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 K ∈ Z n there is a unique Riemann function f ∧ K such that for all d ∈ Z n we have f (d) -f ∧ K (K -d) = deg(d) + 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 } d∈Z 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 } d∈Z n , are-up to isomorphism-independent of all the choices made.

Abstract licence: CC BY-ND

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