How Grothendieck Simplified Algebraic Geometry

How Grothendieck Simplified Algebraic Geometry

Coarse Geometry via Grothendieck Topologies

In the course of the last years several authors have studied index problems for open Riemannian manifolds. The abstract indices are elements in the K-theory of an associated C∗-algebra, which only depends on the ”coarse” (or large scale) geometry of the underlying metric space. In order to make these indices computable J.Roe introduced a new cohomology theory, called coarse cohomology, which is...

Derived Algebraic Geometry XII: Proper Morphisms, Completions, and the Grothendieck Existence Theorem

1 Generalities on Spectral Algebraic Spaces 5 1.1 Scallop Decompositions in the Separated Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Quasi-Finite Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Quasi-Separatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Points of Spectral Algebr...

Introductory Geometry: Algebraic Geometry

Algebraic Geometry

Physical objects and constraints may be modeled by polynomial equations and inequalities. For this reason algebraic geometry, the study of solutions to systems of polynomial equations, is a tool for scientists and engineers. Moreover, relations between concepts arising in science and engineering are often described by polynomials. Whatever their source, once polynomials enter the picture, notio...