A new look at smoothness.
type: Item
instance of: scholarly article
title: A new look at smoothness.
zbMATH Open document ID: 1037.14001
DOI: 10.1016/S0723-0869(02)80029-9
author: Michael Maltenfort
published in: Expositiones Mathematicae
publication date: 2002-01-01T00:00:00Z
review text: In various textbooks of algebraic geometry, the notion of smoothness is defined in different ways under different hypotheses. The aim of the paper is to reconcile these definitions and to give direct proofs and some new formulations of standard results about smoothness under weaker hypotheses. The starting point is the following definition: A morphism \(R\to B\) of rings (commutative, with unit) is called quasi-smooth if, endowing the rings with the discrete topology, it is formally smooth in the sense of A. Grothendieck, i.e., for every \(R\)-algebra \(C\) and every ideal \(J\subset C\) with \(J^2= 0\), every morphism of \(R\)-algebras \(B\to C/J\) can be lifted to a morphism of \(R\)-algebras \(B\to C\). One defines, similarly, using \(\text{Spec\,}C\) instead of \(C\) and reversing arrows, quasi-smooth morphisms of schemes. Now, if \(B= A/I\) for a quasi-smooth algebra \(A\) over \(R\) (for example, a polynomial algebra in possibly infinitely many indeterminates) then \(B\) is quasi-smooth iff the surjection \(A/I^2\to A/I\) has a right inverse in the category of \(R\)-algebras. Moreover, if \(R\to B\) is quasi-smooth, the module \(\Omega_{B/R}\) of differentials is a projective \(B\)-module. Interesting problems occur when one investigates the ``local nature'' of quasi-smoothness. The author introduces the following definition: A \(B\)-module \(M\) is ``sum finitely presented'' if there exists a \(B\)-module \(M'\) such that \(M\oplus M'\) is a (possibly infinite) direct sum of finitely presented \(B\)-modules. For example, the quotient of a projective module by a finitely generated submodule is sum finitely presented. The author shows that if \(\Omega_{B/R}\) is sum finitely presented (for example, if \(B= A/I\) with \(A\) quasi-smooth over \(R\), and \(I/I^2\) is finitely generated) then \(R\to B\) is quasi-smooth iff, for every maximal \(m\) of \(B\) with \(p= m\cap R\), \(R_p\to B_m\) is quasi-smooth and gives an example showing that this is no longer true w…
Mathematics Subject Classification ID: 14B25
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13N05
zbMATH DE Number: 1750376
zbMATH Keywords: formal smoothness
zbMATH Keywords: simple points
zbMATH Keywords: Jacobian criterion
zbMATH Keywords: sum finitely presented
Wikidata QID: Q83562406
reviewed by: Iustin Coandă
MaRDI profile type: Publication
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full work available at URL: s0723-0869(02)80029-9
OpenAlex ID: W2076971510
description: scientific article; zbMATH DE number 1750376