Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Blown-up Čech cohomology and Cartan's Theorem B on real algebraic varieties

Tomasz Kowalczyk

Abstract

We introduce a concept of blown-up Čech cohomology for coherent sheaves of homological dimension $\leq 1$ and some quasi-coherent sheaves on a non-singular real affine variety. Its construction involves a directed set of multi-blowups. We establish, in particular, long exact cohomology sequence and Cartan's Theorem B. Finally, some applications are provided, including universal solution to the first Cousin problem (after blowing up).

Blown-up Čech cohomology and Cartan's Theorem B on real algebraic varieties

Abstract

We introduce a concept of blown-up Čech cohomology for coherent sheaves of homological dimension and some quasi-coherent sheaves on a non-singular real affine variety. Its construction involves a directed set of multi-blowups. We establish, in particular, long exact cohomology sequence and Cartan's Theorem B. Finally, some applications are provided, including universal solution to the first Cousin problem (after blowing up).

Paper Structure

This paper contains 7 sections, 34 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Directed system of multi-blowups
  3. Section functor is right exact after blowing up
  4. Homological dimension of sheaves
  5. Blown-Up Čech Cohomology
  6. A real algebraic version of the first Cousin problem
  7. An Example

Key Result

Theorem 1.1

Let $f : X \rightarrow \mathbb{R}$ be a regular function on a non-singular real algebraic variety $X$. Then there exists a multi-blowup $\sigma : X_\sigma \rightarrow X$ such that $f^\sigma:=f \circ \sigma$ is a simple normal crossing. $\square$

Theorems & Definitions (62)

  • Remark 1.1
  • Theorem 1.1
  • Lemma 1.1
  • Corollary 1.1
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.1
  • ...and 52 more