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'' $\omega=\omega_W$ and a pairing for $i=0,1$, $$ H^i(M_{W,{\bf 0}}\otimes F) \times {\rm Ext}^{1-i}(F,M_{W^\wedge_{\bf L},{\bf K}})\to H^1(\omega)\cong k $$ 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 $\omega_W$ 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.