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STRUCTURE OF LOCAL HOMOMORPHISMS Luchezar L. Avramov, Hans{Bj rn Foxby, and Bernd Herzog

Luchezar L. Avramov, Hans{Bjrn Foxby, and Bernd Herzog
A by now standard and most productive technique for studying local properties of
commutative noetherian rings proceeds in two stages: The corresponding problem is rst
treated in the maximal{ideal{adic completions of a ring, and then the atness of completion
is used in order to descend the information to the initial ring.
The success of the rst step largely depends on a fundamental result of commutative
algebra { I. S. Cohen's Structure Theorem which shows that a complete local ring may be
presented as a homomorphic image of a ring of formal power series over a complete discrete
valuation domain; thus, questions on complete local rings are reduced to questions on ideals
in speci c rings.
The second step has been developed by A. Grothendieck into a full- edged and versatile
theory of at descent. The idea behind it is that if ' : R ! S is a faithfully at homo-
morphism of commutative rings, then the properties of R and of the bers of ' determine
and are determined by the properties of S; this is in accordance with geometric intuition,
which perceives such homomorphisms as algebraic substitutes for ber bundles.
The purpose of this paper is to introduce the local stage of a new approach to the study
of arbitrary homomorphisms of noetherian rings. If ' : R ! S is a local homomorphism,
it involves breaking down the canonically associated homomorphism 


Source: Avramov, Luchezar L.- Department of Mathematics, University of Nebraska-Lincoln


Collections: Mathematics