§ ABSTRACT

Beilinson gave a resolution of the diagonal for complex projective spaces which Bayer-Popescu-Sturmfels generalized to what they refer to as unimodular toric varieties. Here, we generalize Bayer-Popescu-Sturmfels resolution of the diagonal to the case of smooth toric varieties and global quotients of smooth toric varieties by a finite abelian group, considered as a toric Deligne-Mumford stack.

§ INTRODUCTION

In this paper, we study bounded derived categories of coherent sheaves $D^b_{Coh}(X)$ on a smooth projective variety $X$. The derived category $D^b_{Coh}(X)$ forms part of the B-side of Homological Mirror Symmetry (HMS). Beilinson's resolution of the diagonal <cit.> was crucial in understanding $D^b(\PP^n)$. Since $\PP^n$ is a toric variety, we can ask for a generalization of Beilinson's resolution of the diagonal to any toric variety. However, this will be too broad, so we will restrict our attention to simplicial toric varieties. Bayer-Popescu-Sturmfels generalized Beilinson's resolution of the diagonal for unimodular toric varieties. We generalize Bayer-Popescu-Sturmfels' resolution of the diagonal to toric D-M stacks $\mathcal{X}(\boldsymbol{\Sigma}') = \left[ \faktor{X_\Sigma}{\mu} \right]$ associated to the global quotient of a smooth toric variety $X_\Sigma$ by a finite abelian group $\mu$. By deforming the resolution $\mathcal{F}^\bullet_{\mathcal{H}_L/L}$ of a unimouldar toric variety in the case $X_\Sigma$ is smooth and non-unimodular, we  can resolve the diagonal in families corresponding to a parameter $\epsilon$ in the effective cone of $X_\Sigma$. Since smooth toric varieties admit an open cover by affine spaces $\mathbb{A}^m$ as a variety, local diagonal objects for $\left[ \faktor{X_\Sigma}{\mu}\right]$ glue together to give a global diagonal object for $\mathcal{X}(\boldsymbol{\Sigma}')$.

§.§ Results

This paper focuses on the following two theorems.

(A) The complex $(\mathcal{F}^\bullet_{\mathcal{H}_L^\epsilon/L}, \partial^\epsilon)$ of $S$-free modules gives a resolution of the diagonal object $\pi(M_{\Lambda(L)})$ of $D^b_{Coh}(X_\Sigma \times X_\Sigma)$ for $X_\Sigma$ a smooth toric variety.

(B) The complex $(\mathcal{F}^\bullet_{\mathcal{H}_L^\epsilon/\tilde{L}}, \partial^\epsilon)$ resolves the diagonal object $\pi(M_{\Lambda(L)}) = M_{\Lambda(L)} \otimes_{S[\Lambda(\tilde{L})]} S$ of $D^b_{Coh}(X_\Sigma' \times X_\Sigma')$ for a simplicial toric variety $X_\Sigma' = \faktor{X_\Sigma}{\mu}$ viewed as a global quotient toric Deligne-Mumford stack where $X_\Sigma$ is a smooth toric variety and $\mu$ is a finite abelian group.

Serre's theorem for projective space gives an equivalence between quasi-finitely generated graded modules over the homogeneous coordinate ring modulo the equivalence that $M \sim M'$ if there is an integer $d$ such that $M_{\geq d} \cong M'_{\geq d}$ <cit.>. Cox's theorem gives an equivalence between graded modules over the homogeneous coordinate ring of a simplicial toric variety $X_\Sigma$, modulo torsion, and coherent sheaves on $X_\Sigma$ <cit.>. Specifically, Cox's theorem on the relation between coherent sheaves on a simplicial toric variety $X_\Sigma$ and graded modules over the homogeneous coordinate ring, modulo torsion states:

* On a simplicial toric variety $X_\Sigma$, every coherent sheaf is of the form $\tilde{M}$ for some finitely generated module over the homogeneous coordinate ring $S$.

* If $X_\Sigma$ is a smooth toric variety and $F$ is a finitely generated graded $S$-module, then $\tilde{F}=0$ iff there is some $k>0$ with $I_{irr}^k F = 0$ for $I_{irr}$ the irrelevant ideal of $X_\Sigma$.

We rely on part 2 of Cox's theorem to resolve the diagonal for $X_\Sigma$ smooth.

Cox's theorem has the following generalization for simplicial toric D-M stacks. Let the simplicial toric variety $X_\Sigma = \faktor{\left(\A^n \setminus V(I_{irr}) \right)}{ T}$ as a GIT quotient, with $\chi$ the associated toric D-M stack. Let $\mathcal{U}$ be the quasi-affine variety $\A^n \setminus V(I_{irr})$.

* There is an equivalence of categories between finitely generated modules over the homogeneous coordinate ring of $\mathcal{U}$ (modulo torsion) and coherent sheaves on $\mathcal{U}$.

* The category of graded, finitely generated modules over the homogeneous coordinate ring of $\mathcal{U}$ (modulo torsion) is equivalent to the category of equivariant coherent sheaves on $\mathcal{U}$.

* The category of graded, finitely generated modules over the homogeneous coordinate ring (mod torsion) of $\mathcal{U}$ is equivalent to coherent sheaves on the toric D-M stack $\chi$.

If $X_\Sigma$ is a simplicial toric variety, then we view it as a simplicial toric D-M stack.

Bayer-Popescu-Sturmfels<cit.> give a cellular resolution of the Lawrence ideal $J_L$ corresponding to a unimodular lattice $L$. Imposing the requirement that $L$, the lattice of principal divisors of the projective toric variety be unimodular is fairly restrictive: In this case $L$ is given by the image of $B$ in the fundamental exact sequence<cit.>
\[ 0 \rightarrow M \stackrel{B}{\rightarrow} \Z^{|\Sigma(1)|} \stackrel{\pi}{\rightarrow} Cl(X_\Sigma) \rightarrow 0 \]

satisfies the property that $B$ has linearly independent columns and every maximal minor of $B$ lies in the set $\{0,+1,-1\}$ (this is the sense of “unimodularity" used in Bayer-Popescu-Sturmfels<cit.>). This forms a proper subclass of smooth projective toric varieties, as $\PP^2$ blown up at a point and then blown up along the exceptional divisor is no longer unimodular. Also, any non-trivially weighted projective space is no longer unimodular. Throughout, the word “unimodular" refers to this restricted sense of Bayer-Popescu-Sturmfels.

For a unimodular toric variety $X_\Sigma$,  the known resolution of $J_L$ gives a finite and minimal resolution of the ideal sheaf $\mathcal{O}_\Delta$ of the diagonal $\Delta \subset X_\Sigma \times X_\Sigma$ by sums of line bundles
on $X_\Sigma\times X_\Sigma$. In the well-known example for $X_\Sigma=\PP^n_\C$, Beilinson's resolution gives the kernel of a Fourier-Mukai transform inducing the identity on $D^b_{Coh}(X_\Sigma)$. In this paper, we generalize this resolution to simplicial toric varieties $X_\Sigma'$ which are a global quotient of a smooth toric variety $X_\Sigma$ by a finite abelian group $\mu$ by viewing $X_\Sigma'$ as a toric D-M stack in Theorems (A) and (B) above.

For Theorem (B), the cofinite sublattice $\tilde{L}\subset L$ is chosen so that $\faktor{L}{\tilde{L}}\cong \mu$. Now the technique and deformations of the resolution of $J_L$ in Bayer-Popescu-Sturmfels generalize to give a resolution of the diagonal
\[ (\mathcal{F}_{\mathcal{H}_L^\epsilon/\tilde{L}}, \partial) \]

The proof is given in Section <ref> and culminates in Section <ref>.

§ KNOWN RESOLUTIONS: MINIMAL RESOLUTION OF UNIMODULAR LAWRENCE IDEAL $J_L$ (BAYER-POPESCU-STURMFELS)

Here, we recall the conventions of Bayer-Sturmfels<cit.> and Bayer-Popescu-Sturmfels<cit.> to resolve the diagonal for a unimodular toric variety. For $\Sigma$ a complete fan in the lattice $\Z^m$ and $X_\Sigma$ the associated complete normal unimodular toric variety with $\{\textbf{b}_1, \dots, \textbf{b}_n\} \subset \Sigma(1)$ the primitive ray generators of $\Sigma$, and $B$ the $n\times m$ matrix with row vectors $\textbf{b}_i$, each $\textbf{b}_i$ determines a torus-invariant Weil divisor $D_i$ on $X_i$, and the group $Cl(X)$ of torus-invariant Weil divisors modulo linear equivalence is given by<cit.>

\begin{align} 0 \rightarrow \Z^m \stackrel{B}{\rightarrow} \Z^n \stackrel{\pi}{\rightarrow} Cl(X) \rightarrow 0, \end{align}

where $\pi$ takes the $i$-th standard basis vector of $\Z^n$ to the linear equivalence class $[D_i]$ of the corresponding divisor $D_i$.

Definiton: The Lawrence ideal<cit.> $J_L$ corresponding to $L$, a sublattice of $\Z^n$ is

\[ J_L = \left<\textbf{x}^\textbf{a}\textbf{y}^\textbf{b} - \textbf{x}^\textbf{b}\textbf{y}^\textbf{a} \text{ }|\text{ }\textbf{a}-\textbf{b} \in L \right> \subset S = k[x_1, \dots, x_n, y_1, \dots, y_n ] \]

for $k$ a field. Here $\textbf{x}^\textbf{a} = x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$ for $\textbf{a}=(a_1, \dots, a_n) \in \N^n$.

For $X_\Sigma$ unimodular with homogeneous coordinate ring $R = \C[x_\rho\text{ }|\text{ }\rho\in \Sigma(1)] \cong \C[x_1, \dots, x_n]$, the toric variety $X_\Sigma \times X_\Sigma$ has homogeneous coordinate ring

\[ S = R \otimes_k R = k[x_1, \dots, x_n, y_1, \dots, y_n]. \]

The diagonal embedding $X_\Sigma \subset X_\Sigma \times X_\Sigma$ defines a closed subscheme, and is represented by a $Cl(X) \times Cl(X)$-graded ideal $I_X$ in $S$. Here, $I_X = \ker(\psi)$ for $\psi: S = R\otimes_k R \rightarrow k[Cl(X)]\otimes R$ given by $\textbf{x}^{\textbf{u}}\textbf{y}^{\textbf{v}} = \textbf{x}^{\textbf{u}} \otimes \textbf{x}^{\textbf{v}} \mapsto [\textbf{u}]\otimes \textbf{x}^{\textbf{u}+\textbf{v}}$. That is,

\[ I_X = \left< \textbf{x}^\textbf{u}\textbf{y}^{\textbf{v}} - \textbf{x}^\textbf{v} \textbf{y}^\textbf{u} \text{ }|\text{ } \pi(\textbf{u}) = \pi(\textbf{v}) \text{ in } Cl(X) \right> \subset S. \]

First, we have the following

Lemma: Suppose $X = Spec(R)$ is an affine variety. Then the diagonal mapping $\Delta: V \rightarrow V \times V$ corresponds to the $\C$-algebra homomorphism $R \otimes R \rightarrow R$ given by $\sum r_1 \otimes r_2 \mapsto \sum r_1 r_2$, from the universal property of $X \times X$.

Next, locally, for $\Sigma \subset N_\R$, we have that the diagonal map $U_i \stackrel{\Delta}{\hookrightarrow} U_i \times U_i$ corresponds to

\begin{align*}
\C[\sigma_i^\vee \cap M] \otimes_k \C[\sigma_i^\vee \cap M] \rightarrow \C [\sigma_i^\vee \cap M ] \text{ given by} \\
\sum r_1 \otimes r_2 \mapsto \sum r_1 r_2 . \end{align*}

A choice of $\sigma\in \Sigma$ gives<cit.> the monomial
\[ x^{\hat{\sigma} } = \prod_{\rho\not\in\sigma(1) } x_\rho \in S \]
for $S$ the homogeneous coordinate ring<cit.> of $X$. And the map

\begin{align*}
\chi^m \mapsto x^{<m>}& = \prod_\rho x_\rho^{<m, u_\rho> } \end{align*}

induces the isomorphism

\begin{align*}
\pi_\sigma^*: \C[\sigma^\vee \cap M] &\cong ( S_{x^{\hat{\sigma}}})^G \text{ for }G \text{ corresponding to the GIT quotient of } \mathbb{A}^{|\Sigma(1)|}_{ss} \\
&\cong (S_{x^{\hat{\sigma}} } )_0 \end{align*}

which are degree 0 in the $Cl(X)$-grading on $S$.

