Excellent rings in transchromatic homotopy theory

Homotopy Theory of Associative Rings

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that similar to spaces the derived category obtained by inverting the quasiisomorphisms is natura...

Homotopy Theory of Modules over Diagrams of Rings

Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we consider the category of diagrams where the object X(s) at s comes from M(s). We develop model structures on such categories of diagrams and Quillen adjunctions tha...

Chains of Unusual Excellent Local Rings

Let (T,M) be a complete local domain containing the integers. Let p1 ⊆ p2 ⊆ · · · ⊆ pn be a chain of nonmaximal prime ideals of T such that Tpn is a regular local ring. We construct a chain of excellent local domains An ⊆ An−1 ⊆ · · · ⊆ A1 such that for each 1 ≤ i ≤ n, the completion of Ai is T , the generic formal fiber of Ai is local with maximal ideal pi, and if I is a nonzero ideal of Ai th...

Homotopy Theory of Homotopy Algebras

The goal of this paper is to study the homotopy theory of homotopy algebras over a Koszul operad with their infinity morphisms. The method consists in endowing the category of coalgebras over the Koszul dual cooperad with a model category structure. CONTENTS Introduction 1 1. Recollections 2 2. Model category structure for coalgebras 5 3. Conclusion 14 Appendix A. A technical lemma 15 Reference...

Configurations in Homotopy Theory