Euler characteristics and duality in Riemann functions and the graph Riemann-Roch rank

By a Riemann function we mean a function f : Z n Z such that f (d) = f (d 1 , . . ., d n ) is equals 0 for deg(d) = d 1 + + d n sufficiently small, and equals deg(d) + C for a constant, C, for deg(d) sufficiently large.For such an f , for any+ C which we call a generalized Riemann-Roch formula.Our motivation for this definition is that (1) adding 1 to the Baker-Norine rank function of any graph yields a Riemann function; and (2) for the results below, we need to consider non-negative valued functions f .We demonstrate a class of Riemann functions f : Z 2 Z that are modeled by sheaves, M d with d Z 2 over a finite topological space, that models the associated generalized Riemann-Roch formula as expressing the Euler characteristic: the nonzero Betti numbers of M d are the zeroth and first, which respectively equal f (d) and f K (K -d).The sheaves M d satisfy many properties akin to the sheaves that model the classical Riemann-Roch formula as expressing an Euler characteristic.Any Riemann function f : Z 2 Z can be written as the difference of two functions modeled by sheaves, so that the generalized Riemann-Roch formula of f is modeled as an Euler characteristic formula of a family, {M d } dZ 2 , of virtual (i.e., a formal difference of) sheaves.We do the same for any Riemann function f : Z n Z with n 2, by restricting any n -2 of its variables, and varying the remaining two variables.We show that the resulting family of virtual sheaves obtained, {M d } dZ n , are-up to isomorphism-independent of all the choices made.