So locally, we have

\begin{align*}
(S_{x^{\hat{\sigma}}})_0 \otimes_k ( S_{x^{\hat{\sigma}} })_0 \stackrel{\psi}{\rightarrow} (S_{x^{\hat{\sigma}}})_0 \text{ by }\\
\sum \textbf{u}^\textbf{a} \otimes_k \textbf{u}^\textbf{b} \mapsto \sum \textbf{u}^{\textbf{a}+\textbf{b}}. \end{align*}
for $\textbf{u}$ local coordinates in $U_\sigma$.

\begin{align*}
I_X &= \left< \textbf{x}^\textbf{u} \otimes \textbf{x}^\textbf{v} - \textbf{x}^\textbf{v} \otimes \textbf{x}^\textbf{u} \text{ }|\text{ } \pi(\textbf{u})=\pi(\textbf{v}) \in Cl(X) \right> \\
&= \ker(\psi) \end{align*}
for $\psi: R \otimes_k R \rightarrow k[Cl(X)]\otimes_k R \text{ via } \textbf{x}^\textbf{u}\otimes \textbf{x}^\textbf{v} \mapsto [\textbf{u}] \otimes \textbf{x}^{\textbf{u}+\textbf{v}}$, we have that locally:

\begin{align*}
(I_\Delta)_{x^{\hat{\sigma}} \otimes_k x^{\hat{\sigma}}} &= \ker( \textbf{x}^\textbf{u} \otimes_k \textbf{x}^\textbf{v} \mapsto [\textbf{u}]\otimes_k \textbf{x}^{\textbf{u}+\textbf{v}} ) \subset R_{x^{\hat{\sigma}}} \otimes_k R_{x^{\hat{\sigma}}} \end{align*}

\begin{align*}
R_{x^{\hat{\sigma}}} \cong k[ x_1 , \dots, x_r, w_1^{\frac{+}{}}, \dots, w_\ell^{\frac{+}{}} ] \end{align*}

from getting powers of variables to cancel when localizing by  $x^{\hat{\sigma}}$ for some  $r$ and $\ell$, and

\[
(R_{x^{\hat{\sigma}}})_0 =
\dots,
\frac{x_r}{\textbf{w}^{\textbf{$\alpha$}_r}}
] \]

from $X$ unimodular implies that for each $b_k \in \sigma(1)$ there exists $m_k \in M$ such that $m_k \cdot b_i = \delta_i^k \text{ for all } b_k \in \sigma(1)$. Taking degree zero and considering the kernel gives

\begin{align*}
(I_\Delta)_{x^{\hat{\sigma}} \otimes_k x^{\hat{\sigma}}} &= \left< \left\{ \frac{x_i}{w^{\alpha_i}} \otimes 1 - 1 \otimes \frac{x_i}{w^{\alpha_i}}  \right\}_{i=1}^r \right> \end{align*}
which we'll write as

\begin{align*} \left< \{ x^u - y^u \} \right> \end{align*}
on affine charts.

Now<cit.>, the ideal $I_X \subset S$ defining the diagonal embedding $X \subset X \times X$ equals the Lawrence ideal $J_L$ for the lattice $L=\text{Im}(B) = \ker(\pi)$ of principal divisors from the sequence at the beginning of this section <ref>. This latice $L$ is unimodular precisely when $X_\Sigma$ is unimodular, and since the Lawrence lifting

\[ \Lambda(L) = \{ (u, -u)\text{ }|\text{ }u\in L \} \]

is isomorphic to $L$, this implies that $\Lambda(L)$ is unimodular as well<cit.>.

Next, we'll require the following definitions:

Definition: Given a field $k$, we consider the Laurent polynomial ring $T = k[x_1^{\frac{+}{}}, \dots, x_n^{\frac{+}{}}]$ as a module over the polynomial ring $S = k[x_1, \dots, x_n]$. The module structure comes<cit.> from the natural inclusion of semigroup algebras $S = k[\N^n] \subset k[\Z^n] = T$.

Definition: A monomial module is an $S$-submodule of $T$ which is generated by monomials $\textbf{x}^\textbf{a} = x_1^{a_1}\cdots x_n^{a_n}, \textbf{a}\in \Z^n$.

Definition: When the unique generating set<cit.> of monomials of a monomial module $M$ forms a group under multiplication, $M$ coincides with the lattice module

\[ M_L := S\{\textbf{x}^\textbf{a} \text{ }|\text{ }\textbf{a} \in L \} = k\{\textbf{x}^\textbf{b} \text{ }|\text{ } \textbf{b} \in \N^n + L \} \subset T \]

for some lattice $L \subset \Z^n$ whose intersection with $\N^n$ is the origin $\textbf{0}=(0,0,\dots,0)$.

Definition: The $\Z^n/L$ graded lattice ideal $I_L$ is<cit.>

\[ I_L = \left< \textbf{x}^\textbf{a} - \textbf{x}^\textbf{b} \text{ }|\text{ } \textbf{a}-\textbf{b} \in L \right> \subset S. \]

Claim: $I_{\Lambda(L)} = J_L$.

\begin{align*}
I_{\Lambda(L)} &= \left< \mathbb{x}^\textbf{a}-\mathbb{x}^\textbf{b} \text{ }|\text{ }\textbf{a}-\textbf{b}\in \Lambda(L) \right>\\
&= \left<\textbf{x}^{\textbf{a}_1}\textbf{y}^{\textbf{a}_2}-\textbf{x}^{\textbf{b}_1}\textbf{y}^{\textbf{b}_2}\text{ }|\text{ }\textbf{a}=(\textbf{a}_1,\textbf{a}_2), \textbf{b}=(\textbf{b}_1, \textbf{b}_2), \textbf{a}-\textbf{b} \in \Lambda(L) \right>\\
&= \left< \textbf{x}^{\textbf{a}_1}\textbf{y}^{\textbf{a}_2} - \textbf{x}^{\textbf{b}_1}\textbf{y}^{\textbf{b}_2}\text{ }|\text{ }(\textbf{a}_1-\textbf{b}_1, \textbf{a}_2-\textbf{b}_2) = (\textbf{a}_1,\textbf{a}_2) - (\textbf{b}_1,\textbf{b}_2) \in \{(\textbf{u},-\textbf{u})\text{ }|\text{ }\textbf{u}\in L \} \right>
\end{align*}

where the rightmost constraint implies
$$ \textbf{a}_2-\textbf{b}_2 = -(\textbf{a}_1-\textbf{b}_1)=\textbf{b}_1-\textbf{a}_1 \text{ and } \textbf{a}_1-\textbf{b}_1\in L$$

\begin{align*}
&= \left<\textbf{x}^\textbf{u}\textbf{y}^\textbf{v} - \textbf{x}^\textbf{v} \textbf{y}^\textbf{u} \text{ }|\text{ } \textbf{u} - \textbf{v} \in L \right>= J_L
\end{align*}

While the hull resolution of the unimodular Lawrence ideal $I_{\Lambda(L)}=J_L$ is not necessarily minimal, the minimal resolution of $I_{\Lambda(L)}$ does come from a cellular resolution of $M_{\Lambda(L)}$ and is described by a combinatorial construction given by the infinite hyperplane arrangement<cit.> $\mathcal{H}_L$. Here, $\mathcal{H}_L$ comes from intersecting $\R L$, the real span of the lattice $L = \ker(\pi) \subset \Z^n$, with all lattice translates of the coordinate hyperplanes $\{x_i=j\}, 1\leq i \leq n, j\in \Z$. Bayer-Popescu-Sturmfels<cit.> prove that each lattice point in $L$ is a vertex of the affine hyperplane arrangement $\mathcal{H}_L$, and that there are no additional vertices in $\mathcal{H}_L$ iff $L$ is unimodular. Furthermore, the quotient complex $\mathcal{H}_L/L$ of the hyperplane arrangement modulo $L$ supports<cit.> the minimal $S$-free resolution of $J_L$, precisely because $\R L \cap \mathcal{H}_L = L$.

§.§ Example of cellular resolution of $\mathcal{H}_L/L$ for $\PP^2$

For the unimodular toric variety $X_\Sigma=\PP^2$, we have the fundamental toric exact sequence defining $Cl(X_\Sigma)$ via

\[ 0 \rightarrow \Z^2 \stackrel{B}{\rightarrow} \Z^3 \stackrel{\pi}{\rightarrow} \Z \rightarrow 0 \]

\[ B = \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ -1 & -1 \end{matrix} \right] \]

and $B(a,b)$ given by

\begin{align*}
\left[\begin{matrix} 1 & 0 \\ 0 & 1 \\ -1 & -1 \end{matrix}\right]\left[\begin{matrix} a \\ b \end{matrix}\right] &= \left[\begin{matrix} a \\ b \\ -a-b \end{matrix}\right]  \end{align*}

so that $\ker(\pi)=\text{Im}(B)$ contains

\[ \{e_1 - e_2, e_1-e_3, e_2-e_3 \} \]
but can be generated by
\[ \left\{ e_1-e_2, e_2-e_3 \right\}. \]

Now $L = \Z \cdot \left< \left[\begin{matrix} 1 \\ -1 \\ 0 \end{matrix}\right], \left[\begin{matrix} 0 \\ 1 \\ -1 \end{matrix}\right] \right> $ is rank 2.

If we label $\mathcal{H}_L$ with elements of the monomial module $M_{\Lambda(L)}$ and build the corresponding quotient cellular complex on $\mathcal{H}_L/L$ labeled with one up triangle and one choice of down triangle, subject to the orientation below, we see<cit.>:

Cellular Complex $\mathcal{H}_L/L$ for $\PP^2$

Here, the edges corresponding to the basis vectors $e_1-e_2$ and $e_2-e_3$ are labeled in blue, the edge ordering is given by the number of arrows (1-3), and the upward-pointing face of $\mathcal{H}_L/L$ is oriented counterclockwise while the downward-facing face is oriented clockwise in Figure <ref>. We order the 2-cells via $\{\text{``up-triangle," ``down-triangle"}\}$. The cellular complex $\mathcal{H}_L/L$ has one vertex, three edges, and two 2-cells. This choice of fundamental domain and face ordering gives the S-free resolution<cit.>  $(\textbf{F}_{\mathcal{H}_L/L}^\bullet, \partial)$ resolution of $S/J_L$:

\begin{equation}\label{eqn:3.2}
\begin{array}{ccccccccc}
0 &\rightarrow & S^2 &\stackrel{\left(\begin{matrix} y_2 & x_2 \\ y_1 & x_1 \\ y_3 & x_3 \end{matrix}\right) }{\longrightarrow}& S^3 &\stackrel{\left(\begin{matrix} y_1x_3 - x_1y_3 \\ x_2y_3 - x_3y_2\\ x_1y_2 - x_2y_1 \end{matrix}\right) }{\longrightarrow}& S^1 & \rightarrow & 0 \\ & & & & & & \downarrow\\ & & & & & & \faktor{S}{J_L} & \rightarrow & 0
\end{array}
\end{equation}

\[\partial(F) = \sum_{\makecell{F'\subset F,\\ codim(F',F)=1}}\epsilon(F,F')\frac{m_F}{m_{F'}} \cdot F' \]

for $m_F$ the monomial label of the face $F$ and $\epsilon(F,F')\in \{\frac{+}{}1\}$ with $\epsilon(F,F')=1$ if the orientation of $F'$ agrees with that induced by $F$. Here, $m_F$ is the lowest common multiple of $m_{F'}$ over the facets of $F$, starting with a labeling of the vertices.

That is, the cellular resolution of $\faktor{S}{J_L}$ coming from $M_{\Lambda(L)}$ in Equation <ref> resolves the ideal $I_{\PP^2}$ of the diagonal $\PP^2 \subset \PP^2\times \PP^2$.

