Problems in algebra inspired by universal algebraic geometry

un 2 00 4 Problems in algebra inspired by universal algebraic geometry

Let Θ be a variety of algebras. In every Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We consider also a special categorical invariant K Θ (H) of this geometry. The classical algebraic geometry deals with the variety Θ = Com − P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the g...

Local Algebra in Algebraic Geometry

Introduction 1 0.1. Associated primes 2 0.2. Depth 2 1. Regular local rings and their cohomological properties 4 1.1. Notions of dimension 4 1.2. Dualizing functor 5 1.3. Local rings 6 1.4. Serre’s theorem on regular local rings 6 1.5. Cohomology of D when global dimension is finite 8 2. Depth 8 2.1. Property (Sk) 8 2.2. Serre’s criterion of normality 10 2.3. Cohen-Macaulay modules 11 3. Diviso...

Experiments in Commutative Algebra and Algebraic Geometry

This is an extended set of lecture notes for the course given by the author at the CoCoA school in Sardegna in May, 2005. These notes are not intended to be exhaustive. Instead, the goal is to present the student with an overview of certain areas of algebraic geometry and commutative algebra which are extremely active, and to start seeing how almost all of the constructions and theory can be ca...

Derived Algebraic Geometry III: Commutative Algebra

1 ∞-Operads 4 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Fibrations of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Cartesian Monoidal Structures . . . . . . . . . . ....

Derived Algebraic Geometry III: Commutative Algebra