6 Introduction to Affine Schemes 2 6.1 Rings and Modules of Fractions . . . . . . . . . . . . . . . . . 2 6.2 The Spectrum of a Unital Commutative Ring . . . . . . . . . 5 6.3 The Spectrum of a Quotient Ring . . . . . . . . . . . . . . . . 7 6.4 The Spectrum of a Ring of Fractions . . . . . . . . . . . . . . 9 6.5 Intersections of Prime Ideals . . . . . . . . . . . . . . . . . . . 11 6.6 Topolo...

One feature which clearly distinguishes the theory of schemes from the older theory of varieties is the possibility of having nilpotent elements in the structure sheaf of a scheme. In particular, if Y is a closed subvariety of a variety X, defined by a sheaf of ideals I , then for any n ≥ 1 we can consider the closed subscheme Yn defined by the nth power I n of the sheaf of ideals I . For n ≥ 2...

For any smooth projective variety X of dimension n over an algebraically closed field k of characteristic p > 0 with μ(Ω X ) > 0. If T(Ω X ) (0 < l < n(p − 1)) are semi-stable, then the sheaf B X of exact 1-forms is stable. When X is a surface with μ(Ω X ) > 0 and Ω X is semi-stable, the sheaf B X of exact 2-forms is also stable. Moreover, under the same condition, the sheaf Z X of closed 1-for...

Sheaf theoretic notions and methods have entered model theory during recent years and important constructions like reduced powers and products, unions of chains, Boolean valued models and models for non-classical logics appear here in a unified setting. If one looks at a sheaf as a generalized structure, then forcing becomes the genuine extension of the validity relation between models and form...

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,...

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalisation of this fact and prove a converse of the generalisation. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf repr...

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...