Duality and a Canonical Sheaf in Periodic Riemann Functions

Abstract

Let $f\colon{\mathbb Z}^2\to{\mathbb Z}$ be a Riemann function whose weight $W$ is a perfect matching. Then there is a family of sheaves of $k$-vector spaces $\{{M}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ on a five-point topological that models $f$ in that $f({\bf d})=b^0({M}_{W,{\bf d}})$ and that $$ b^1({M}_{W,{\bf d}})= f^\wedge_{\bf K}({\bf d}-{\bf K}) $$ for any ${\bf K}\in{\mathbb Z}^2$. Hence a Riemann-Roch formula for $f$ is equivalent to an Euler characteristic computation of ${M}_{W,{\bf d}}$. If $f$ and $W$ are $r$-periodic, then the sheaves ${M}_{W,{\bf d}}$ become ${O}_r$-modules of finite type for a natural sheaf of rings ${O}={O}_r$. We show that in this case there is a ``canonical ${O}$-module'' $ω=ω_W$ and a pairing for $i=0,1$, $$ H^i(M_{W,{\bf 0}}\otimes F) \times {\rm Ext}^{1-i}(F,M_{W^\wedge_{\bf L},{\bf K}})\to H^1(ω)\cong k $$ that is perfect when ${\bf L}={\bf K}+{\bf 1}$ and ${F}$ is a certain type of line bundle or a certain type of skyscraper sheaf. In particular when ${F}$ is a line bundle, we realize the above formula for $b^1({M}_{W,{\bf d}})$ as a duality theorem akin to Serre duality. We show that canonical ${O}$-module $ω_W$ is a rather exceptional element in a family of tensor products of two modules ${M}\otimes_{O}{M}'$, where ${M}$ and ${M}'$ vary over ${O}_r$-modules of the form ${M}_{W',{\bf d}}$. This article doesn't assume any background in sheaf theory; rather we describe all our sheaves as a ``diagrams of vector spaces,'' where each diagram is essentially a sheaf of vector spaces on a fixed topological space of five points.