mathbb A^1$ charts. When restricted, this resolution exactly matches Equation (12).

Example 2.31 (Weighted projective space).

We now consider the stacky fan associated with a weighted projective space. Consider the fan $\Sigma $ on $\mathbb Z^2$ so that $X_\Sigma =\mathbb A^2\setminus \{0\}$ . We take $\underline {\beta } \colon \mathbb Z^2\to \mathbb Z$ which is given by the matrix $\begin {pmatrix}1 & -2\end {pmatrix}$ . This corresponds to the action of $G_\beta \curvearrowright X_\Sigma $ by $(z_1, z_2)\mapsto (tz_1, t^2 z_2)$ . The action has finite stabilizers, and the quotient $X_{\Sigma }/G$ is the $(1,2)$ weighted projective space, denoted $\mathbb P(1, 2)$ . Note that sheaves on $[X_{\Sigma }/G]$ do not agree with sheaves on the quotient.

The weighted projective space can also be described via a covering by smooth stacky charts. The first chart, which we call $U_0$ , comes from the toric morphism which descends to the quotient as

$$ \begin{align*} \mathbb A^1 \to& \mathbb P(1,2) \\ z_1\mapsto& [z_1:1]_{(1, 2)}. \end{align*} $$

The second chart, which we call $U_1$ , is parameterized by a morphism of toric stacks $[\mathbb A^1/(\mathbb Z/2\mathbb Z)]\to [X_\Sigma /G_\beta ]$ where $[\mathbb A^1/(\mathbb Z/2\mathbb Z)]$ is realized as a toric stack with $L=\mathbb Z, N=\mathbb Z, \beta '=(2)$ and $\Sigma =\{\langle 0 \rangle ,\langle 1\rangle \cdot \mathbb N \}$ . The morphism is given by

Support functions for $[X_\Sigma /G_\beta ]$ are determined by the integer values

$$ \begin{align*} a_0=F(\underline{\beta}(\langle 1, 0\rangle)) = F(\langle 1\rangle ) & & a_1=F(\underline{\beta}(\langle 0, 1\rangle))=F(\langle 2 \rangle) \end{align*} $$

Let $e\in [X_\Sigma /G_\beta ]$ be the identity point. We claim that the following chain complex is a resolution of ${\mathcal O}_{e}$

To check that this is a resolution of ${\mathcal O}_e$ , we restrict to the $U_0$ and $U_1$ charts. The pullback of this resolution to the $U_1$ chart is Equation (9), while the pullback of the resolution to the $U_0$ charts is chain homotopic to Equation (12). Therefore, this is a resolution of ${\mathcal O}_e$ in every toric chart and therefore a resolution of ${\mathcal O}_e$ .

2.6 Equivariant codimension two complements

In this section, we will address a few additional technical aspects of toric stacks that we will need to verify that our resolutions behave as expected.

First, we verify that immersions of toric stacks behave well under further quotients by a toric subgroup. Let $(\Sigma , \underline {\beta }\colon L\to N)$ be a stacky fan. Given a saturated subgroup $N_G\to N$ , we obtain a new stacky fan $(\Sigma , \underline {\beta }'\colon L\to N/N_G)$ , along with a morphism of stacky fans $(\underline \phi , \underline \Phi )\colon (\Sigma , \underline {\beta })\to (\Sigma , \underline {\beta }')$ . This construction identifies the stack $ [{\mathcal X}_{\Sigma , \underline \beta }/\mathbb T_{G}]$ as ${\mathcal X}_{\Sigma , \underline \beta '}$ .

Proposition 2.32. Let $(\underline \phi , \underline \Phi )\colon (\Sigma _Y, \underline \beta _Y\colon L_Y\to N_Y) \to (\Sigma _X, \underline \beta _X\colon L_X\to N_X)$ be an immersion of a closed toric substack inducing a morphism of toric stacks $\phi : \mathcal Y \to {\mathcal X}$ . Let $\mathbb T_G\subset \mathbb T_N$ be a toric subgroup corresponding to a saturated sublattice $N_G\subset N$ . If $\mathcal Y$ is $\mathbb T_G$ -invariant, there is an immersion $\phi '$ , making the following diagram commute:

Proof. For brevity of notation, we set ${\mathcal X}' = [{\mathcal X}/\mathbb T_G]$ and $\mathcal Y' = [\mathcal Y/\mathbb T_G]$ . As $\mathcal Y$ is a closed toric substack, the maps $\underline \phi , \underline \Phi $ are inclusions. The condition that $\mathcal Y$ is a G-invariant substack is equivalent to $N_G\subset \phi (N_Y)$ We therefore obtain a saturated sublattice $\underline \phi ^{-1}(N_G)\subset N_Y$ . The toric fan $(\Sigma _{\mathcal Y'},\underline \beta _{\mathcal Y'})$ for $\mathcal Y'$ is defined using the construction preceding the proposition. The map $\underline \phi \colon N_Y\to N_X $ descends to a map $\underline \phi ' \colon N_{Y'}\to N_{X'}$ . The map $\underline \Phi ' \colon L_{Y'}\to L_{X'} $ is taken so it agrees with $\underline \Phi $ .

It remains to show that $\phi ' \colon \mathcal Y'\to {\mathcal X}' $ is an immersion of a closed substack. Since $\underline \Phi '=\underline \Phi $ , we only need to check that the map $\underline \phi ' \colon N_{Y'}\to N_{X'} $ is an injection. This follows as $\underline \phi $ is an injection and $\underline \phi (\underline \phi ^{-1}(N_G))=N_G$ .

