Let X be a projective scheme and let M and N be two coherent OX -modules. Given an integer m, we present an algorithm for computing the global extension module ExtX (M,N ). As a consequence, one may calculate the sheaf cohomology Hm(X,N ) and construct the sheaf corresponding to an element of the module ExtX(M,N ). This algorithm depends only on the computation of Gröbner bases and syzygies and...

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...

This is an introduction to the notion of local subgroupoid introduced by the author and R. Brown. It can also serve as an introduction to an application of sheaf theory, and so could be useful to beginners in that theory. The main results are the construction of the holonomy groupoid and the notion of s-sheaf for certain local subgroupoids s. 2000 Mathematics Subject Classification: 18F20, 18F0...

In this paper, we establish results of a basic nature for the the K-theory and G-theory of algebraic stacks, i.e. Artin stacks. At the same time, we enlarge the framework a bit more so that these results not only hold for stacks, but also for what are called dg-stacks, i.e. algebraic stacks where the usual structure sheaf is replaced by a sheaf of dgas.

We study the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces with respect to morphisms of a particular kind. We obtain upper bounds for the arithmetic self-intersection number of the dualizing sheaf on minimal regular models of the modular curves associated with congruence subgroups Γ0(N) with square free level, as well as for the modular curves X(N) and the Fe...

Abstract The aim of this article is to study the geometry Bott-Chern hypercohomology from bimeromorphic point view. We construct some new invariants involving cohomology for sheaf germs pluriharmonic functions, truncated holomorphic de Rham cohomology, and cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish blow-up formula together with canonical morphism h...

T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $Z/p^2Z$. We study same construction $Z_{(p)}$ and prove regularity criterion. For local ring, tensor product with residue field is constructed in different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we use sheaf FW-differentials to define cotangent bundle mic...

A generalization of the usual ideles group is proposed, namely, we construct certain adelic complexes for sheaves of K-groups on schemes. More generally, such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated to the presheaf of a homology theory with certain natural axioms, satisfied by K-theory. In this case it is proven that the adelic ...

We describe a canonical L∞ structure on the total complex of a semicosimplicial differential graded Lie algebra and give an explicit descriprion of the Maurer-Cartan elements and of the associated deformation functor in the particular case of semicosimplicial Lie algebras. We use these results to investigate the deformation functor associated to a sheaf of Lie algebras L and to show that it is ...

This article presents several numerical algorithms for computations in sheaf cohomology. Let X be an algebraic set defined by a system of homogeneous multivariate polynomials with coefficients in C. Let C be a union of reduced, irreducible pure-dimensional curve components of X. The first algorithm computes the dimension of the first cohomology of any twist of the ideal sheaf of C. Let D be a r...