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Schemes of Associative Algebras

Arvid Siqveland

Abstract

We give a definition of associative schemes, schemes of associative rings, over a field $k,$ using the definition of completion of an associative $k$-algebra in a finite set of simple modules. We start by giving a weaker but sufficient definition of ordinary schemes of commutative rings which can be generalized to associative rings.

Schemes of Associative Algebras

Abstract

We give a definition of associative schemes, schemes of associative rings, over a field using the definition of completion of an associative -algebra in a finite set of simple modules. We start by giving a weaker but sufficient definition of ordinary schemes of commutative rings which can be generalized to associative rings.

Paper Structure

This paper contains 7 sections, 17 theorems, 14 equations.

Table of Contents

  1. Algebraic Geometry
  2. The Category of Schemes Over $k$
  3. Integral Noetherian Schemes
  4. Associative Schemes over $k$
  5. Augmented $k$-algebras
  6. Completions of Algebras
  7. The Ring of Local Functions

Key Result

Lemma 1

Let $X$ be a topological space and let $F$ be a presheaf on $X,$ that is a contravariant functor $F:\operatorname{Top}X\rightarrow\mathbf{Rings}.$ Then the functor is a sheaf on $X.$

Theorems & Definitions (47)

  • Definition 1
  • Example 1: Algebraic Varieties
  • Example 2: Projective Varieties
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • ...and 37 more