When $X^\circ \hookrightarrow X$ has a complement of codimension at least 2, then the pushforward of a line bundle is a coherent sheaf and is still a line bundle if X is smooth (Hartog’s principle). If G acts freely on X, and $X^\circ \to X$ is G-equivariant, it is clear that the dimension of $(X\setminus X^\circ )/G$ is the relevant quantity to study (as opposed to the codimension of $X^\circ $ in X). When G does not act freely, it is not clear to us in general what quantity is appropriate to examine. For our applications, the following criterion will be useful.

An important feature of open inclusions with complements of equivariant codimension two is that we can ‘by hand’ define a correspondence from line bundles on ${\mathcal X}^\circ =[X^\circ /G]$ to line bundles on ${\mathcal X}=[X/G]$ . This can be thought of as a version of Hartog’s principle for equivariant line bundles. Let $(\underline i, \underline I)\colon (\Sigma ^\circ , \underline {\beta }^\circ )\to (\Sigma , \underline {\beta })$ be an open immersion with a complement of equivariant codimension 2 where $\Sigma $ is smooth. Write $I_\Sigma \colon \Sigma ^\circ (1)\hookrightarrow \Sigma (1)$ for the map on rays induced by $\underline i$ . Given $F^\circ \colon L^\circ \to \mathbb R $ a support function, we obtain a support function $\underline I_{\flat } F \colon L \to \mathbb R $ whose values on primitive generators for $\Sigma ^\circ (1)$ are defined by

$$\begin{align*}\underline I_{\flat} F(u_{\rho})= \begin{cases} F\left(u_{\underline I^{-1}(\rho)}\right) &\text{ if } \rho\in I_\Sigma(\Sigma^\circ(1))\\ 0 & \text{otherwise}. \end{cases} \end{align*}$$

We now show that this construction comes from a functor between sheaves on $[X^\circ /G]$ and $[X/G]$ .

Example 2.36. Consider the toric stack $[X/G]=[\mathbb A^1/\mathbb T^1]$ . The inclusion $i: [X^\circ /G]:=[\mathbb T^1/\mathbb T^1]\to [X/G]$ is an example of an inclusion with equivariant codimension two. The pushforward $i_*{\mathcal O}_{[X^\circ /G]}(0)$ is not a line bundle on $[X/G]$ . This can be seen, for instance, by letting ${\mathcal F}_0$ be the skyscraper sheaf of the point $0 \in \mathbb A^1$ with the trivial group action and computing

$$\begin{align*}\hom(i_*{\mathcal O}_{[X^\circ/G]}(0), {\mathcal F}_0)=0.\end{align*}$$

However, when one takes the $\mathbb T^1$ -invariant sections of ${\mathcal O}_{[X^\circ /G]}(0)$ first, we observe that ${\mathcal O}^G_{[X^\circ /G]}(0)$ is the constant ${\mathbb K}$ -sheaf. Its pushforward to $[X/G]$ is again the constant ${\mathbb K}$ -sheaf. After tensoring with ${\mathcal O}_{[X/G]}$ , we obtain our desired answer.

Let ${\mathcal F}$ be a sheaf on $[X^\circ /G]$ . For $U\subset X$ , define

$$\begin{align*}i_{\flat} {\mathcal F}(U):= {\mathcal F}^{G}(U\cap X^\circ)\otimes_{{\mathcal O}_{[X^\circ/G]}^G(U\cap X^\circ)} {\mathcal O}_{[X/G]}(U).\end{align*}$$

For $f\in \hom _{[X^\circ /G]}({\mathcal F}, {\mathcal G})$ , we similarly define the morphism $i_{\flat } f$ on sections $x\in i_{\flat } {\mathcal F}(U)$ as

$$\begin{align*}i_{\flat} f(x)= f(y)\otimes r\end{align*}$$

after writing $x=y\otimes r$ , where $y\in {\mathcal F}^{G}(U\cap X^\circ )$ and $r \in \mathcal {O}_{[X/G]}(U)$ .

Proof. We first prove that this is a functor of topological sheaves; this follows as all operations (taking invariant sections, pullback along open inclusion and tensor product) are functors of topological sheaves. By a similar argument, $i_{\flat } {\mathcal F}(U)$ is an ${\mathcal O}_{[X/G]}$ module, and $i_{\flat } f$ is a morphism of ${\mathcal O}_{[X/G]}$ modules.

We now check Item 1 by exhibiting an isomorphism of sheaves $i_{\flat }{\mathcal O}_{[X^\circ /G]}(F)={\mathcal O}_{[X/G]}(\underline I_{\flat } F)$ . Let $[X_\sigma /G]$ be a smooth stacky chart for X, with the one-dimensional cones of $\Sigma _\sigma $ generated by $\{u_{\rho _i}\}_{i=1}^k$ . Then a section $s\in {\mathcal O}_{[X^\circ G]}^{G}(F)(U\cap X^\circ )$ can be written as a sum of monomials $s=\sum _{m'\in M'} a_{m'} \chi ^{\underline {\beta }^{*}m'}$ , where the $k'=\underline {\beta }^{*}m'$ satisfy the following.

• • Because the section is G-invariant, $\langle k', l'\rangle =0$ for all $l'\in \ker (\underline \beta ^\circ )$ . Further, $(\Sigma _\sigma (1)\setminus \Sigma ^\circ (1))\subset \ker (\underline {\beta }^\circ )$ implies that $\langle u_{\rho _i}, k'\rangle \geq 0$ for all $\rho _i\in \Sigma _\sigma (1)\setminus \Sigma ^\circ $ .

