© Publications mathématiques de l’I.H.É.S., 1993, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/etale-cohomology-for-non-archimedean-analytic-spaces-128r2xxirl.pdf

Étale cohomology for non-Archimedean analytic spaces

1 Basic Definitions 2 Examples 3 Projective Schemes 4 Classical Constructions 5 Local Constructions 6 Schemes and Functors

/pdf/the-geometry-of-schemes-2w8x4acrtn.pdf

The geometry of schemes

This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic $K$-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization. (GSM/114)

/pdf/advanced-modern-algebra-xt5ld7apas.pdf

Advanced Modern Algebra

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.234.6263&rep=rep1&type=pdf

Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories

This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d’Estudis Catalans, July 15–26, 1996

Miles Reid

Papers

Commutative ring theory: Regular rings