§.§ Discussion

Here, the cellular resolution for $M_{\Lambda(L)}$ works precisely because the vertices of a unimodular lattice are exactly the vertices of $\mathcal{H}_L$. In order to resolve the diagonal for a toric D-M stack associated to a given simplicial toric variety, we require the Morita theory of Section <ref>.

We generalize the Cellular Resolution of $\mathcal{H}_L/L$ for toric D-M stacks associated to some simplicial toric varieties in Section <ref>.

§ RESOLUTION OF DIAGONAL FOR GLOBAL QUOTIENT OF UNIMODULAR TORIC VARIETY BY FINITE ABELIAN GROUP

§.§ Resolution of diagonal in bimodules

Let $A$ be an algebra over $R$, a commutative algebra over $k$ a field. $A$ is not necessarily commutative. We consider $I_\Delta \leq A \otimes_k A$ as a submodule given by $\ker \mu: A\otimes_k A \rightarrow A$ of multiplication. Here, $A \otimes_k A \geq A$ contains an isomorphic copy of $A$. We consider $A \otimes_k A$ as an $A \otimes_k A^{op}$-bimodule, not as an algebra. Here,
$$ A_\Delta = \faktor{A \otimes_k A}{I_\Delta}$$

To consider derivations and differentials of elements in $A$, consider the map given by

\[ \alpha \in A \mapsto \tilde{d}\alpha = \alpha \otimes 1 - 1 \otimes \alpha \in I_\Delta \]

We do not consider $I/I^2$ here.

Claim: $\ker\mu = I_\Delta \cong <\tilde{d}\alpha \text{ }|\text{ }\alpha \in A>$ as $A\otimes A^{op}$-module.

($\Rightarrow$): We construct $g: \ker \mu \rightarrow \left< \tilde{d}\alpha: \alpha\in A\right>$ via $g(\sum_i z_i\otimes_k w_i) = \sum_i z_i \tilde{d}w_i$ as follows:

\begin{align*}
\sum_i z_i\otimes w_i &\in \ker \mu \implies \sum_i z_iw_i = 0 \text{ so that }\\
\sum_i z_i\otimes w_i &= \sum_i z_i(1\otimes w_i - w_i\otimes 1) \in \left< \tilde{d}\alpha \text{ }|\text{ }\alpha\in A \right>. \end{align*}

($\Leftarrow$): We clearly have that $\tilde{d}\alpha = 1\otimes_k \alpha - \alpha\otimes_k 1 \in \ker\mu$.

Relative to $R$, if we suppose that $A$ is finitely generated as an $R-$module over $R$ by $\{f_1, \dots, f_r\}$ then
\[ \left< \tilde{d}a \text{ } |\text{ }a \in A \right> \cong \left< \tilde{d}r \text{ }|\text{ }r\in R \right> \otimes \{f_1, \dots, f_r \} \]

so that this gives a finite set of generators for $I_\Delta$.

Additionally, we observe that the Leibniz rule     $\tilde{d}(\alpha\beta) = \tilde{d}(\alpha)\beta + \alpha\tilde{d}(\beta)$ holds here.

§.§ Constructing local diagonal object via Morita theory

Consider $\faktor{\A^n}{\mu}$ where $\mu$ is a finite group of the torus. We have that $D(\faktor{\A^n}{\mu}) = D(k[z_1,\dots,z_n]-\text{Mod}^\mu)$ where modules are considered with grading. The goal here is to understand $D(\faktor{\A^n}{\mu} \times \faktor{\A^n}{\mu})$ in terms of $D(\faktor{\A^n}{\mu})$, and more specifically, to understand $\mathcal{O}_\Delta$ of $D(\faktor{\A^n}{\mu}\times\faktor{\A^n}{\mu})$ in terms of $\mathcal{O}_\Delta$ on $\faktor{\A^n}{\mu}$. That is, we would like a description of $\mathcal{O}_\Delta$ of $\faktor{\A^n}{\mu}\times \faktor{\A^n}{\mu}$ in terms of $\Delta$ for affine space. There is always the cohomological grading present, but now we also have an extra grading on $k[\A^n]-\text{Mod}^\mu$. Here, let $S = k[\A^n]$. Note that $D(\faktor{\A^n}{\mu})$ is generated by line bundles $\mathcal{O}(\xi)$ for characters $\chi\in \mu^*$, so we have the generator

\[ G = \sum_{\chi \in \mu^*}\mathcal{O}(\chi) \]

Note: $Hom(\mathcal{O}(\chi_i),\mathcal{O}(\chi_j))$ is $S$ with the right grading. Ignoring grading, we have

\begin{align*}
G &= \sum_{|\mu|}  S \end{align*}
so that

\begin{align*}    End^\bullet(G) &= Mat_{|\mu|}(S).
\end{align*}

In order to incorporate the grading, we would then take the degree $0$ part. Here we use Morita theory.

§.§ Background on Morita Equivalence

Note: Notation in this sub-subsection follows Weibel, and differs from notation used in later sections.

Definition: Two unital rings $R,S$ are said to be Morita equivalent if there exists an $R-S$ bimodule $P$ and $S-R$ bimodule $Q$ such that $P\otimes_S Q \cong R$ as $R-R$ bimodules and $Q\otimes_R P \cong S$ as $S-S$ bimodules. Then

$(-)\otimes_R P$ gives a functor from $\text{mod}-R \rightarrow \text{mod}-S$ and

$(-)\otimes_S Q$ gives a functor from $\text{mod}-S \rightarrow \text{mod}-R$

which are inverse categorical equivalences, since for all $M\in \text{ mod }-R$, we have

\begin{align*}
M\otimes_R P \otimes_S Q &\cong M\otimes_R (P\otimes_S Q) \cong M \end{align*}

\begin{align*}
M\otimes_S Q \otimes_R P &\cong M\otimes_S (Q\otimes_R P)\cong M
\end{align*}

for all $M\in \text{mod}-S$.

§.§ Choosing cofinite lattice $\tilde{L}\subset L$ for global quotient of unimodular toric variety

For the toric D-M stack associated to $X_\Sigma'$, a simplicial toric variety which is a global quotient $\faktor{X_\Sigma}{G}$ of a unimodular toric variety $X_\Sigma$ with $G$ a finite abelian group, we can choose an isomorphism of the lattice $L$ of principal divisors of $X_\Sigma$ which is unimodular such that $\tilde{L}\subset L$ is cofinite, and $\faktor{L}{\tilde{L}} \cong G$. Furthermore, locally, $X_\Sigma' \cong \faktor{\A^n}{G}$, so that locally we can consider $L \cong \Z^n$.

§.§ Diagonal Object in Mod-A

Here, let $R = k[\A^n]$ and $S = R\otimes_k R \cong k[x_1,\dots,x_n,y_1,\dots,y_n]$.
In $D(\faktor{\A^n}{G})$, we have generator $H=\bigoplus_{\chi\in\hat{G}}\mathcal{O}(\chi)$, with endomorphism algebra $A=End(H)$. Now, $A\simeq \Delta \in A\otimes A^{op}-Mod$. We have functors given in Figure <ref>

Morita Equivalence Functors Between $R-Mod$ and $Mat(R)-Mod$, and their Graded and $\hat{G}$-Invariant Counterparts Between $R-Mod^{\hat{G}}$ and $A-Mod$

Here, $P \in (R^{\hat{G}}, Mat(R))-Mod$ is a row vector with grading on generators

\[ P = \left< e_\chi\text{ }|\text{ } |e_\chi|=-\chi\right>, \]

$Q\in (Mat-R, R^{\hat{G}})-Mod$ is a column vector with grading on generators

\[ Q = \left< e_\chi\text{ }|\text{ }|e_\chi|=\chi\right> \]

and we have

\[ Hom_{Mat}(Q,-) \simeq P\otimes_{R^{\hat{G}}} ( - ) \]

For $P\otimes_A (-): A-Mod \rightarrow R-Mod^{\hat{G}}$ applied to $\Delta \cong A \in Obj(A\otimes A^{op}-Mod)$, we have that

\[ P\otimes_A A = \left< z^\alpha e_{\rho_1\rho_2} \text{ }|\text{ } |z^\alpha e_{\rho_1\rho_2}| = |z^\alpha| + \rho_1 - \rho_1 + \rho_2 \right> \]

with grading

\begin{align*}
|z^\alpha e_{\rho_1\rho_2}| &= |z^\alpha| + \rho_1-\rho_1 + \rho_2\\
&= |z^\alpha| + \rho_2. \end{align*}

For $\pi(M_{\Lambda(L)}) = M_{\Lambda(L)} \otimes_{S[\Lambda(\tilde{L})]} S$, we have

\begin{align*}
\pi(M_{\Lambda(L)}) &= S \cdot \{ \frac{x^u}{y^u} \text{ }|\text{ } u \in L \} \otimes_{S[\Lambda(\tilde{L})]} S
\end{align*}

with $S[\Lambda(\tilde{L})]= \left< x^{u_1}y^{u_2} \otimes z^{(v,-v)} \text{ }|\text{ } (u_1,u_2) \in \mathbb{N}^{2n}, v\in \tilde{L} \right>$ induces an $\faktor{L}{\tilde{L}}$ grading left on, say, $\{x_1,\dots,x_n\}$ from cancelling $y^u$ in the denominator. Here, the map

\[ \phi: S[L]\rightarrow M_L \text{ via } x^{u_1}y^{u_2} \otimes z^{(v_1,v_2)} \mapsto x^{u_1 + v_1}y^{u_2+v_2} \]
induces an $S[\Lambda(L)]$-module structure on $M_L$, hence an $S[\Lambda(\tilde{L})]$-structure on $M_{\Lambda(L)}$. In Bayer-Sturmfels<cit.>, the map $\phi: S[L]\rightarrow M_L$ was used to show that

\[ \pi(M_L) \cong \faktor{S[L]}{\ker\phi} \otimes_{S[L]} S \cong \faktor{S}{I_L}\]

by using the fact that $\pi(M_L)$ is cyclic when we only have the unimodular lattice $L$ to consider.

We can also view

\[ (R-Mod)^{\hat{G}} \rightarrow Mod-A \]

\begin{align*}
\bigoplus_{\rho\in\hat{G}} M_\rho \cong M_\cdot \mapsto Hom(\bigoplus_{\rho\in \hat{G}} R(-\rho), M) \cong \bigoplus_\rho Hom(R(-\rho),M) \cong \bigoplus_\rho M_\rho \end{align*}

where $\bigoplus_\rho M_\rho$ contains idempotents $e_{\rho\rho}\in A$ on the right-hand side.

We also have $\pi(M_{\Lambda(L)}) \in (S-Mod)^{\hat{G}\times\hat{G}}$ with

\begin{align*}
\pi(M_{\Lambda(L)}) \simeq +_{\overline{u}\in\hat{G}} M_{\tilde{L}} \cdot \tilde{z}^{\overline{u}}
\end{align*}

is a sum which is not necessarily direct, and

\begin{align*}
S\cdot \left\{ \frac{x^u}{y^u} \text{ }|\text{ }u\in L \right\} \subset S_{(x_1\dots x_n y_1 \cdots y_n )} \cong k[\Z^{2n}]. \end{align*}

Here, the localized ring $S_{(x_1\dots x_n y_1\dots y_n)}$ admits a $\Z^{2n}$ grading descending to an $\faktor{L}{\tilde{L}} \oplus \faktor{L}{\tilde{L}}$ grading. While $\pi(M_{\Lambda(L)})$ is no longer cyclic, we do have

\begin{align*}
S[L] \cong \bigoplus_{\overline{u} \in \hat{G}} S[\tilde{L}]\cdot z^{\overline{u}} \end{align*}

\begin{align*}
S[\Lambda(L)] \cong \bigoplus_{u\in \hat{G}} S[\Lambda(\tilde{L})] \cdot z^{(u,-u)} \end{align*}

carries an anti-diagonal grading. Now
$$S[L] = \left< x^u \otimes z^v \text{ }|\text{ } v \in L\right>$$
and in
$$ S[\Lambda(L)] = \left< x^{u_1}y^{u_2} \otimes z_1^u z_2^{-u} \text{ }|\text{ } u \in L \right>$$
$S$ has an $L \times L$ grading since $L\cong \Z^n$, and taking the quotient of $L$ by $\tilde{L}$ induces a quotient on $\Lambda(L)$ by $\Lambda(\tilde{L})$ which gives an $\faktor{L}{\tilde{L}} \times \faktor{L}{\tilde{L}}$-grading on $S[L]$. Now $S[\tilde{L}]$ has a degree 0 grading on $z'$s with respect to $\faktor{L}{\tilde{L}}$.