• • For each $\rho _i\in \Sigma _\sigma (1)\cap \Sigma ^\circ (1)$ , we have the relation $\langle u_{\rho _i}, k'\rangle \geq F(u_{\rho _i})$ .

It follows that for all $u_{\rho _i}\in \Sigma _\sigma (1)$ , we have $\langle u_{\rho _i}, \underline {\beta }^{*}m \rangle \geq F(u_{\rho _i})$ . Therefore, s corresponds to a section of ${\mathcal O}_{[X/G]}^G(\underline I_{\flat } F)(U)$ . The result follows from

$$ \begin{align*} {\mathcal O}_{[X^\circ/G]}^G(\underline I_{\flat} F)(U\cap X^\circ)\otimes_{{\mathcal O}_{[X^\circ/G]}^G(U\cap X^\circ)}{\mathcal O}_{[X/G]}(U) &={\mathcal O}_{[X/G]}^G(\underline I_{\flat} F)(U)\otimes_{{\mathcal O}_{[X/G]}^G(U)}{\mathcal O}_{[X/G]}(U)\\&={\mathcal O}_{[X^\circ/G]}(\underline I_{\flat} F)(U). \end{align*} $$

We now prove Item 2 which allows us to compare the stalks of $(\pi ^{*}\circ i_{\flat }){\mathcal F}$ over $X\setminus X^\circ $ to stalks in $X^\circ $ . Pick $x_0\in X$ and $x_1\in X^\circ $ so that $x_0$ is in the closure of $G\cdot x_1$ . Because $x_0$ is in the closure of $G\cdot x_1$ , we can take a sequence of subgroups $G_0\supset G_1\supset \cdots $ of G and sequence of open sets $V_i\ni x_1$ satisfying

• • for all i, there exists $U_i\ni x_0$ an open subset of X so that $U_i\cap X^\circ = \bigcup _{g\in G_i}V_i$ and;

• • for every $U\ni x_0$ and open subset of X, there exists j sufficiently large so that for all $i>j$

  $$\begin{align*}\bigcup_{g\in G_i} g\cdot V_i\subset U\cap X^0.\end{align*}$$

We then compute the stalk

$$ \begin{align*} (\pi^{*}\circ i_{\flat} F)_{x_0}=& \lim_{U\ni x_0} (\pi^{*}\circ i_{\flat} {\mathcal F})(U)\\ =& \lim_{i\to\infty} {\mathcal F}^{G}(U_i\cap X^0)\otimes_{{\mathcal O}^G_{[X^\circ/G](U_i\cap X^\circ)}}{\mathcal O}_X(U_i)\\ \cong& \lim_{i\to\infty} {\mathcal F}^{G} \left(\bigcup_{g\in G_i} g\cdot V_i\right)\otimes_{{\mathcal O}^G_{[X/G]}(\bigcup_{g\in G_i} g\cdot V_i)}{\mathcal O}_X(U_i) \end{align*} $$

Since ${\mathcal F}^{G}$ and ${\mathcal O}^G_{[X^\circ /G]}$ are sheaves of G invariant sections,

$$ \begin{align*} &\cong \lim_{i\to\infty} {\mathcal F}^{G} \left(V_i\right)\otimes_{{\mathcal O}^G_{[X^\circ/G]}(V_i)}{\mathcal O}_X(U_i)\\&={\mathcal F}^G_{x_1}\otimes_{({\mathcal O}^G_{[X^\circ /G]})_{x_1}}\otimes \mathcal (O_X)_{x_0}\\&\cong {\mathcal F}^G_{x_1}\otimes_{({\mathcal O}^G_{[X^\circ /G]})_{x_1}}\otimes \mathcal (O_X)_{x_1}\cong (\pi^{*}\circ i_{\flat} F)_{x_1}. \end{align*} $$

From Item 2, it follows that $(\pi ^{*}\circ i_{\flat })$ is exact, as exactness can be computed on stalks, and for points $x_1\in X\setminus X^\circ $ , we have an isomorphism of stalk functors

$$\begin{align*}(\pi^{*}\circ i_{\flat})(-)_{x_1}= (I_*\circ (\pi^\circ)^{*})(-)_{x_1}.\end{align*}$$

The latter functor is exact, as $(\pi ^\circ )^{*}$ is an equivalence of categories, $I_*$ is an open inclusion and $x_1$ is in the image of $I_*$ .

Finally, it remains to show Item 4. Since $\pi ^{*}\circ i_{\flat }$ is exact, and sends ${\mathcal O}_{[X^\circ /\mathbb T_Y]}$ to ${\mathcal O}_X$ , it sends coherent sheaves to coherent sheaves.

We will make use of Lemma 2.37 in conjunction with the following local construction of an open inclusion with equivariant codimension two complement from a toric subvariety.

We represent this data in the form of the following diagram:

We do not have a good characterization for when the third property of Proposition 2.39 holds over fields of positive characteristic.

2.7 Returning to resolutions of toric subvarieties in $\mathbb A^2$

Now that we have established the background on toric stacks and the tools needed to apply our general strategy, we conclude this section with two more examples of putting those tools to work.

Example 2.42 (Resolving $z_1^2=z_2$ equivariantly).

