Duality and a Canonical Sheaf in Periodic Riemann Functions

Let $f\text{\hspace{0.17em}⁣}:{\mathbb{Z}}^2\to \mathbb{Z}$ be a Riemann function whose weight $W$ is a perfect matching. Then there is a family of sheaves of $k$-vector spaces $\{M_{W,\mathbf{d}}{\}}_{\mathbf{d}\in {\mathbb{Z}}^2}$ on a five-point topological that models $f$ in that $f(\mathbf{d})=b^0(M_{W,\mathbf{d}})$ and that $$b^1(M_{W,\mathbf{d}})=f_{\mathbf{K}}^{\wedge }(\mathbf{d}-\mathbf{K})$$ for any $\mathbf{K}\in {\mathbb{Z}}^2$. Hence a Riemann-Roch formula for $f$ is equivalent to an Euler characteristic computation of $M_{W,\mathbf{d}}$. If $f$ and $W$ are $r$-periodic, then the sheaves $M_{W,\mathbf{d}}$ 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.