A new look at smoothness.
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 without any hypothesis on \(\Omega_{B/R}\). He also shows that if \(\Omega_{B/R}\) is a direct sum of finitely generated \(B\)-modules, if \(b_1,\dots, b_n\) generate the unit ideal of \(B\) and if all \(R\to B_{b_i}\) are quasi-smooth, then \(R\to B\) is quasi-smooth. It is an open question whether this remains true without assuming anything about \(\Omega_{B/R}\). This is related to the following question: If \(M\) is a \(B\)-module such that every \(M_{b_i}\) is a projective \(B_{b_i}\)-module, does it follow that \(M\) is a projective \(B\)-module? Finally, the author proves the Jacobian criterion and shows that a finitely presented morphism \(R\to B\) is quasi-smooth iff it is flat with regular geometric fibres. In this case, the morphism is called smooth.
scientific article; zbMATH DE number 1750376
scientific article