Take the parabola $\phi \colon \mathbb A^1 \to \mathbb A^2 $ parameterized by $z\mapsto (z, z^2)$ . Let $i\colon \mathbb A^2\setminus \{0\}\to \mathbb A^2$ be the inclusion. Consider the action of $\mathbb T_\beta \curvearrowright \mathbb A^2$ given by $(z_1, z_2)\mapsto (tz_1, t^2z_2)$ . We can take the quotient $\pi _\beta \colon \mathbb A^2\setminus \{0\}\to [\mathbb A^2\setminus \{(0,0)\}/\mathbb T_\beta ]$ . There is a resolution $C_{\bullet }(S^{\phi _e}, {\mathcal O}^{\phi _e})$ for a point on the quotient stack so that $\pi ^{*}i_{\flat } C_{\bullet }(S^{\phi _e}, {\mathcal O}^{\phi _e})$ is

which is homotopic to our original resolution from Example 2.2. Note that the line bundles used in all phases of the construction are in a Thomsen collection.

Example 2.43 (Resolving $z_0z_1=1$ ).

Take the toric subvariety $\phi _{(-1, 1)} \colon \mathbb {G}_m \to \mathbb A^2 $ parameterized by $z\mapsto (z, z^{-1})$ . We now look at the map $\pi _{(-1, 1)} \colon \mathbb A^2\setminus \{(0,0)\}\to X_\beta :=[\mathbb A^2\setminus \{(0,0)\}/\mathbb T_\beta ] $ ; the latter is the toric stack corresponding to the nonseparated line. Then applying $\pi ^{*}i_{\flat }$ to our resolution from Equation (10) of ${\mathcal O}_e$ is the standard resolution of $z_0z_1=1$

3.1 Resolving points in toric varieties

We first lay out the steps for resolving the identity point in a smooth toric variety.

In Section 3.4, we describe the (a priori not exact) complex of sheaves $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ , which is defined combinatorially from the morphism of stacky fans $(\underline \phi , \underline \Phi )\colon (\Sigma _Y, \underline \beta _Y)\to (\Sigma _X, \underline \beta _X)$ whenever ${\mathcal X}$ is covered by smooth stacky charts; the version needed for Theorem 3.1 comes from specializing $\mathcal Y=e$ and ${\mathcal X}=X$ . From the construction of $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ , it is immediate that there is a morphism $\alpha ^\phi \colon {\mathcal O}_{{\mathcal X}}\to C_0(S^\phi , {\mathcal O}^\phi ) $ and $C_k(S^\phi , {\mathcal O}^\phi )=0$ whenever $k<0$ or $k> \dim ({\mathcal X})-\dim (\mathcal Y)$ . Before we get to the definition of $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ in Section 3.4, we list some of the properties that this complex will satisfy. We state the lemmas for the general setting of toric substacks, as we will use them later. Proofs are delayed until Section 4.

Recall that given a ray $\rho \in \Sigma (1)$ , the star of $\rho $ is $\operatorname {\mathrm {star}}(\rho ) = \{ \sigma \in \Sigma : \rho < \sigma \}$ . Removing $\operatorname {\mathrm {star}}(\rho )$ from $\Sigma $ corresponds to taking the complement of the divisor corresponding to $\rho $ . We then prove that $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ transforms under deletion of a toric boundary divisor in the following way.

Lemma 3.3 (Functoriality along restrictions up to homotopy).

Let $\phi : \mathcal Y_{\Sigma ', \underline \beta '} \hookrightarrow {\mathcal X}_{\Sigma , \underline \beta }$ be an immersion of a closed toric substack. For any $\rho \in \Sigma (1)$ , let $\Sigma _{\rho }=\Sigma \setminus \operatorname {\mathrm {star}}(\rho )$ , giving us the toric stack ${\mathcal X}_{\rho }:={\mathcal X}_{\Sigma _{\rho },\underline \beta }$ , and let $\Sigma ^{\prime }_{\rho }=\Sigma '\setminus \{\alpha \operatorname {\mbox { | }} \phi (\alpha )\cap \operatorname {\mathrm {star}}(\rho )\neq \emptyset \},$ giving us the toric stack $\mathcal Y_{\rho }:= \mathcal Y_{\Sigma _{\rho }', \underline \beta '}$ fitting into the diagram

where $\phi _{\rho }$ is induced by the same morphism of stacky fans as $\phi $ and $i_{\rho }, i_{\rho }'$ are the open inclusions. Suppose that ${\mathcal X}_{\Sigma _{\rho },\underline {\beta }}$ is covered by smooth stacky charts. There is a homotopy equivalence $\Psi \colon i^{*}_{\rho } C_{\bullet }(S^\phi , {\mathcal O}^\phi )\to C_{\bullet }(S^{\phi _{\rho }}, {\mathcal O}^{\phi _{\rho }})$ . Furthermore, the diagram

(11)

commutes where the left vertical arrow and upper right vertical arrow are the natural restriction morphisms.

Given these two lemmas, we can prove Theorem 3.1.

Proof of Theorem 3.1.

The theorem is proved in two steps:

Step 1: Resolution of points in charts. By Example 3.11, the theorem holds for $\phi \colon e\to \mathbb A^1 $ , and furthermore, $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is the Koszul resolution. The case $\phi \colon e\to \mathbb A^n $ follows from applying Lemma 3.2.