For $\phi: S[L]\rightarrow M_L$ via $x^u \otimes z^v \mapsto x^{u+v}$, we have that $\phi$ is surjective, $\ker\phi = \left< x^u \otimes z^v - x^{u+v}\otimes 1\right>$ and
$$I_L = \left< x^u - x^v \text{ }|\text{ }u-v\in L \right> \leq S $$
comes from $\ker\phi$ by setting $z's$ equal to 1. Here, we have that

\begin{align*}
M_L \cong \faktor{S[L]}{\ker\phi} \end{align*}
as $S[L]$ modules. As $S[\tilde{L}]$-modules, we still have

\[ S[L] \cong \bigoplus_{\overline{u}\in\hat{G}} S[\tilde{L}] \cdot z^{\overline{u}} \]

For $\phi: S[\Lambda(L)] \rightarrow M_{\Lambda(L)}$, we have that

$$ \ker\phi = \left< y_i \otimes z^{(e_i, -e_i)} - x_i \right>$$
is homogeneous of degree $(1,0)$ is one generating set, and $\ker\phi = \left< x_i \otimes z^{(-e_i, e_i)} - y_i\right>$ is another generating set of homogeneous elements of degree $(0,1) \in (\hat{G}, \hat{G})$. Furthermore, we have that

\begin{align*}I_{\Lambda(L)} = \left< x_j - y_j \right> \end{align*}
since $L \cong \Z^n$, and considering $\phi$ as a map of $S[\tilde{L}]$-modules gives that $\ker\phi$ still specializes to $I_{\Lambda(L)}$ when we set the $z$'s equal to $1$.

To show that

\[ \begin{array}{ccccc}
A \cong &\pi(M_{\Lambda(L)}) = M_{\Lambda(L)} \otimes_{S[\Lambda(\tilde{L})]} S \in (S-Mod)^{\hat{G}\times\hat{G}}\\
& || \\
& M_L \otimes_{S[\tilde{L}]} S \end{array}
\]

we note that $A$ is generated over $k$ by

\[ A/k = \left< w^\alpha e_{\rho_1\rho_2} \text{ }|\text{ } |\alpha| = \rho_1 - \rho_2 \right> \]

so we construct a map $\Psi: A \rightarrow \pi(M_{\Lambda(L)})$ by describing its image on generators of $A/k$:

\begin{align*}
\psi(w^\alpha e_{\rho_1 \rho_2}) = x^\alpha \otimes z^{\Lambda(\rho_2)}. \end{align*}

Given $M\in R-Mod$, we can use the projectors $e_{\rho_i\rho_i}$ to get
\[ e_{\rho_i\rho_i} M = M_{\rho_i} \in k-v.s. \]

\[ M = \bigoplus_i M_{\rho_i} \]
as a graded $R-$module.

To get a $\hat{G}\times \hat{G}$-grading on $A\in (A\otimes A^{op})-Mod$, we use projectors
\[ e_{\rho \rho} A e_{\rho'\rho'}\]
to pick up the $(\rho, -\rho')$-homogeneous component of $A$ in the $\hat{G}\times \hat{G}$-grading.

To get a $\hat{G}\times \hat{G}$-grading on $\pi(M_L)$, we have that

\[ |x^{\alpha_1}y^{\alpha_2}\otimes z_1^\beta z_2^{-\beta}| = (\alpha_1+\beta, \alpha_2-\beta). \]

Here, the grading on generators of $A$ over $k$ comes from Figure <ref>.

Quiver associated to $w^\alpha e_{\rho_1\rho_2}$

Since $\pi(M_{\Lambda(L)}) \in (S-Mod)^{\hat{G}\times\hat{G}}$ and $A \simeq \Delta \in (A \otimes A^{op})-Mod$, we put an $(S-Mod)^{\hat{G}\times \hat{G}}$ structure on $A$ via

\[ S = R\otimes_k R \]

and use the matrix representation representing multiplication by $r\in R$ to give $R\hookrightarrow A$ so that both $A$ and $\pi(M_{\Lambda(L)})\in (S-Mod)^{\hat{G}\times \hat{G}}$.

Now to check that $\psi$ is compatible with the $(S-\text{ Bimod })^{\hat{G}\times\hat{G}}$ structure, we note that

\begin{align*}
x^\beta w^\alpha e_{\rho_1\rho_2} &= w^{\alpha+\beta} e_{\rho_1+|\beta|, \rho_2} & \text{ Left action}\\
\end{align*}

\begin{align*}
y^\beta w^\alpha e_{\rho_1\rho_2} &= w^{\alpha+\beta} e_{\rho_1, \rho_2-|\beta|} &\text{ Right Action}
\end{align*}

gives that we have an $S-Mod$ homomorphism:
\[ f(s\cdot m) = s\cdot f(m) \]

\begin{align*}
\psi(x^\beta w^\alpha e_{\rho_1\rho_2}) &= \psi(w^{\alpha+\beta}e_{\rho_1+|\beta|, \rho_2} )\\
&= x^{\alpha+\beta} \otimes z^{\lambda(\rho_2)}\\
&= x^\beta \cdot (x^\alpha \otimes z^{\lambda(\rho_2)}\\
&= x^\beta \psi(w^\alpha e_{\rho_1\rho_2} )\\
\end{align*}

\begin{align*}
\psi(y^\beta w^\alpha e_{\rho_1\rho_2}) &= \psi(w^{\alpha+\beta} e_{\rho_1, \rho_2-|\beta|})\\
&= x^{\alpha+\beta}\otimes z^{(\rho_2-|\beta|, -\rho_2+|\beta|)}\\
&= (x^\beta\otimes z^{(-\beta,\beta)})\cdot(x^\alpha \otimes z^{\lambda(\rho_2)})\\
&= y^\beta (x^\alpha\otimes z^{\lambda(\rho_2)})\\
&= y^\beta \psi(w^\alpha e_{\rho_1\rho_2} ) \end{align*}

Lastly, to show that this is an isomorphism, we construct an inverse to $\Psi$. Since $\pi(M_{\Lambda(L)}) \cong R[\hat{G}]$ as $k-v.s.$, we construct

\[ \Phi: x^\alpha \otimes z^{\overline{v}} \mapsto w^\alpha e_{\alpha+\overline{v},\overline{v}} \]

as an inverse to $\psi$.

This shows that for global quotients, we tensor over $S[\Lambda(\tilde{L})]$ rather than over $S[\Lambda(L)]$ to get $\Delta$.

§.§ Global diagonal object and resolution of the diagonal for $X_\Sigma' = \faktor{X_\Sigma}{G}$ a global quotient of unimodular $X_\Sigma$ by $G$ a finite abelian group

Here, we'll give a global diagonal object for the example of $\faktor{\PP^n}{G}$ and a proof for why this is a global diagonal object for $\faktor{\PP^n}{G}$ which depends only on the fact that $\PP^n$ is unimodular, hence giving a global diagonal object for the more general case of $X_\Sigma' = \faktor{X_\Sigma}{G}$ a global quotient of unimodular $X_\Sigma$ by $G$ a finite abelian group.

Claim: We have a global diagonal object for $\faktor{\PP^n}{G}$ for $G$ a finite abelian group.

Locally, $\faktor{\PP^n}{G}$ is isomorphic to $\faktor{\A^n}{G}$. So locally, we have $U_\sigma' = \faktor{U_\sigma}{G}$ for $U_\sigma = \A^n$ open in $X_\Sigma = \PP^n$. For $\A^n$, we have presentation of $Cl(\A^n)=0$ via

\[ 0 \rightarrow M \stackrel{B}{\rightarrow} \Z^{\Sigma(1)} \stackrel{\pi}{\rightarrow} Cl(X) \rightarrow 0 \]

\[ 0 \rightarrow \Z^n \stackrel{Id}{\rightarrow} \Z^n \stackrel{\pi}{\rightarrow} 0 \rightarrow 0 \]

and $L= \ker(\pi)=Im(B)\cong \Z^n$. This gives $\Lambda(L)= \{(u,-u) \text{ }|\text{ } u\in \Z^n \}$. We have $\Delta$ for $\faktor{\A^n}{G}$ as

\begin{align*}
\Delta \cong \pi(M_{\Lambda(L)}) &= \faktor{S[\Lambda(L)]}{\ker\phi} \otimes_{S[\Lambda(\tilde{L})]} S \\
&= \faktor{\left( k[\A^{2n}]\otimes \left< z^{\Lambda(e_i)\frac{+}{}} \right> \right) }{\left< y_i - x_i \otimes z^{\Lambda(e_i)} \right> } \otimes_{S[\Lambda(\tilde{L})]} k[\A^{2n}]\\
&\cong R[\hat{G}]
\end{align*}

where $R[\hat{G}]$ is the group-ring with variables $w_i$ and $G \cong \faktor{L}{\tilde{L}}$. Now the transition maps for coordinates $x_i$ on local charts of $\A^n = U_\sigma \subset X_\Sigma=\PP^n$ induce transition maps on transition maps for $U_\sigma \times U_\sigma$, and hence on $U_\sigma' \times U_\sigma' = \faktor{U_\sigma}{G} \times \faktor{U_\sigma}{G}$. This induces transition maps on the $z^{\Lambda(e_i)}$'s from locally writing $x_i = y_i \otimes z^{\Lambda(e_i)}$ in $M_L$ in $U_\sigma'$. This implies that the transition maps for the $z^{\Lambda(e_i)}$'s glue together, and the diagonal object $\Delta$ for $\faktor{\A^n}{G}$ glues together to give a global diagonal object for $\faktor{\PP^n}{G}$.

Note: This proof is highly dependent on the fact that locally, $\PP^n$ is isomorphic to $\A^n$ in the sense that $\PP^n = \cup_i U_{\sigma_i}$ with each $U_{\sigma_i}\cong \A^n$. Since this also holds for any unimodular toric variety by the Lemma below, an analogous argument gives a global diagonal object for $X_\Sigma' = \faktor{X_\Sigma}{G}$ with $X_\Sigma$ unimodular.

Lemma: A complete unimodular toric variety $X_\Sigma$ has a cover by open charts which are all isomorphic to $\A^n$.

Let the complete unimodular toric variety $X_\Sigma$ have fan $\Sigma$ with maximal-dimensional cone $\sigma \in \Sigma(n)$. Then $\sigma^\vee \cap M$ is generated as a semigroup by normal vectors to the facets of $\sigma$, which is to say by the ray generators of $\sigma^\vee$. But there are $n$ such normal vectors, and

$X_\Sigma$ unimodular impies that $\sigma $ is unimodular, so subsets of $\rho_i \in \sigma(1)$ up to size $n$ are linearly independent over $\Z$. This implies that the $n$ ray generators of $\sigma^\vee$ are also linearly independent over $\Z$. So $U_\sigma \cong \A^n$.

