Categories and Commutative Algebra

Nous fixerons une fois pour toutes un corps K algébriquement clos de caractéristique arbitraire; toutes les varietés algébriques seront supposées irreductibles sur K, i. e. des schémas séparés, intègres et de type fini sur K.

1. Soient j: Y′ ↪ Y une immersion fermée des variétés algébriques, J l'idéal de Y′ dans Y, φ : X → Y l'èclatement de Y le long de Y′, X′ = φ ⁻¹ (Y′), i: X′↪ l'immersion fermée correspondante, I = O_X (1) l'idéal de X′ dans X, de telle sor-te qu'on peut considérer le diagramme suivant cartésien:

The purpose of these lectures is twofold. On the one hand, we shall study further the notion of satellites of functors in order to understand more generally what is going on in homological algebra. In particular, we shall try to see what happens if we consider functors defined on abelian categories when the domain category does not necessarily have projective or injective objects. Further, we shall investigate in what sense the notion of satellite still survives if we remove the condition that the domain and/or range of our functor is abelian.

Our second purpose is to look at concrete categories and see what the general methods of homological algebra yield in particular cases. More precisely, we shall look at the category of finitely generated modules over a local ring and, using ideas suggested by Ext, obtain some useful results some of which are directly connected with more classical problems.

Cli anelli locali henscliani,che prendono il nome dal classico lemma di Hensel, sono stati introdotti da Azumaya in [4],e successivamente studiati da Nagata in [25], [26], [27]. Lafon in [23] ha iniziato la teoria delle coppie henseliane, che include la precedente; di tale teoria esporremo alcuni aspetti elementari,ottenuti rielaborando e semplificando vari fatti contenuti principalmente in [10], [11], [l3], [23] e [29].

Abbiamo cercato di indicarc i punti essenziali di tutte le dimostrazioni,rimandando,se possibile, alla letteratura per i dettagli.

La nostra trattazione riguarda solo la parte puramente algebrica della teoria. Per quanto riguarda gli aspetti topologici della stessa si riraanda ai lavori di Valabrega [32], [33], [34], mentre per avere un'idea delle numerose applicazioni alla georaetria algebrica si possono vedere i lavori [1], [2], [17], [22] e le relative bibliografie.

Dans ce qui suit exposons les fondements de la théorie des esquisses, c'est-à-dire de la théorie des structures algébriques décrites au moyen d'un graphe multiplicatif, telle qu'elle a été introduite par C. Ehresmann dans [E. T. S. A ] Elle constitue une généralisation des théories de Lawvere et Bénabou.

Nous développons, d'autre part, deux résultats récents qui résolvent le problème, pour une catégorie donnée, de trouver une description (i.e. une esquisse) de ses unités et de ses morphiames, lorsqu'ils sont considérés comme des structures et des homomorphismes algébriques d'un certain type. Ces résultats généralisent ceux obtenus par Gabriel et Ulmer dans [L. P. L. G.].

The purpose of these notes is to provide as rapid an introduction to category theory and homological algebra as possible without overwhelming the reader entirely unfamiliar with these subjects. We begin with the definition of a category, and end with the basic properties of derived functors, in particular, Tor and Ext. This was the spirit of the four lectures on which thenotes are based, although there is, needless to say, much more material contained herein an was touched on in the lectures. For example, we have included a fairly complete treatment of the basic facts pertaining to adjoint functors, including Freyd's adjoint functor theorems. Application of category theory in the direction of topos theory and logic were treated in the accompanying lectures of Tierney, and Buchsbaum in his lectures indicated some outlets for homological algebra in commutative algebra and local ring theory. We have therefore not felt compelled to emphasize any specific topic. We have, nevertheless, presented module theory as something associated with ringoids (small, additive categories) rather than with the more conventional and restrictive notion of a ring. This point of view has enabled us recently to incorporate several new examples into the traditional setting of homological algebra as found in the book of Cartan-Eilenberg [2]. One can consult [15] in this regard.

Soit A un anneax commutatif unitaire intègre. On désignera par K son corps des fractions. Nous associons à A l'anneau local a [[X]_(X)

Proposition 1-1 : L'anneau local B = A [[X]]_(X) est un an-neau de valuation discrète, d'égale caractéristique de corps résiduel K.

La K-tcoria algebrica ha avuto un nolevolc sviluppo noll'ultimo decennio I risultati più importanti sono stati raccolti ed esposti sistematicamen-te nel trattato [3] di Hyman Bass, nel quale sono anche poste le necessarie premesse di algebra categoriale, omologica, commutativa assleme ai loro legami con altri rami della matematica quali, ad esempio, la teoria dei numeri e la geometria algebrica.

In queste lezioni mi propongo di mostrare come alcuni risultati della K-teoria algebrica possono essere di grande aiuto nello studio di certi problemi di algebra commutativa,in cui le nozioni classiche di teoria de-gli ideali non sono sufficienti per un'analisi approfondita.

Salvo menzione contraria, consideriamo sempre anelli commutativi con identità.

After having thought for some time about what it was that I wanted to say in these lectures, I finally decided they should serve two purposes, and two purposes only. First, I wanted to provide a leisurely, and reasonably complete, introduction to the axiomatic theory of sheaves developed recently by F. W. Lawvere and myself. Grothendieck, and those around him, have long maintained (see [11] ) that in sheaf theory it is the topos itself - i.e., the whole category of sheaves - that is important, and not the site, or small category, from which it is derived. He himself, however, had never consequently developed this point of view. Thus, this was our first goal, and it is the one I would like to concentrate on here. Later, we began to think of the notion of topos as a kind of set theory useful for dealing with many kinds of “sets” other than just “abstact” sets. Though this is perhaps the most interesting aspect of topos, I shall hardly mention it here; I hope the interested reader will consult [4] or [9] for more information. After developing the basic general theory, we will turn to the more specialized topics whose exposition was my second aim. Here, among several possible ways of constructing topos, I would like to describe thesis from the topos point of view. Thus, this section, as well as the preceding, may be considered background material Tor [10]. The Continuum Hypothesis forms the subject of the last section, so these lectures are independent of [10], though the reader might want to consult this for more details.

To simplify, let us work with the category Aff of affine schemes. This is the dual category to the category of commutative rings. One thinks of affine schemes as topological spaces of the form Spec A where morphisms are those induced from ring homomorphisms by the functorality of Spec.