Step 2: Restricting to smooth toric charts. Let $\mathcal Y=\{e\}$ and let X be a toric variety which can be covered with smooth toric charts $\{X_\sigma \}_{\sigma \in \Sigma (n)}$ . Let $j_\sigma \colon X_{\sigma }\to X$ be the corresponding inclusion of a coordinate chart. For each $\sigma $ , let $\rho _{\sigma ,1}, \ldots , \rho _{\sigma ,k}\in \Sigma (1)$ be the one-dimensional cones which are disjoint from $\sigma $ . Consider the iterated inclusion

where $X_{\rho _{\sigma ,1}, \ldots , \rho _{\sigma , l}}$ is the toric stack whose fan consists of cones which do not contain rays $\rho _{\sigma ,1}, \ldots , \rho _{\sigma , l}$ . By repeated application of Lemma 3.3, we obtain that $j_\sigma ^{*}C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is chain homotopic to $C_{\bullet }(S^{\phi _{\rho _1,\ldots ,\rho _k}},{\mathcal O}^{\rho _1,\ldots ,\rho _k})$ . The latter is the Koszul resolution of the skyscraper sheaf of $\{e\}$ by the previous step. This implies that $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ resolves $\phi _*{\mathcal O}_{e|_{X_{\sigma }}}$ on every smooth toric chart. Since these charts cover X, it is a resolution of $\phi _*{\mathcal O}_e$ .

3.2 Resolving points in toric stacks

We will now generalize Theorem 3.1 to toric stacks. Note that we use this generalization even to resolve some smooth toric subvarieties of a smooth toric variety (see, for instance, Examples 2.42 and 2.43).

We will need an additional lemma in this setting.

Lemma 3.4 (Pushforward functoriality along finite group quotients).

Let $\pi _X \colon {\mathcal X}\to {\mathcal X}'=[{\mathcal X}/G]$ be a finite group quotient and let $\phi '\colon \mathcal Y'\to {\mathcal X}'$ be an inclusion of a toric substack. Suppose that ${\mathcal X}$ is covered with smooth stacky charts. Then there exists $\mathcal Y\to {\mathcal X}$ an inclusion of a toric substack and $\pi _Y\colon \mathcal Y\to \mathcal Y'$ a finite group quotient so that the diagram

commutes.Footnote ⁵ Additionally assume $|G|\in {\mathbb K}^\times $ and that ${\mathcal X}$ is covered with smooth stacky charts. Observe that $(\phi '\circ \pi _Y)_*{\mathcal O}_{\mathcal Y}=\bigoplus _{q\in \ker (\tilde \pi _Y)}(\phi ^{\prime }_*){\mathcal O}_{\mathcal Y'}(\textbf {F}'(q))$ . The pushforward $(\pi _X)_*C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is isomorphic to $\bigoplus _{q\in \ker (\tilde \pi _Y)}C_{\bullet }(S^{\phi '},{\mathcal O}^{\phi '})\otimes {\mathcal O}_{{\mathcal X}'}(\textbf {F}'(q))$ with the augmentation map splitting across the decomposition.

The proof of Theorem 3.1' proceeds in a similar way to the proof of Theorem 3.1.

Step 1: Resolution of points in smooth stacky coordinate charts We show that when ${\mathcal X}$ is a smooth stacky coordinate chart (Definition 2.26) and $\phi \colon e\to {\mathcal X} $ is the inclusion of the identity, $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is a resolution of ${\mathcal O}_e$ .

As ${\mathcal X}$ is a smooth stacky coordinate chart, ${\mathcal X}$ is a finite quotient ${\mathcal X}=[\mathbb A^n/G]$ . Let $\pi _G \colon \mathbb A^n\to {\mathcal X} $ , and $\phi ' \colon e\to \mathbb A^n$ . The statement then follows from our earlier calculation on $\mathbb A^n$ and Lemma 3.4. By Nakayama’s isomorphism, $(\pi _X)_* C_{\bullet }(S^\phi , {\mathcal O}^\phi ) =\bigoplus _{q\in \ker (\tilde \pi _Y)}C_{\bullet }(S^{\phi '},{\mathcal O}^{\phi '})\otimes {\mathcal O}_{\mathcal Y'}(\textbf {F}'(q))$ is an exact resolution of $(\phi ' \circ \pi _Y)_* {\mathcal O}_{\mathcal Y} = \bigoplus _{q\in \ker (\tilde \pi _Y)}(\phi ^{\prime }_*){\mathcal O}_{\mathcal Y'}(\textbf {F}(q))$ . Since the augmentation respects the splitting, we conclude $C_{\bullet }(S^{\phi '}, {\mathcal O}^{\phi '})$ resolves $\phi _*'{\mathcal O}_Y$ .

Step 2: Restricting to smooth stacky charts We can apply exactly the same argument as in Step 2 above simply by replacing X with ${\mathcal X}$ , and smooth chart with smooth stacky chart.

3.3 Resolution of toric substacks

We now move to the most general case that we handle here – resolving the structure sheaf of a closed toric substack in a toric stack covered by smooth stacky charts.

To prove this more general version of the theorem, we add a few more lemmas to our repertoire.

Lemma 3.6 (Pullback functoriality along toric quotients).

Suppose that we have a subtorus $\mathbb T\subset \mathbb T_N$ acting on ${\mathcal X}$ , and that $\mathcal Y$ is $\mathbb T$ -equivariant. Additionally, suppose that the action of $\mathbb T$ has trivial stabilizers. From Proposition 2.32, we obtain the diagram

Then, $\pi ^{*}_{\mathbb T} C_{\bullet }(S^{\phi /\mathbb T}, {\mathcal O}^{\phi /\mathbb T})= C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ , and $\pi ^{*}_{\mathbb T}\circ \alpha ^{\phi /\mathbb T}= \alpha ^\phi $ .