Claim: $\pi(M_{\Lambda(L)})$ is resolved by the cellular complex associated to $\faktor{\mathcal{H}_L}{\tilde{L}}$, which we view as
\[ \faktor{\mathcal{H}_L}{\tilde{L}} \cong \bigoplus_{u\in G} \faktor{\mathcal{H}_L}{L}\]

together with a covering map of

\[ \begin{array}{c}
\faktor{\mathcal{H}_L}{\tilde{L}}\\
\downarrow \\
\faktor{\mathcal{H}_L}{L}
\end{array}\]

Since $M_{\Lambda(L)}$ is a monomial module, $\mathcal{H}_L$ resolves $M_{\Lambda(L)}$. The complex $(\mathcal{F}_{\mathcal{H}_L}^\bullet, \partial)$ is not $S$-finite, but has finite length $m=\text{rank}(L)$. It is also a minimal $\Z^{2n}$-graded free $S$-resolution of the lattice module $M_{\Lambda(L)}$ <cit.>.
Since $\faktor{\mathcal{H}_L}{L}$ resolves
\[ M_{\Lambda(L)} \otimes_{S[\Lambda(L)]} S\]

we have that $\faktor{\mathcal{H}_L}{\tilde{L}}$ resolves
\[ \pi(M_{\Lambda(L)}) = M_L \otimes_{S[\Lambda(\tilde{L})]} S. \]

§.§ Example of resolution for $\faktor{\PP^1}{\mu_6}$

Recall that for $\PP^1$ we have a presentation of $Cl(\PP^1)\cong \Z$ given by

\[\begin{array}{ccccccccc}
0 &\rightarrow & M & \stackrel{B}{\rightarrow} & \Z^{|\Sigma(1)|} &  \stackrel{\pi}{\rightarrow} & Cl(\PP^1) & \rightarrow 0\\
& & ||\wr & & ||\wr & & ||\wr \\
0 & \rightarrow & \Z & \rightarrow  & \Z^2 & \rightarrow & \Z & \rightarrow & 0
\end{array}\]

with maps $B$ and $\pi$ given by

\[ 0 \rightarrow \Z  \stackrel{ \left(\begin{matrix} 1 \\ -1 \end{matrix} \right)}{\rightarrow} \Z^2 \stackrel{(1 \text{ }1 ) }{\rightarrow} \Z \rightarrow 0 \]

so that $L = \ker(1 1)$ with $L = \Z\left< \left( \begin{matrix} 1 \\ -1 \end{matrix} \right) \right> \subset \Z^2$. If we fix $L \cong \Z$, then $\tilde{L} = 6\Z$ and $\faktor{L}{\tilde{L}}\cong \mu_6=G$. Now $\PP^1$ has monomial labels on $\faktor{\mathcal{H}_L}{L}$ given by Figure <ref>:

$\faktor{\mathcal{H}_L}{L}$ for $\PP^1$

For $G = \mu_6$, we have the cellular complex $\faktor{\mathcal{H}_L}{\tilde{L}}$ and is given by Figure <ref>.

$\faktor{\mathcal{H}_L}{\tilde{L}}$ for $\faktor{\PP^1}{\mu_6}$

The decomposition

\[ \faktor{\mathcal{H}_L}{\tilde{L}} = \bigoplus_{u\in G} \faktor{\mathcal{H}_L}{L} \]
and corresponding covering map $h$ is given by Figure <ref>

$\faktor{\mathcal{H}_L}{\tilde{L}}$ for $\faktor{\PP^1}{\mu_6}$

with corresponding complex and resolution

\[
\begin{array}{ccccc}
S^6 & \stackrel{\left( \begin{matrix} -x_1y_2 & 0 & \cdots & x_2y_1 \\ x_2y_1 & -x_1y_2 & \cdots & 0 \\
0 & x_2y_1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & -x_1y_2 \end{matrix} \right)}{\longrightarrow} & S^6 & \rightarrow & 0 \\
& &  \downarrow & \\
& &  \pi(M_{\Lambda(L)}) & \rightarrow & 0 \end{array} \]

The covering map $h$ corresponds to a subdivision of the maximal torus of $\PP^n$ by $G$.

§.§ Upgrading the grading on $(S-Mod)^{\hat{G} \times \hat{G}}$

If we add in a $(\Z\oplus \hat{G}) \times (\Z\oplus \hat{G})$-grading to $(S-Mod)^{\hat{G}\times \hat{G}}$, and view $\pi(M_{\Lambda(L)}) \in Ob\text{ }(S-Mod)^{\hat{G}\times \hat{G}}$, for instance, for $\faktor{\PP^1}{\mu_4}$ with $R = \C[x_0,x_1]$, exact sequence

\begin{align*}
0 \rightarrow \Z \stackrel{\left( \begin{matrix} 1 \\ -1 \end{matrix} \right)}{\rightarrow} \Z^2 \stackrel{\left( \begin{matrix} 4& 4 \end{matrix} \right)}{\rightarrow} \Z \rightarrow G \rightarrow 0
\end{align*}

with map

\begin{align*}
\Z \oplus \faktor{\Z}{4\Z} \leftarrow \Z^2\\
(a+b,\overline{a}) \leftarrow (a,b) \end{align*}

\begin{align*}
|x_0^{a_0}x_1^{a_1}| = (a_0+a_1, \overline{a_0})
\end{align*}

with $a_0=-a_1$, $a_0=4b$ from $L=Im(B)\subset \Z^2$ gives a $\Z\oplus G$-grading on $R$, inducing a $(\Z\oplus G)\times(\Z\oplus G)$-grading on $S$.

Now, with
\[ S[\Lambda(L)] \cong \bigoplus_{\overline{u} \in \hat{G}} S[\Lambda(\tilde{L})]\cdot z^{\Lambda(\overline{u})} \]

allows us to write the resolution for $\faktor{\PP^1}{\mu^4}$ as

\[
\begin{array}{cccccc}
0 \rightarrow S^4 & \stackrel{\left( \begin{matrix} -x_1y_2 & 0 & 0 & x_2y_1 \\ x_2y_1 & -x_1y_2 & 0 & 0 \\ 0 & x_2y_1 & -x_1y_2 & 0 \\ 0 & 0 & x_2y_1 & -x_1y_2 \end{matrix} \right)}{\longrightarrow} & \bigoplus_{\chi\in \hat{G}} \mathcal{O}(\chi, -\chi) & \rightarrow & 0 \\
& & \downarrow \\
& & \Delta & \rightarrow & 0
\end{array}
\]

where the vertical map downwards takes the basis element $e_\chi$ for $\mathcal{O}(\chi,-\chi)$ to $z^{\Lambda(\overline{u})} \in \pi(M_{\Lambda(L)})$.

§.§ Weighted projective space $\PP(1,2)$

For $\mathcal{L}, \mathcal{L}'$ line bundles on normal toric variety $X_\Sigma$, if $\mathcal{L}$ has sheaf of sections $\mathcal{O}(D)$ and $\mathcal{L}'$ has sheaf of sections $\mathcal{O}(E)$, then to compute

\[ Ext^i(\mathcal{O}_{X_\Sigma}(D), \mathcal{O}_{X_\Sigma}(E)) \]

we note that for $i=0$,

\begin{align*}
\mathcal{H}om(\mathcal{O}(D), \mathcal{O}(E)) \simeq \mathcal{O}(E-D)
\end{align*}

and $Ext^i(\mathcal{O}(D), \mathcal{O}(E)) = R\Gamma(\mathcal{O}(E-D)) = H^*(X_\Sigma, \mathcal{O}(E-D))$.

Now, for

$$\PP(1,2) = \left[ \faktor{\C^2\setminus\{(0,0)\}}{\C^*} \right]$$

where $\C^*$ acts via $\lambda\cdot(z_1,z_2) = (\lambda z_1, \lambda z_2)$, the set
\[ \mathcal{O}, \mathcal{O}(1), \mathcal{O}(2) \]
is an exceptional collection.

Note that

\begin{align*}
H^*(\mathcal{O}(-1)) &= R\Gamma Hom(\mathcal{O}(2),\mathcal{O}(1))\\
&= R\Gamma Hom(\mathcal{O}(1),\mathcal{O}).
\end{align*}

To find $H^*\mathcal{O}(n)$ on $\PP(1,2)$, we form a $\check{C}$ech cover on $\C^2$, noting that $z_1$ has degree $1$ and $z_2$ has degree $2$. On $U_1$, we localize at $z_1$ to get $\mathcal{O}(U_1)$. In Figure <ref> each lattice point is labeled by the total degree of monomial. On $U_2$, localizing at $z_2$ gives $\mathcal{O}(U_2)$. Here, lattice points in turqoise are again labeled by total degree of monomial. Level sets of total degree are highlighted in increments of 2. In the $\check{C}$ech cochain complex, $\check{C}^0(U_1 \cup U_2)$ is given by $\Gamma(U_1) \times \Gamma(U_2)$ in Figure  <ref>. Note that both $\Gamma(U_1)$ and $\Gamma(U_2)$ contain $(0,0)$. For
\[H^*\left(\left[ \faktor{\C^2\setminus\{(0,0)\}}{\C^*}\right]\right)\]
we consider elements only of degree $0$.

$\mathcal{O}(U_1)$ and $\mathcal{O}(U_2)$ on $\PP(1,2)$

$d^0(\check{C}^0)$ for $\mathcal{O}(-1)$ on $\PP(1,2)$

To build the $\check{C}$ech complex for $\mathcal{O}(n)$ on $\PP(1,2)$, we map $\mathcal{O}(U_1)$ to $\mathcal{O}(U_2)$ using $d^0$, which shifts $\mathcal{O}(U_2)$ $n$ steps to the left. For $\mathcal{O}(-1)$, and $H^*(\mathcal{O}(-1))$, $d^0(\check{C}^0)$ and $\check{C}^1$ are given in Figure <ref>. Here, $H^1$ is the kernel of $d^1$ by a dimension argument. Any lattice points labeled by both red and turquoise have elements in the kernel. But this does not occur for any points with degree 0, so $H^1(\PP(1,2),\mathcal{O}(-1))=0$. Here, $H^0$ is the cokernel: but all degree 0 elements lie in the image of $\check{C}^0$, since each degree 0 lattice point is labeled either red or turquoise. Therefore,
\[ H^i(\PP(1,2), \mathcal{O}(-1)) = 0 \]
for all $i$.

For $\mathcal{O}(-2)$, we would move the turquoise dots over to the right by $1$ in Figure <ref> (a total of 2 steps to the right). However, we still have no lattice points of degree $0$ labeled by both red and turqoise lattice points, so $H^1(\PP(1,2),\mathcal{O}(-2))=0$. Similarly, all degree $0$ lattice points are labeled by either red or turqoise dots, so the cokernel is 0, and $H^0(\mathcal{O}(-2))=0$ on $\PP(1,2)$.

$d^0(\check{C}^0)$ for $\mathcal{O}(-2)$ on $\PP(1,2)$

\begin{align*}
Ext(\mathcal{O}(2),\mathcal{O}(1)) \cong Ext(\mathcal{O}(1),\mathcal{O}) &\cong R\Gamma(\mathcal{O}(-1))\\
&\cong H^*(\mathcal{O}(-1))  = 0 \end{align*}

\begin{align*}
Ext(\mathcal{O}(2),\mathcal{O}) = R\Gamma(\mathcal{O}(-2)) = H^*(\mathcal{O}(-2))=0. \end{align*}

Note that $d^0(\check{C}^0)$ for $\mathcal{O}(-2)$ is given in Figure <ref>. To see that $\mathcal{O},\mathcal{O}(1),\mathcal{O}(2)$ generate $D^b(\PP(a,b))$ for $a,b>0$ and coprime, we note exactness of
\[ 0 \rightarrow \mathcal{O}(-a-b) \rightarrow \mathcal{O}(-a)\oplus\mathcal{O}(-b) \rightarrow \mathcal{O}\rightarrow 0 \]

on $\PP(a,b)$.

§ RESOLVING THE DIAGONAL FOR SMOOTH TORIC VARIETY $X_\SIGMA$

§.§ Resolving the diagonal via deformation: $Bl_{[1:0:0]}(\PP^2)$ and deforming to $\mathcal{H}_L^\epsilon$ for projective toric varieties which are smooth and non-unimodular

For toric varieties which are smooth and non-unimodular in the sense of Bayer-Popescu-Sturmfels, we'll need to deform the cellular complex $\mathcal{H}_L$ by an element $\epsilon\in \R^A$. We use $\epsilon$ to give transversal intersections at the vertices in $\mathcal{H}_L^\epsilon(0)$. Here, we deform the cellular complex $\mathcal{H}_L$ to $\mathcal{H}_L^\epsilon$ for $Bl_{[1:0:0]}\PP^2$ which is smooth, Fano, and unimodular as an illustrative example.

