Ext and Tor on two-dimensional cyclic quotient singularities

Abstract

Given two torus invariant Weil divisors $D$ and $D^{\prime }$ on a two-dimensional cyclic quotient singularity $X$, the groups ${\mathrm{Ext}}_X^i(\mathcal{O}(D),\mathcal{O}(D^{\prime }))$, $i>0$, 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 ${\mathrm{Ext}}^1$-groups in the polyhedra of global sections of the Weil divisors involved. Higher ${\mathrm{Ext}}^i$-groups are then reduced to the case of ${\mathrm{Ext}}^1$ via a quiver. We use this description to show that ${\mathrm{Ext}}_X^1(\mathcal{O}(D),\mathcal{O}(K-D^{\prime }))={\mathrm{Ext}}_X^1(\mathcal{O}(D^{\prime }),\mathcal{O}(K-D))$, where $K$ denotes the canonical divisor on $X$. Furthermore, we show that ${\mathrm{Ext}}_X^{i+2}(\mathcal{O}(D),\mathcal{O}(D^{\prime }))$ is the Matlis dual of ${\mathrm{Tor}}_i^X(\mathcal{O}(D),\mathcal{O}(D^{\prime }))$.

16 pages

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• Monomial Resolutions