Lemma 3.8 (Functoriality along embeddings with equivariant codimension 2 complement).

Suppose we have a diagram

where the image of Y is invariant under $\mathbb T$ and i is an inclusion of equivariant codimension two (in the sense of Definition 2.33). Then, $\pi ^{*}i_{\flat } C_{\bullet }(S^{\phi _{[Y/\mathbb T]}}, {\mathcal O}^{\phi _{[Y/\mathbb T]}})= C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ .

Proof of Theorem 3.5.

As before, we want to be able to resolve toric substacks in local models and restrict to that setting.Step 3: Resolution of ${\mathbb K}$ -admissible toric subvarieties of smooth charts Let $X=\mathbb A^n$ be a smooth toric chart, and let Y be a closed ${\mathbb K}$ -admissible toric subvariety (Definition 2.40). By Proposition 2.39, we can construct $I\colon X^\circ \hookrightarrow X$ so that $[X^\circ /\mathbb T_Y]$ is covered by smooth stacky charts and $Y\cap X^\circ =\mathbb T_Y$ . We reprint the diagram of stacks here for readability:

Let $\phi _e \colon \{e\} \to [X^\circ /\mathbb T_Y] $ be the inclusion of the identity point. By Theorem 3.1', we have $C_{\bullet }(S^{\phi _e}, {\mathcal O}^{\phi _e})$ , which resolves $(\phi _e)_*{\mathcal O}_{e}$ . By Lemma 3.8, we obtain

$$\begin{align*}\pi_*i_{\flat} C_{\bullet}(S^{\phi^\circ}, {\mathcal O}^{\phi^\circ})=C_{\bullet}(S^\phi, {\mathcal O}^\phi).\end{align*}$$

For any point $x_0\in \mathbb A^n$ , there exists a point $x_1\in X^\circ $ so that $x_0$ is in the closure of $\mathbb T_Y \cdot x_1$ . Then by Lemma 2.37, we can compute the stalks of $H_k(C_{\bullet }(S^\phi , {\mathcal O}^\phi ))$ on $\mathbb A^n$ by comparing them to stalks of points in $X^\circ $ . We obtain

$$\begin{align*}H_k(C_{\bullet}(S^\phi, {\mathcal O}^\phi))_{x_0}= H_k(C_{\bullet}(S^\phi, {\mathcal O}^\phi))_{x_1}= H_k( C_{\bullet}(S^{\phi^\circ}, {\mathcal O}^{\phi^\circ}))_{i^{-1}\pi(x_1)}.\end{align*}$$

Since $C_{\bullet }(S^{\phi ^\circ }, {\mathcal O}^{\phi ^\circ })$ is a resolution of ${\mathcal O}_{\mathbb T_Y}$ , we learn that

• • $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is a resolution

• • (i.e., it is exact except at $k=0$ );

• • As the closure of the $\mathbb T_Y$ orbit in X is Y, we obtain that $H_0(C_{\bullet }(S^\phi , {\mathcal O}^\phi ))$ has the same stalks as $\phi _*{\mathcal O}_Y$ . We have a morphism $\alpha \colon {\mathcal O}_{\mathbb A^n}\to H_0(C_{\bullet }(S^\phi , {\mathcal O}^\phi ))$ . By another application of Lemma 2.37, this morphism is an epimorphism on stalks for all $x\in Y$ and is zero otherwise. We conclude that $H_0(C_{\bullet }(S^\phi , {\mathcal O}^\phi ))\cong \phi _*{\mathcal O}_Y$ .

Step 4: Resolution of closed toric substacks of smooth stacky charts As ${\mathcal X}$ is a smooth stacky coordinate chart, ${\mathcal X}$ is a finite quotient ${\mathcal X}=[\mathbb A^n/G]$ . Just as in Step $1$ , the statement follows from the previous step and Lemma 3.4.

Step 5: Resolution of toric substacks By repeated application of Lemma 3.3 (as in Step 2 to reduce to Step 4), $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ restricts to a resolution of $\phi _*{\mathcal O}_{\mathcal Y}$ on every smooth stacky toric chart. This is not enough to determine that $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is a resolution of ${\mathcal O}_{\mathcal Y}$ . Let $C_{\bullet }(S^\phi , {\mathcal O}^\phi )\to {\mathcal F}$ be exact. The morphism $\alpha ^\phi $ determines a map ${\mathcal O}_{\mathcal X} \to {\mathcal F}$ . Since ${\mathcal F}$ agrees with $\phi _* {\mathcal O}_{\mathcal Y}$ on every chart for a cover of ${\mathcal X}$ , it is a twist of $\phi _*{\mathcal O}_{\mathcal Y}$ by a line bundle. The existence of an epimorphism ${\mathcal O}_{\mathcal X}\to {\mathcal F}$ determines that ${\mathcal F} \cong \phi _*{\mathcal O}_{\mathcal Y}$ .

