The semantics of Petri Nets are discussed within the "Objects are sheaves" paradigm. Transitions and places are represented as sheaves and nets are represented as diagrams of sheaves. Both an interleaving semantics, and a non-interleaving semantics are shown to arise as the limit of the sheaf diagram representing the net.

Jean Leray (November 7, 1906–November 10, 1998) was confined to an officers’ prison camp (“Oflag”) in Austria for the whole of World War II. There he took up algebraic topology, and the result was a spectacular flowering of highly original ideas, ideas which have, through the usual metamorphism of history, shaped the course of mathematics in the sixty years since then. Today we would divide his...

Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the hig...

A formula due to Grothendieck, Ogg, and Shafarevich gives the EulerPoincaré characteristic of a constructible sheaf of Fl-modules on a smooth, proper curve over an algebraically closed field k of characteristic p > 0, as a sum of a global term and local terms, where l 6= p. A previously known result removes the restriction on l in the case of p-torsion sheaves trivialized by p-extensions. The a...

This note states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a multiplier ideal sheaf. This new conjecture trivially implies earlier conjectures in Nevanlinna theory or diophantine approximation, and in fact is equivalent to these conjectures. Although it does ...

Let X be a normal scheme and F a coherent sheaf on X . The reflexive hull or double dual of F is the sheaf F ∗∗ := HomX(HomX(F,OX),OX). The natural map F → F ∗∗ kills the torsion subsheaf of F and the support of coker[F → F ] has codimension ≥ 2. This establishes a functor from the category of coherent sheaves on X to the category of reflexive coherent sheaves on X . The same construction works...

Let X be a nonsingular algebraic variety. Suppose Z ⊆ X is a closed subscheme of X, with ideal sheaf IZ . When Z has codimension one in X, everything is as nice as it could be: IZ is a locally free sheaf, in fact a line bundle, and Z can locally be defined by a single equation. But starting in codimension two, all these pleasant things are usually false. To begin with, not every closed subschem...

We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A-homotopy type. Arithmetic considerations involving the sheaf of A-connected components lead us to introduce several di...