– For a base scheme X and an effective (Cartier) divisor D on X defined as a closed subscheme by an ideal I, there is code to compute an invertible sheaf corresponding to the class of D. Here, computing the explicit divisor map is essentially the same as computing the Riemann-Roch space if X is a variety. In fact, the Riemann-Roch space can be recovered during the computation of the associated ...

We reformulate Hecke’s open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota’s formulations of n-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate condit...

Awareness of a conscious entity can exist without elements; therefore, the general notion of an object of a category is employed. One of the characterization of understanding is: for a given local infonnation (awareness) there exists a global information whose restriction is the given information. For such mental activities, category and sheaf theories are employed to formulate consciousness. W...

Let X be a smooth curve over a finite field of characteristic p , let l 6= p be a prime number, and let L be an irreducible lisse Ql -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number l 6= p , there exists an irreducible lisse Ql′ -sheaf L ′ on X which is compatible with L , in the sense that at every closed point x of X , the characteristic po...

Let Xbe an irreducible algebraic variety defined over a field k. let ¡X be a sheaf of (noncommutative) noetherian A-algebras on X containing the sheaf of regular functions 0 and let R be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of R obtained from the local sections of ifî) there is an equivalence...

This sketch is intended as a summary of my thoughts during the fall of 2005, attempting to understand a conversation with Kontsevich that summer as well as a little bit of his preprint [1]. The theme is to explore further the geometric meaning of the p-curvature, which can provide an analogy to the characteristic variety in microlocal analysis. Let S be a scheme and let X/S be a smooth S-scheme...