3.4 Exit paths and sheaves

Let $(\Sigma _X, \underline \beta _X)$ and $(\Sigma _Y, \underline \beta _Y)$ be stacky fans associated to toric stacks ${\mathcal X}$ and $\mathcal Y$ , respectively. Suppose further we have a closed immersion $\phi \colon \mathcal Y\to {\mathcal X}$ induced by an immersion of stacky fans $(\underline \Phi \colon L_Y\to L_X, \underline \phi \colon N_Y\to N_X )$ . As $(\underline \phi , \underline \Phi )$ is an immersion (Definition 2.25), we have that the cones of $\Sigma _Y$ are $\underline \Phi ^{-1}(\Sigma _X)$ . We have a dual map $\underline \phi ^{*} \colon M_X\to M_Y$ inducing a map $\tilde \phi : M_X\otimes \mathbb R/ M_X\to M_Y\otimes \mathbb R/ M_Y$ on tori. Consider now the real torus $T^{\phi }:=\ker ( \tilde {\phi })$ , whose dimension is the codimension of $\mathcal Y$ .Footnote ⁶ For every ray $\rho \in \Sigma _X(1)$ , we obtain a map

$$ \begin{align*} T^\phi\to& \, \mathbb R/\mathbb Z\\ [m]\mapsto& \, \underline \beta_X^{*}m(u_{\rho}) \end{align*} $$

where, as before, $u_{\rho }$ is the primitive generator of $\rho $ . We denote the kernel of this map by $T^{\phi }(\rho )$ , which is either all of $T^\phi $ (when $\underline {\beta }_X(\rho )\in \text {Im}(\underline \phi )$ ) or a disjoint union of toric hyperplanesFootnote ⁷ .

The set of subtori $\{T^{\phi }(\rho )\operatorname {\mbox { | }} \underline {\beta }_X(\rho ) \not \in \text {Im}(\underline \phi )\}$ is a toric hyperplane arrangement. Let $S^\phi $ be the corresponding stratification of $T^\phi $ .

By applying Equation (7) and working under the assumption that $\Sigma _X$ is smooth, we associate to each stratum $\sigma \subset T^\phi \subset M_{X, \mathbb R}/M$ of $S^\phi $ a line bundle ${\mathcal O}^\phi (\sigma )\in \operatorname {\mathrm {Pic}}({\mathcal X})$ . For each point $p\in T^{\phi }$ , let $\dim _{S}(p)$ be the dimension of the stratum containing p. If $\sigma , \tau $ are strata of $S^\phi $ such that $\sigma> \tau $ , we let $\operatorname {\mathrm {EP}}(\sigma , \tau )$ denote the homotopy classes of paths $\gamma \colon I\to T^{\phi }$ satisfying

$$ \begin{align*} \gamma(0)\in \sigma, & & \gamma(1)\in \tau, \text{ and} & & t_1\leq t_2 \Rightarrow \dim_{S}(\gamma(t_1))\leq \dim_{S}(\gamma(t_2)). \end{align*} $$

The exit path category $\operatorname {\mathrm {EP}}(S^\phi )$ has objects equal to strata of $S^\phi $ , morphisms given by $\hom (\sigma , \tau ) = \operatorname {\mathrm {EP}}(\sigma , \tau )$ and composition defined by concatenation of paths. Given $\gamma \in \operatorname {\mathrm {EP}}(\sigma , \tau )$ , there exist lifts $\tilde \sigma , \tilde \tau \in S_{\Sigma _X, \beta _X}$ satisfying the properties

$$ \begin{align*} (\tilde \sigma \cap \ker(\underline\phi^{*})_{\mathbb R})/\ker(\underline\phi)=\sigma && (\tilde \tau\cap \ker(\underline\phi^{*})_{\mathbb R})/\ker(\underline\phi)=\tau && \tilde \sigma>\tilde \tau \end{align*} $$

By Proposition 2.14, this determines a boundary morphism

$$\begin{align*}\delta^\phi_\gamma: {\mathcal O}^\phi(\sigma)\to\mathcal {\mathcal O}^\phi( \tau),\end{align*}$$

which is independent of the lift chosen. In summary,

Proposition 3.9. There is a functor ${\mathcal O}^\phi \colon \operatorname {\mathrm {EP}}(S^\phi )\to \operatorname {\mathrm {Coh}}(X) $ which sends

$$ \begin{align*} \sigma\mapsto & {\mathcal O}^\phi(\sigma) \\ \gamma \mapsto & \delta^\phi_\gamma. \end{align*} $$

3.5 Definition of $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$

Whenever the one-dimensional cones of $\Sigma $ project to an $\mathbb R$ -basis for $N_X$ under $\underline {\beta }$ , which occurs when we assume that ${\mathcal X}$ is covered by smooth stacky charts, the strata of $S^\phi $ are simply connected. If $S^\phi $ is a regular CW complex, then $\operatorname {\mathrm {EP}}(S^\phi )$ is a posetal category (i.e., $|\operatorname {\mathrm {EP}}(\sigma ,\tau )| \in \{ 0,1 \}$ ). In our setting, it is possible that $S^\phi $ is not a regular CW complex – for example, when $\phi \colon \{\bullet \}\hookrightarrow \mathbb A^1 $ , as in Example 3.11. From $S^\phi $ , we build quiver $Q^\phi $ whose vertices are the strata of $S^\phi $ and whose edge sets are

$$\begin{align*}E(\sigma, \tau) =\left\{\begin{array}{cc} \operatorname{\mathrm{EP}}(\sigma, \tau) & \text{if } \dim(\tau)=\dim(\sigma)-1\\ \emptyset & \text{otherwise.}\end{array} \right.\end{align*}$$

We will let $Q^\phi $ denote this quiver, and $\gamma $ denote the edges in the quiver. We retain a map $|-|:Q^\phi \to \mathbb N$ which assigns to each stratum its dimension. Observe that when $S^\phi $ is a regular CW complex, $Q^\phi $ is the Hasse diagram of the poset associated with $S^\phi $ .