Recall that a sublattice $L\subset \Z^n$ is called unimodular if $L$ is the image of an integer matrix $B$ with linearly independent columns, such that all maximal minors of $B$ lie in the set $\{0,1,-1\}$ <cit.>. If we blow-up the torus-invariant point $[1:0:0]$ of $\PP^2$ (following Cox-Little-Schenck, I will call $X_\Sigma=Bl_{[1:0:0]}\PP^2$ the toric variety associated to the fan in $N_\R$ given in Figure <ref>), then the lattice $L$ given by the image of $B$ in

\[ 0 \rightarrow M \stackrel{B}{\longrightarrow} \Z^4 \stackrel{\pi}{\rightarrow}  Cl(X_\Sigma) \rightarrow 0 \]

where $B$ given by $\left[\begin{matrix} 1 & 0 \\ 1 & 1 \\ 0 & 1 \\ -1 & -1 \end{matrix}\right]$ is unimodular.

The fan $\Sigma$ and the nef cone for $Bl_{[1:0:0]}\PP^2$

\[ Cl(X_\Sigma) \cong \faktor{\left<D_1, \dots, D_4\right>}{(D_1+D_2-D_4, D_2+D_3-D_4)}\]

Since $D_2$ corresponds to the exceptional divisor in the blow-up, $D_2$ has self-intersection number -1:
\[ D_2\cdot D_2=-1\]
while $D_1, D_3$, and $D_4$ have self-intersection number $1$, so the nef cone of $X_\Sigma$ is given by the span of $D_1$ and $D_4$. This description of the nef cone comes from a map

\[ (\R^2)^\vee \stackrel{\left[\begin{matrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \end{matrix}\right]}{\longleftarrow} \R^4 \]

corresponding to a fan in the first orthant of $\R^4$. Here, the element $D_1+D_4$ in the nef cone corresponds to $\left[\begin{matrix} \epsilon \\ 0 \\ 0 \\ \epsilon\end{matrix}\right] = \epsilon \left[\begin{matrix} 1 \\ 0 \\ 0 \\ 1 \end{matrix}\right] \in \R^4$.

§.§.§ Deforming the cellular complex $\mathcal{H}_L$

Here, $X_\Sigma$ is assumed to be smooth.
If we consider $\R L = L\otimes_\Z \R \subset \R^4$ as before from the unimodular case, then the intersection of the infinite hyperplane arrangement $\mathcal{H}_L$ with $\R L$ will have more lattice points than just what appears in $L$ <cit.>. That is, for $X_\Sigma$ smooth and non-unimodular,

\[ (\mathcal{H}_L \cap \R L) \setminus L \neq \emptyset. \]

$\R L\cap \mathcal{H}_L$ is given in Figure <ref> for $Bl_{[1:0:0]}\PP^2$, which is unimodular (hence no extra vertices appear). Here, vertical lines in yellow correspond to integer values of $x_1$, and $x_1$ increases in value as we move to the right in the diagram. Horizontal lines in black correspond to integer values of $x_3$, which increase as we move up. Diagonal lines in blue give integer values in $x_2$ and $x_4$, which increase up and to the right, and down and to the left, respectively.

Since more than two hyperplanes intersect in $\R L$ at each point, the cellular complex $\mathcal{H}_L$ does not have transversal intersections at each vertex.

$\R L$, showing $\R L \cap \mathcal{H}_L \subset \R^4$

To remedy the lack of transversal intersection at each vertex $v \in \mathcal{H}_L(0)$, we adjust the infinite cellular complex $\mathcal{H}_L$ by deforming the hyperplane arrangement in a manner governed by our choice of $\epsilon$ in the nef cone, given by

\[ x \left[ \begin{matrix} 1 \\ 1 \\ 0 \\ -1 \end{matrix}\right] + y\left[\begin{matrix} 0 \\ 1 \\ 1 \\ -1 \end{matrix}\right] + \left[\begin{matrix} \epsilon \\ 0 \\ 0 \\ \epsilon \end{matrix}\right] = 0 \]

corresponding to row $1$ and row $2$ of $B$, giving a basis for $L\subset \R^4$.

This gives the deformed hyperplane arrangement $\mathcal{H}_L^\epsilon$ corresponding to the shifts $e_1 + \epsilon \in \Z, e_4 + \epsilon \in \Z$ for $0<\epsilon \ll 1$. We construct the Hull resolution $(\mathcal{F}^\bullet_{\mathcal{H}_L^\epsilon}, \partial^\epsilon)$ below. The deformed complex $\mathcal{H}_L^\epsilon$ is given in Figure <ref>, together with the monomial labelings given by taking the vertex labeling associated to the floor function for each coordinate of a given point in $\mathcal{H}_L^\epsilon \subset \R^4$. Vertices with common color labeling are given the same monomial vertex labeling in Figure <ref>. The arrows off of hyperplanes indicate direction of increase of $x_i$ for $1\leq i \leq 4$, as in Figure <ref>.

$\mathcal{H}_L^\epsilon$ for $Bl_{[1:0:0]}\PP^2$ deformed by $\mathcal{O}(D_1)+\mathcal{O}(D_4)$

Here, we emphasize that $\R L$ is a plane in $\R^4$ along which we are considering the intersections with the hyperplane arrangement $\mathcal{H}_L = \{ x_i \in \Z \text{ }|\text{ } 1\leq i \leq 4 \}$.

Taking the quotient of $\mathcal{H}_L^\epsilon$ by $L$ and giving monomial labels from $M_{\Lambda(L)}$ given by the integer floor function in each component are given in Figure <ref>.

Here, the differential

\[ \partial^\epsilon_1: \bigoplus_{e \in \mathcal{H}_L^\epsilon(1)}S \rightarrow \bigoplus_{v\in \mathcal{H}_L^\epsilon(0)} S \]
is given by

\[\begin{array}{c|cccccccccc|}
v_4 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & x_3y_4 & 0 & -x_3y_1\\ v_3 & 0 & 1 & -y_2 & 0 & 0 & 1 & 0 & 0 & -x_3y_1 & 0 \\
v_2 & 0 & 0 & 0 & 1 & -1 & 0 & -x_1y_4 & 0 & 0 & x_1y_3\\
v_1 & y_2 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & x_1y_3 & 0 \\
v_0 & -x_2 & 0 & x_2 & 0 & 0 & 0 & x_4y_1 & -x_4y_3 & 0 & 0 \\
\hline
& E_0 & E_1 & E_2 & E_3 & E_4 & E_5 & E_6 & E_7 & E_8 & E_9 \end{array} \]

subject to the labeling in Figure <ref>. The color labeling of vertices and edges in Figure <ref> corresponds to the color labeling in Figure <ref>. Exactness away from homological index $i=0$ follows as before for the unimodular case by the same convexity argument used in Theorem 3.1 in Bayer-Popescu-Sturmfels<cit.>. We investigate the cokernel of $\partial^\epsilon_0$ in the following section.

The quotient cellular complex $\faktor{\mathcal{H}_L^\epsilon}{L}$ with monomial labelings from $M_{\Lambda(L)}$

§.§ Cokernel of $\partial^\epsilon_0$

Here, we consider the map

\[ (\mathcal{F}_{\mathcal{H}_L^\epsilon}, \partial^\epsilon) \stackrel{f}{\rightarrow} S_{\prod x_i}\]

given by $\bigoplus_{v\in \mathcal{H}_L^\epsilon}S(-m_F) \rightarrow S_{\prod x_i}$ where a vertex in $\mathcal{H}_L^\epsilon$ maps to its vertex monomial label in $M_{\Lambda(L)}$.

For $Bl_{[1:0:0]}\PP^2$, we have that $L = \left< \left[\begin{matrix} 1 \\ 1 \\ 0 \\ -1\end{matrix}\right] , \left[\begin{matrix} 0 \\ 1 \\ 1 \\ -1 \end{matrix}\right]\right>$,

\begin{align*}
I_L &= \left< x^{v_+}-x^{v_-} \text{ }|\text{ } v\in L \right> \\
&= \left< x_1x_2-x_4, x_2x_3-x_4, x_1-x_3\right>, \end{align*}

\begin{align*}
J_L &= \left< x^uy^v - x^vy^u \text{ }|\text{ }u-v\in L\right>\\
&= \left<x_1x_2y_4 - x_4y_1y_2, x_2x_3y_4 - x_4y_2y_3, x_1y_3-x_3y_1\right> \end{align*}

and $M_{\Lambda(L)}$ as an $S-$submodule of $T$ has the infinite generating set $\{ x^uy^{-u}\text{ }|\text{ }u\in L \}$. In the quotient by $L$, the cellular complex $\mathcal{H}_L^\epsilon$ carries monomial labels given in Figure <ref>. When we consider monomial labels on vertices in the quotient $\mathcal{H}_L^\epsilon/L$, we only need to consider the monomial labels on vertices given in Figure <ref> due to which vertices in $\mathcal{H}_L^\epsilon(0)$ carry the same monomial label.

Here, $L\cap \N^n = \{0\}$ implies that $M_{\Lambda(L)}\ni 1$, so that $M_{\Lambda(L)} \subset Im(f)$ above, by considering the action of $L$ on the quotient from Figure <ref>. Modulo the action of $L$, there is one additional monomial $\frac{y_2}{x_2}$ in $Im(f)$ which is not contained in $M_{\Lambda(L)}$. Note that since

\[ \{ \frac{x_1x_2 y_4}{x_4y_1y_2}, \frac{x_2x_3y_4}{x_4y_2y_3}, \frac{x_1y_3}{x_3y_1} \} \subset M_{\Lambda(L)},\]

we also have

\[ \{\frac{x_4y_1y_2}{x_1x_2y_4}, \frac{x_4y_2y_3}{x_2x_3y_4}, \frac{x_3y_1}{x_1y_3} \} \subset M_{\Lambda(L)}. \]

Now for $X_\Sigma = Bl_{[1:0:0]}\PP^2$,

\begin{align*}
I_{irr} &= \left< x_3x_4, x_1x_4, x_1x_2, x_2x_3\right> \\
&= \left<x_1, x_3\right> \cap \left<x_2,x_4\right> \end{align*}

and $I_{irr}$ for $X_\Sigma \times X_\Sigma$ is given by
\[ \left<x_1, x_3\right>\cap\left<x_2,x_4\right> \cap \left<y_1,y_3\right> \cap \left<y_2, y_4\right>. \]

Now, for any monomial $q$ in $I_{irr}$ for $X_\Sigma \times X_\Sigma$, we must have that either $x_2$ or $x_4$ divides $q$. If $x_2$ divides $q$, then

\[ x_2 \left( \frac{y_2}{x_2} \right) = y_2 \in M_{\Lambda(L)} \]

since $M_{\Lambda(L)} \ni 1$ shows that $q \cdot \frac{y_2}{x_2} \in M_{\Lambda(L)}$. If $x_4 | q$, then

\begin{align*}
y_3x_4 \cdot \left( \frac{y_2}{x_2}\right) &= \left(\frac{x_4y_2y_3}{x_2x_3y_4}\right) \cdot x_3y_4 \in M_{\Lambda(L)} \end{align*}

shows that $q \cdot \left(\frac{y_2}{x_2}\right)$ is in the image of the action of $S$ on $q\cdot \left( \frac{y_2}{x_2}\right)$. Therefore, the cokernel of the inclusion of $M_{\Lambda(L)} \hookrightarrow Im(f)$ is torsion with respect to $I_{irr}$ for $X_\Sigma \times X_\Sigma$. In general, we say that an $R-$module $M$ is $I$ torsion iff there exists $k\in \Z_+$ such that $I^k M = 0$. Here, $k=1$ suffices to show that the cokernel of the inclusion of $M_{\Lambda(L)} \hookrightarrow Im(f)$ is torsion with respect to $I_{irr}$ for $X_\Sigma \times X_\Sigma$.

Monomial labels on vertices $v\in \mathcal{H}_L^\epsilon(0)$ modulo $L$

§.§ Moving around in the effective cone

Here we consider $\text{coker }(M_{\Lambda(L)} \hookrightarrow Im(f))$
for $\epsilon$ in the interior of the other chamber of the effective cone, spanned by $\mathcal{O}(D_2)$ and $\mathcal{O}(D_4)$. (Again, $D_2$ corresponds to the exceptional divisor of the blow-up of $Bl_{[1:0:0]}\PP^2$.) Our previous discussion of $\text{coker}(M_{\Lambda(L)}\hookrightarrow Im(f))$ from Section <ref>  took place inside the nef cone of $N^1(X_\Sigma)$, with $N^1(X_\Sigma)\cong Cl(X_\Sigma)\otimes_\Z \R$ since $X_\Sigma$ is smooth. More generally, we can ask about the cokernel of the inclusion of $M_{\Lambda(L)}\hookrightarrow Im(f)$ inside of the other chamber of the effective cone, given in Figure <ref>.

