Euler Characteristics and Duality in Riemann Functions and the Graph Riemann-Roch Rank

Abstract

By a {\em Riemann function} we mean a function $f\colon{\mathbb Z}^n\to{\mathbb Z}$ such that $f({\bf d})=f(d_1,\ldots,d_n)$ is equals $0$ for ${\rm deg}({\bf d})=d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$ -- the {\em offset of $f$} -- for ${\rm deg}({\bf d})$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions. For such an $f$, for any ${\bf K}\in{\mathbb Z}^n$ there is a unique Riemann function $f^\wedge_{\bf K}$ such that for all ${\bf d}\in{\mathbb Z}^n$ we have $$ f({\bf d}) - f^\wedge_{\bf K}({\bf K}-{\bf d}) = {\rm deg}({\bf d})+C $$ which we call a {\em generalized Riemann-Roch formula}. We show that any such equation can be viewed as an Euler charactersitic equation of sheaves of a particular simple type that we call {\em diagrams}. This article does not assume any prior knowledge of sheaf theory. To certain Riemann functions $f\colon{\mathbb Z}^2\to{\mathbb Z}$ there is a simple family of diagrams $\{\mathcal{M}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ such that $f({\bf d})=b^0({\mathcal{M}}_{W,{\bf d}})$ and $f^\wedge_{\bf K}({\bf K}-{\bf d})=b^1({\mathcal{M}}_{W,{\bf d}})$. Furthermore we give a canonical isomorphism $$ H^1({\mathcal{M}}_{W,{\bf d}})^* \to H^0({\mathcal{M}}_{W',{\bf K}-{\bf d}}) $$ where $W'$ is the weight of $f^\wedge_{\bf K}$. General Riemann functions $f\colon{\mathbb Z}^2\to{\mathbb Z}$ are similarly modeled with formal differences of diagrams. Riemann functions ${\mathbb Z}^n\to{\mathbb Z}$ are modeled using their restrictions to two of their variables. These constructions involve some ad hoc choices, although the equivalence class of virtual diagram obtained is independent of the ad hoc choices.