Now arbitrarily assign orientations $o_\sigma $ to every stratum $\sigma \in S^\phi $ . To each edge $\gamma \in Q^\phi $ , we say that $\operatorname {\mathrm {sgn}}(\gamma )=+1$ if the boundary orientation of $\sigma $ (with outwards direction assigned by $\dot \gamma $ ) agrees with the boundary orientation of $\tau $ , and $-1$ otherwise.

By Equation (18), we obtain a chain complex $C_{\bullet }(Q^\phi , {\mathcal O}^\phi )$ . For the purpose of streamlining notation, we will write

$$\begin{align*}C_{\bullet}(S^\phi, {\mathcal O}^\phi):= C_{\bullet}(Q^\phi, {\mathcal O}^\phi)\end{align*}$$

instead. The stratification $S^\phi $ has a special stratum $\sigma _0$ corresponding to the identity of the torus. The stratum $\sigma _0$ is always zero-dimensional under our assumptions and $\textbf {F}(\sigma _0)={\mathcal O}_{{\mathcal X}}$ . We define the morphism $\alpha \colon {\mathcal O}_{{\mathcal X}}\to C_{\bullet }(S^\phi , {\mathcal O}^\phi ) $ to map via the identity to that direct summand.

3.6 Examples of resolutions

Example 3.11 (Resolution of point in $\mathbb A^1$ ).

For $\phi \colon e\to \mathbb A^1 $ the inclusion of the identity point, the real torus is $T^\phi =S^1$ . Observe that the dimension of the torus is the codimension of the point. The stratification comes from a single point corresponding to the one toric divisor of $\mathbb A^1$ .

(12)

The complex is the standard resolution for the point in $\mathbb A^1$ .

Example 3.12 (Resolution of point in $\mathbb P^2$ ).

We now consider the identity point $e = (1:1:1)$ in $\mathbb P^2$ included by $\phi \colon e \to \mathbb P^2$ . The stratification $S^\phi $ is drawn in Figure 10a. We label each stratum with a sheaf and each exit path with a morphism of sheaves in Figure 10.

Figure 10 Stratification and associated diagram of sheaves for resolving the identity point in $\mathbb P^2$ .

For this example, the orientation on the 2-strata is given by the standard orientation of the plane, while the orientation on the 1-strata is given by the arrows indicated in Figure 10a. The resulting complex $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is given by the top row of the following commutative diagram.

We first check ‘by hand’ that this is a resolution of ${\mathcal O}_e$ . The bottom row provides a resolution of ${\mathcal O}_e$ (as the intersection of the lines $x_0-x_1$ and $x_1-x_2$ ). The morphisms between the top and bottom rows are chain homotopy equivalences.

We now instead sketch how to use Lemma 3.3 to prove that $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ resolves ${\mathcal O}_e$ following our general proof strategy. Let $i_{\rho }\colon \mathbb A^2\to \mathbb P^2$ be the toric inclusion of the $x_0x_1$ -plane. Then we can then look at $i^{*}_{\rho }C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ which is a diagram of sheaves on $\mathbb A^2$ (see Figure 11a).

The highlighted red arrow is now an invertible morphism, and we can construct a chain homotopy that simplifies the diagram by contracting along this arrow. This homotopy is the content of Lemma 3.3. The resulting diagram, drawn in Figure 11, is the Koszul resolution of the skyscraper sheaf on $\mathbb A^2$ . Since a similar story applies to the $x_0x_2$ and $x_1x_2$ planes, we obtain that $C_{\bullet }(S^\phi , {\mathcal O}^\phi )$ is a resolution of the skyscraper sheaf by line bundles.

Figure 11 Restriction to a chart in $\mathbb P^2$ is homotopic to the diagram for $\mathbb A^2$ .

Example 3.13 (Resolution of the diagonal on $\mathbb P^1\times \mathbb P^1$ ).

We now consider the diagonal inclusion $\phi \colon \mathbb P^1\to \mathbb P^1\times \mathbb P^1 $ .

On the left-hand side of Figure 12, we have the map of fans for the diagonal inclusion $\phi \colon \mathbb P^1\to \mathbb P^1\times \mathbb P^1 $ . On the bottom of the right-hand side, we have the torus $M_{\mathbb P^1, \mathbb R}\times M_{\mathbb P^1, \mathbb R}$ with the stratification $\mathcal {S}_{\mathbb P^1 \times \mathbb P^1}$ . The kernel of $\underline \phi ^{*}$ , on the bottom right, inherits a stratification labeled by line bundles on $\mathbb P^1\times \mathbb P^1$ . The corresponding diagram gives a resolution for $\phi _*{\mathcal O}_{\mathbb P^1}$ . This can be seen, for instance, by restricting to the four toric charts (setting some of the $x_i, y_i$ to $1$ ) as in the previous example.

Figure 12 Lattices and real tori which play a role in the construction of the resolution of the diagonal in $\mathbb P^1\times \mathbb P^1$ .

Example 3.14 (Resolution of the diagonal on $\mathbb P^2\times \mathbb P^2$ ).

While it is beyond our ability to draw the map of fans for the diagonal embedding $\phi \colon \mathbb P^2\to \mathbb P^2\times \mathbb P^2 $ , the torus $T^\phi $ is two-dimensional. The stratification with the associated resolution overlayed is drawn in Figure 13. The red (respectively blue) lines represent the one-dimensional cones from the first (respectively second) fan in $\Sigma _{\mathbb P^2}\times \Sigma _{\mathbb P^2}$ .

Figure 13 The resolution of the diagonal of $\mathbb P^2\times \mathbb P^2$ (signs omitted).