Chambers of the Effective Cone

Inside of this cone spanned by $\mathcal{O}(D_2)$ and $\mathcal{O}(D_4)$ in $Cl(X_\Sigma)\otimes_\Z \R$, we have a corresponding fan for which the ray $\rho_2$ is removed. This gives the fan for $\PP^2$, though we retain the information that $\rho_2$ no longer lives in any maximal cone so that the irrelevant ideal for $X_\Sigma$ becomes

\[ I_{irr} = \left<x_1,x_3,x_4\right> \cap (x_2) \]

and the irrelevant ideal for $X_\Sigma \times X_\Sigma$ in this case is

\[ I_{irr} = \left<x_1,x_3,x_4\right> \cap (x_2)\cap \left<y_1, y_3, y_4\right> \cap (y_2). \]

Taking $\epsilon = \left[ \begin{matrix} 0 \\ \epsilon \\ 0 \\ \epsilon \end{matrix}\right]$ corresponding to $\mathcal{O}(D_2)+\mathcal{O}(D_4)$ in the interior of the chamber of the effective cone spanned by $\mathcal{O}(D_2)$ and $\mathcal{O}(D_4)$, we have the deformed cellular complex $\mathcal{H_L^\epsilon}$ whose quotient by $L$ is given in Figure <ref>. Here, $x_2=0$ and $x_4=0$ are labeled in blue and yellow, respectively, since those planes are deformed in $\R L$.

Deformed complex $\mathcal{H}_L^\epsilon$ for $\epsilon$ corresponding to $\mathcal{O}(D_2) + \mathcal{O}(D_4)$

In this case, $I_{irr}$ for $X_\Sigma \times X_\Sigma$ is given above and $\text{coker}(M_{\Lambda(L)}) \ni \frac{y_1}{x_1}, \frac{y_3}{x_3}$ and is spanned by translates of these elements by $L$. Note that in this case, at least one of $x_1, x_3$, or $x_4$ must divide any monomial $q$ in the irrelevant ideal $I_{irr}$ of $X_\Sigma\times X_\Sigma$.

For $L$-translates of $\frac{y_1}{x_1}$, we note that if $x_1$ divides $q$, then

\begin{align*}
x_1 \cdot (\frac{y_1}{x_1})=y_1 \in M_{\Lambda(L)}\end{align*}
since $1\in M_{\Lambda(L)}$, and $M_{\Lambda(L)}$ is an $S-$submodule of $T= k[x_1^{\frac{+}{}}, \dots, x_n^{\frac{+}{}}, y_1^{\frac{+}{}}, \dots, y_n^{\frac{+}{}}]$. If $x_3$ divides $q$, then

\begin{align*}
x_3 \cdot \frac{y_1}{x_1} &= y_3 \cdot \left(\frac{x_3y_1}{x_1y_3}\right) \in M_{\Lambda(L)}. \end{align*}

If $x_4$ divides $q$, then

\begin{align*}
x_4y_2 \left(\frac{y_1}{x_1}\right) &= x_2y_4\left(\frac{x_4y_1y_2}{x_1x_2y_4}\right) \in M_{\Lambda(L)}\end{align*}

so that $\overline{\frac{y_1}{x_1}} = \overline{0} \in \text{coker}(M_{\Lambda(L)}\hookrightarrow Im(f) )$.

Next, for ($L-$translates of) $\frac{y_3}{x_3}$, we again have that at least one of $x_1, x_3$, or $x_4$ divides any monomial $q$ in $I_{irr}$, the irrelevant ideal for $X_\Sigma\times X_\Sigma$. If $x_1$ divides $q$, then

\begin{align*}
\frac{y_3}{x_3}\cdot x_1 &= \left(\frac{x_1y_3}{x_3y_1}\right)\cdot y_1 \in M_{\Lambda(L)}\end{align*}

If $x_3$ divides $q$, then

\begin{align*} \frac{y_3}{x_3}\cdot x_3 &= y_3\in M_{\Lambda(L)} \end{align*}
since $1 \in M_{\Lambda(L)}$. Lastly, if $x_4 | q$, then
\begin{align*} \left(\frac{y_3}{x_3}\right)x_4y_2 &= \left(\frac{x_4y_2y_3}{x_2x_3y_4}\right)x_2y_4 \in M_{\Lambda(L)}. \end{align*}

This shows that $\overline{\frac{y_3}{x_3}} = 0 \in \text{coker}(M_{\Lambda(L)}\hookrightarrow Im(f))$. Hence,

\[ \text{coker}(M_{\Lambda(L)}\hookrightarrow Im(f)) = 0 \]

modulo $I_{irr}$ for $X_\Sigma \times X_\Sigma$ for $\epsilon = \mathcal{O}(D_2)+\mathcal{O}(D_4)$ in the interior of the chamber spanned by $\mathcal{O}(D_2)$ and $\mathcal{O}(D_4)$ in the effective cone of the secondary fan.

§.§ General argument for $\text{coker}(M_{\Lambda(L)}\hookrightarrow Im(f))$ for $\epsilon$ in the effective cone

Given $X_\Sigma$ smooth and projective with no torus factors, we have the following exact sequence:

\[
\begin{array}{ccccccccc}
0 &\leftarrow &Cl(T)_\R &\leftarrow &\R^A & \leftarrow & L_\R &\leftarrow &0 \\
& &  & & \rotatebox[origin=c]{90}{$\in$}\\
& & & & \epsilon\end{array}\]

where $\epsilon\mapsto \overline{\epsilon}$ in the effective cone of $\mathcal{F}_{\Sigma(A)}$. Now $\overline{\epsilon}$ gives a convex function $g_\epsilon: N_\R \rightarrow \R$ which is linear on all cones of a corresponding fan $\Sigma_\epsilon$ such that the maximal domains of linearity correspond to maximal cones in $\Sigma_\epsilon$. So $g_\epsilon$ gives the fan $\Sigma_\epsilon$ with corresponding irrelevant ideal $I_{irr}$ for $\Sigma_\epsilon\times\Sigma_\epsilon$ given by
\[ \left< \prod_{i\in I_\emptyset} x_i \prod_{\rho\not\in\sigma} x_\rho \text{ }|\text{ } \sigma \in \Sigma_{max} \right> \cap \left< \prod_{i\in I_\emptyset} y_i \prod_{\rho\not\in \sigma} y_\rho \text{ } \sigma \in \Sigma_{max}\right> \]

Continuing with the map $(\mathcal{F}_{\mathcal{H}_L^\epsilon}^\bullet, \partial_\epsilon^\bullet) \stackrel{f}{\rightarrow} S_{\prod x_i}$ and our consideration of $coker(M_{\Lambda(L)} \hookrightarrow Im(f))$ modulo $I_{irr}$ for $\Sigma_\epsilon \times \Sigma_{\epsilon}$, we consider $p\in \R^{\mathcal{A}}$ such that $p \mapsto \overline{p}\in\mathcal{H}_L^\epsilon(0)$ near $\overline{0}$ in $\R L$. We have previously referred to $\overline{p}\in \R L$ as $p$. Here, we take

\[ \overline{p} = \bigcap_{\rho\in \mathcal{B}_p} \{x_\rho = 0 \} \]

as a definition of $\mathcal{B}_p$. Now transversality of $\mathcal{H}_L^\epsilon$ implies that $\mathcal{B}_p$ gives a basis for $\R L$. Since $p$ is near $\vec{0}$ in $\R^\mathcal{A}$, the monomial label $m_p$ is squarefree, and $\epsilon \in \R^\mathcal{A}_{\geq 0}$ implies that $m_p$ can be written $\frac{y_{i_1} \cdots y_{i_\ell}}{x_{i_1}\cdots x_{i_\ell}}$.

By the definition of $\mathcal{H}_L^\epsilon$, the monomial label on $p$ is given by
\[ m_p = \frac{x^{\vec{u}}}{y^{\vec{u}}} \text{ }|\text{ } \vec{u} = \lfloor p \rfloor. \]
We now adjust $g_\epsilon$ by subtracting a linear functional $\ell_p$ determined by $\ell_p(u_\rho)=\epsilon_\rho$ for $\rho\in \mathcal{B}_p$. $\ell_p$ is determined by this information, since $\mathcal{B}_p$ gives a basis for $\R L$ and for $N$. That is, we construct
\[ \tilde{g}_\epsilon := g_\epsilon - \ell_p \]
so that we have
\[ p = \tilde{g}_\epsilon(\{u_\rho\}) \]

except for coordinates in $I_\emptyset$. Since any variable corresponding to an element of $I_\emptyset$ always appears in the support of any monomial in the irrelevant ideal, we may assume without loss of generality that $p = \tilde{g}_\epsilon$. To show that $coker (M_{\Lambda(L)}\hookrightarrow Im(f)) =0$ modulo $I_{irr}$, let $\sigma$ be any maximal cone in $\Sigma_{max}$. From convexity of $g_\epsilon$, it follows that $\tilde{g}_\epsilon$ is also convex. This gives a subdivision of maximal cones in $\Sigma$ such that either:

(i) $\sigma \subseteq \tilde{g}_\epsilon^{-1}(\R_{\geq 0})$

(ii) $\sigma \subseteq \tilde{g}_\epsilon^{-1}(\R_<0)$

(iii) Neither (i) nor (ii) holds

In case (i), since we only need to worry about what's not in $I_\emptyset$, we can suppose that $g_\epsilon(u_\rho)=\epsilon_\rho$ so that $p = \tilde{g}_\epsilon(\{u_\rho\})$. If we choose $\epsilon$ to be small so that $|\epsilon|<<1$, then $\ell_p$ remains small as a continuous function of $\epsilon$ which gives that the coordinates of $p\in (-1,1)$ and the floor function of all coordinates is in the set $\{-1,0\}$. Therefore, the set of indices $\{i_j\}$ for which $x_{i_j}$ is in the support of $m_p$ are exactly the coordinates for which the floor function returns $-1$. Since $x^{\hat{\sigma}} = \prod_{\rho\not\in\sigma} x_\rho$ is a generator of $I_{irr}$ (ignoring $I_\emptyset$), we have that all variables cooresponding to rays of $\sigma$ do not lie in the support of $m_p$, so that $x^{\hat{\sigma}}$ can clear denominators to give $x^{\hat{\sigma}}\cdot m_p \in M_{\Lambda(L)}$.

In case (ii), we have that $\sigma \subseteq \tilde{g}_\epsilon^{-1}(\R_{<0})$, $m_p = \frac{y_{i_1}\cdots y_{i_\ell}}{x_{i_1} \cdots x_{i_\ell}}$ and $x^{\hat{\sigma}} = \prod_{\rho\not\in \sigma} x_\rho$. For $p$ near $\vec{0}$ we show that there exists $k>0$ such that $I_{irr}^k \cdot m_p \in M_{\Lambda(L)}$ and $coker(M_{\Lambda(L)}\hookrightarrow Im(f)) = 0$ by adding in the assumption that $X_\Sigma$ is Fano. If $X_\Sigma$ is Fano, then the anticanonical divisor $D = \sum_{\rho\in\Sigma(1)} D_\rho $ is very ample so that we can define $v\in L$ to be $v(u_\rho)=-1$ for $\rho\in \sigma$ and away from $\sigma$, $v(\rho') > -1$ for $\rho'\not\in \sigma$. Since $v\in L$ is an element of the lattice $L$, $v(u_\rho) \in \Z$ for all $\rho\in \Sigma(1)$ implies $v(\rho')\geq 0$ for all $\rho'\not\in \sigma$.

