Ext and Tor on two-dimensional cyclic quotient singularities

TL;DR

This paper provides a combinatorial description of Ext groups between torus invariant Weil divisors on two-dimensional cyclic quotient singularities, revealing symmetries and dualities with Tor groups using toric geometry methods.

Contribution

It introduces a combinatorial approach to compute Ext groups on cyclic quotient singularities and establishes new duality relations involving these groups.

Findings

Combinatorial description of Ext^1 groups via polyhedra

Symmetry of Ext^1 groups involving the canonical divisor

Ext^{i+2} groups are Matlis duals of Tor_i groups

Abstract

Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional cyclic quotient singularity $X$, the groups $\mathop{Ext}\nolimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are naturally $\mathbb{Z}^2$-graded. We interpret these groups via certain combinatorial objects using methods from toric geometry. In particular, it is enough to give a combinatorial description of the $\mathop{Ext}\nolimits^1$-groups in the polyhedra of global sections of the Weil divisors involved. Higher $\mathop{Ext}\nolimits^i$-groups are then reduced to the case of $\mathop{Ext}\nolimits^1$ via a quiver. We use this description to show that $\mathop{Ext}\nolimits^1_{X}(\mathcal{O}(D),\mathcal{O}(K-D')) = \mathop{Ext}\nolimits^1_{X}(\mathcal{O}(D'),\mathcal{O}(K-D))$, where $K$ denotes the canonical divisor on $X$. Furthermore, we show that $\mathop{Ext}\nolimits^{i+2}_{X}(\mathcal{O}(D),\mathcal{O}(D'))$ is the Matlis dual of $\mathop{Tor}\nolimits_{i}^{X}(\mathcal{O}(D),\mathcal{O}(D'))$.