Given a sheaf of unital commutative and associative algebras A, first we construct the k-th Grassmann sheaf GA(k, n) of An, whose sections induce vector subsheaves of An of rank k. Next we show that every vector sheaf over a paracompact space is a subsheaf of A. Finally, applying the preceding results to the universal Grassmann sheaf GA(n), we prove that vector sheaves of rank n over a paracomp...

Let X be a projective scheme over a noetherian base scheme S, and let F be a coherent sheaf on X. For any coherent sheaf E on X, consider the set-valued contravariant functor Hom(E,F) on S-schemes, defined by Hom(E,F)(T ) = Hom(ET ,FT ) where ET and FT are the pull-backs of E and F to XT = X ×S T . A basic result of Grothendieck ([EGA] III 7.7.8, 7.7.9) says that if F is flat over S then Hom(E,...

Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representation turns out to have many equivalent guises making it analogous to the sign ...

We study rational Lagrangian immersions in a cotangent bundle, based on the microlocal theory of sheaves. construct sheaf quantization immersion and investigate its properties Tamarkin category. Using quantization, we give an explicit bound for displacement energy Betti/cup-length estimate number intersection points Hamiltonian image by purely sheaf-theoretic method.

Given a complex space X, we cosidered the problem of finding a hyperbolic model of X. This is an object Ip(X) with a morphism i : X → Ip(X) in such a way that Ip(X) is “hyperbolic” in a suitable sense and i is as close as possible to be an isomorphism. Using the theory of model categories, we found a definition of hyperbolic simplicial sheaf (for the strong topology) that extends the classical ...

Recall that every commutative ring R determines an affine scheme in algebraic geometry. This consists of two components: a topological space Spec(R) (the spectrum) and a sheaf of local rings OR (the structure sheaf). In this way, a scheme encodes both geometric and algebraic data. In this work, we present a construction of “logical schemes,” geometric entities which represent logical theories i...

Let L be the function field of a projective space P n k over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf V on P n k is a collection of isomorphisms V −→ g * V for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semi-linear representations...

Let L be the function field of a projective space P n k over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf V on P n k is a collection of isomorphisms V −→ g * V for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semi-linear representations...

This paper is an expanded version of notes for a set of lectures given at the Isaac Newton Institute for Mathematical Sciences during a NATO ASI Workshop entitled “Homotopy Theory of Geometric Categories” on September 23 and 24, 2002. This workshop was part of a program entitled New Contexts in Stable Homotopy Theory that was held at the Institute during the fall of 2002. The intent for the lec...