Now $\frac{x^{\vec{v}}}{y^{\vec{v}}} \in M_{\Lambda(L)}$ and we need only consider variables in the support of $m_p$ for which $\rho\in \sigma$ from our previous description of $I_{irr}$. Here, there exists some $\vec{w}\in \N^\mathcal{A}$ with $y^{\vec{w}}\in S$ and some $\{s_\rho\} \subset \N$ so that

\begin{align*}
\prod_{\rho\not\in \sigma} x_\rho^{s_\rho} \cdot m_p &= y^w \cdot \frac{x^{\vec{v}}}{y^{\vec{v}}} \in M_{\Lambda(L)}
\end{align*}

\[ (x^{\hat{\sigma}})^{\text{max}\{s_\rho\}} \cdot m_p = q \cdot \prod_{\rho\not\in \sigma} x_\rho^{s_\rho} m_p \in M_{\Lambda(L)} \]
for some $q\in S$. Here we use the fact that $|\mathcal{A}|<\infty$ to take a maximum, and the fact that $I_{irr}$ has a finite generating set since $S$ is Noetherian.

We generalize statements (i)-(iii) in the next subsection without the Fano assumption and therefore omit consideration of (iii) here.

§.§.§ For $p\in \mathcal{H}_L^\epsilon(0)$ away from $\vec{0}$

For $p$ away from $\vec{0}$ and $X_\Sigma$ smooth (not assumed to be Fano here), we make use of the following lemmas:

Lemma 1: Given $p\in \mathcal{H}_L^\epsilon(0)$ and $\sigma_1, \sigma_2 \in \Sigma(n)$, there exists $k>0$ with $(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^k \cdot m_p \in M_{\Lambda(L)}$ iff there exists $v\in L$ and an $\ell>0$ such that $(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^\ell \cdot m_{p+v} \in M_{\Lambda(L)}$.

Any $v\in L$ has integer coordinates so that $\lfloor v \rfloor = v$ and $m_v = \frac{x^{\lfloor v \rfloor}}{y^{\lfloor v \rfloor }} = \frac{ x^v}{y^v}$. This implies that adding $v\in L$ to $p$ shifts all coordinates by only integer amounts, so that no cancellation occurs when we take a floor function:
\[ \lfloor p+v \rfloor = \lfloor p \rfloor + v \]

In particular, we note that if both $p$ and $v$ are in $L$, then $m_{p+v}=m_p \cdot m_v$ since $p, v \in L \implies p+v\in L$. So if there exists $k>0$ with $(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^k \cdot m_p \in M_{\Lambda(L)}$, then $(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^k\cdot m_{p+v} = (x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^k\cdot m_p m_v \in M_{\Lambda(L)}$ since $M_{\Lambda(L)}$ is closed under the action of $S[L]$. Now closure of $M_{\Lambda(L)}$ under the action of $S[L]$ also proves the reverse direction.

For the following Claim, we require the Separation Lemma<cit.>, which separates convex sets by a hyperplane:

Separation Lemma: If $\sigma$ and $\sigma$' are convex polyhedral cones whose intersection $\tau$ is a face of each, then there is a $u$ in $\sigma^\vee \cap (-\sigma')^\vee$ with
\[ \tau = \sigma \cap u^\perp = \sigma' \cap u^\perp. \]

Generalizing the previous cases (i)-(iii) from Section <ref>, we make the following claim.

Claim: Given $p\in \mathcal{H}_L^\epsilon(0)$, for all $\sigma_1, \sigma_2\in \Sigma(n),$ there exists $u\in L, k\in \mathbb{N}$ and $\alpha,\beta\in \mathbb{N}^{\Sigma(1)}$ such that $(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^km_p = x^\alpha y^\beta m_u \in M_{\Lambda(L)}$.

The Claim holds iff there exists $u\in L, k\in \mathbb{N}$ and $\alpha,\beta\in \mathbb{N}^{\Sigma(1)}$ such that

\begin{align*}
(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^k m_p m_{-u} &= x^\alpha y^\beta \in S \end{align*}

Let $\lfloor p_{\rho_i}\rfloor := \tilde{a}_{\rho_i}$ for all $i$ so that

\begin{align*}
\lfloor p \rfloor &= (\lfloor p_{\rho_1} \rfloor, \dots, \lfloor p_{\rho_{\Sigma(1)}}\rfloor)\\
&= (\tilde{a}_{\rho_1}, \dots, \tilde{a}_{\rho_{|\Sigma(1)|}}) \end{align*}

and define the sets $A_+, A_-$ by

\begin{align*}
m_p &= \frac{ \prod_{A_+}x_\rho^{a_\rho} \prod_{A_-} y_\rho^{a_\rho}}{\prod_{A_+} y_\rho^{a_\rho} \prod_{A_-} x_\rho^{a_\rho}} \end{align*}
with $a_\rho = |\tilde{a_\rho}|$ for all $\rho$. Since $X_\Sigma$ is smooth, there exists $u_1\in L$ such that $u_1(e_\rho)=a_\rho$ for all $\rho \in (\sigma_1 \cap \sigma_2)(1) = \sigma_1(1) \cap \sigma_2(1)$. By the Separation Lemma, there exists $u_2\in L$ such that
\[u_2\bigg|_{\sigma_1\cap \sigma_2}=0, u_2\bigg|_{\sigma_1\setminus \sigma_2}>0, u_2\bigg|_{\sigma_2\setminus \sigma_1}<0. \]

Since $x^{\hat{\sigma}_1}y^{\hat{\sigma}_2}$ is a generator of $I_{irr}$, up to torsion by $I_{irr}$ it suffices to consider only factors (over $S$) in the numerator and denominator of $m_p$ for variables $x_\rho$ and $y_\rho$ for $\rho \in \sigma_1\cup \sigma_2$. Now there exists $k_0 \in \mathbb{N}$ such that

\begin{align*}
(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^{k_0} m_p &= \frac{ f_1}{\prod_{\sigma_2\cap A_+}y_\rho^{a_\rho} \prod_{\sigma_1 \cap A_-}x_\rho^{a_\rho}} \\
&= \frac{f_2}{\prod_{\sigma_1\cap\sigma_2\cap A_+}y_\rho^{a_\rho} \prod_{(\sigma_2\setminus \sigma_1) \cap A_+}y_\rho^{a_\rho}\prod_{\sigma_1\cap \sigma_2\cap A_-}x_\rho^{a_\rho}\prod_{(\sigma_1\setminus\sigma_2)\cap A_-}x_\rho^{a_\rho}} &[ \text{subdivision of sets }A_+, A_-] \end{align*}

for some monomials $f_1, f_2\in S$. Now $m_pm_{-u_1} = m_{p-u_1}$ since $u_1\in L$ gives that there exists $k_1\in \mathbb{N}$ such that

\begin{align*}
(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^{k_1} m_{p-u_1} &= \frac{ f_3}{\prod_{(\sigma_2\setminus \sigma_1)\cap A_+}y_\rho^{b_\rho}\prod_{(\sigma_1\setminus \sigma_2)\cap A_-}x_\rho^{b_\rho}} \end{align*}

for some $\{b_\rho\} \subset \N$ and some monomial $f_3\in S$, so that there exists $\ell\in \mathbb{N}$ such that $m_{p-u_1}m_{-\ell u_2} = m_{p-u_1-\ell u_2}$ since $\ell u_2\in L$ gives

\begin{align*}
(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^k m_{p-u_1-\ell u_2} &= f_4 \end{align*}
for some monomial $f_4 = x^\alpha y^\beta\in S$, for some $\alpha,\beta\in \mathbb{N}^{|\Sigma(1)|}$.

By Lemma 1, this implies there exists $k\in \mathbb{N}$ such that $(x^{\hat{\sigma}_1}y^{\hat{\sigma}_2})^k\cdot m_p \in M_{\Lambda(L)}$ for any choice of $\sigma_1, \sigma_2\in \Sigma(n)$ so that $coker(M_{\Lambda(L)} \hookrightarrow Im(f))=0$ for $\epsilon$ in the Effective Cone.

In fact, this argument shows that for $p\in \mathcal{H}_L(0)$, $coker(M_{\Lambda(L)} \hookrightarrow Im(f))=0$ so that we have a resolution of the diagonal for any smooth toric variety $X_\Sigma$, without requiring the deformation $\mathcal{H}_L^\epsilon$.

§ RESOLUTION OF DIAGONAL FOR TORIC D-M STACK ASSOCIATED TO GLOBAL QUOTIENT OF SMOOTH TORIC VARIETY BY FINITE ABELIAN GROUP

The resolution of the diagonal for $X_\Sigma$ smooth from the previous section also strengthens the construction of the diagonal object giving the kernel of a Fourier-Mukai transformation inducing the identity in $D^b_{Coh}(X_\Sigma')$ for the case that $X_\Sigma'=\faktor{X_\Sigma}{\mu}$ is a global quotient of a smooth toric variety $X_\Sigma$ by a finite abelian group $\mu$.

That is, since smooth toric varieties have an open cover by affine spaces:
\[ X_\Sigma = \cup_{\sigma \preceq \Sigma(n)} U_\sigma \]
with $U_\sigma = \text{Spec }\C[\sigma^\vee \cap M] \cong \A^m$ for $m=\text{rank }(M) = \text{rank } (N)$, we have that the same diagonal object
\[ \pi(M_{\Lambda(L)}) = M_{\Lambda(L)} \otimes_{S[\Lambda(\tilde{L})]}S \]
and corresponding resolution $(\mathcal{F}_{\mathcal{H}_L^\epsilon/\tilde{L}}, \partial)$ carry through in the case of a global quotient
\[ X_\Sigma' = \faktor{X_\Sigma}{\mu}\]
with $X_\Sigma$ smooth (not necessarily unimodular) and $\mu$ a finite abelian group.

Lemma 2 from Section <ref> for $X_\Sigma$ smooth, Fano, not necessarily unimodular: For any $p \in \mathcal{H}_L^\epsilon(0)$ and $\sigma \in \Sigma(n)$, there exists $v_\sigma \in L$ with $(p+v_\sigma)_\rho <0$ only on $\sigma(1)$, and $(p+v_\sigma)_\rho >0$ for $\rho\not\in\sigma(1)$.

We previously noted that since $X_\Sigma$ is Fano, there exists $v\in L$ such that

\[ \begin{cases} v(u_\rho)<0 & \forall \text{ }\rho\in\sigma(1),\\ v(u_{\rho'}) \geq 0 & \forall \rho' \not\in\sigma(1) \end{cases} \]

Now there exists $r\in \Z$ such that $(r\cdot v)_\rho <<0$ for all coordinates with $\rho\in\sigma(1)$ so that all $\rho\in \sigma(1)$ become much less than any coordinate $\rho'\not\in\sigma(1)$. Additionally, there exists a unique $\tilde{v}\in L$ such that
\[ \begin{cases}  (p+\tilde{v})(u_\rho)= 0 & \forall\text{ }\rho\in\sigma(1)\\ (p+\tilde{v})(u_{\rho'}) > 0 & \forall\text{ }\rho'\not\in \sigma(1). \end{cases} \]

Now adding on sufficiently large multiples of $v$ and $\tilde{v}$ ensure that there exists $r, k\in \Z$ such that $v_\sigma = rv + k\tilde{v}$ gives that $p+v_\sigma$ satisfies

\[ \begin{cases} (p + v_\sigma)_\rho < 0 & \text{ iff } \rho\in\sigma(1),\\ (p+v_\sigma)_\rho  >0 & \text{ otherwise}. \end{cases} \]

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

seriesGraduate Texts in Mathematics
titleAlgebraic Geometry

journaltitleFunctional Analysis and Its Applications
titleCoherent Sheaves on Pn and Problems of Linear Algebra

titleErratum to "The Homogeneous Coordinate Ring of a Toric Variety", along with the original paper
Algebraic Geometry (math.AG),FOS: Mathematics,FOS: Mathematics,14M25

Princeton University Press

abstractToric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.
titleIntroduction to Toric Varieties. (AM-131)

titleCellular Resolutions of Monomial Modules
Algebraic Geometry (math.AG),FOS: Mathematics,FOS: Mathematics

titleSyzygies of Unimodular Lawrence Ideals
Algebraic Geometry (math.AG),Combinatorics (math.CO),FOS: Mathematics,FOS: Mathematics

Springer New York, NY

titleCombinatorial Commutative Algebra
volume1st Ed.
given=David A.,
given=John B.,
given=Henry K.,

English (US)

United States

American Mathematical Society

seriesGraduate Studies in Mathematics
titleToric varieties