##
**Evolutions, symbolic squares, and Fitting ideals.**
*(English)*
Zbl 0912.13010

Let \(\Lambda\) be a ring and \(T\) the localization of a finitely generated \(\Lambda\)-algebra. An evolution of \(T\) is the localization \(R\) of a finitely generated \(\Lambda\)-algebra together with a surjection \(R\to T\) of \(A\)-algebras inducing an isomorphism \(\Omega_{R/ \Lambda} \otimes T\to \Omega_{ T/ \Lambda}\). It is trivial if \(R\to T\) is an isomorphism. Similar notions were first studied in the context of analytic algebras by G. Scheja and U. Storch [Math. Z. 114, 101-112 (1970; Zbl 0193.00502)]. In the work of Wiles and Taylor and Wiles on the Taniyama conjecture it is a crucial point to show that certain evolutions are trivial.

Using a result of H. Lenstra the authors show: Let \(\Lambda\) be a regular ring, let \((P,{\mathfrak M})\) be the localization of a polynomial ring in finitely many variables over \(\Lambda\) and let \(I\) be an ideal in \(P\) such that \(T:=P/I\) is reduced and generically separable over \(\Lambda\). Then every evolution of \(T\) is trivial iff the symbolic square \(I^{(2) }\) is contained in \({\mathfrak M}I\). For generically complete intersections \(I\) the elements of \(I^{(2)}\) (and more generally those of \(I^{(d)})\) are characterized with the help of certain Fitting ideals. – Applying these results (and a theorem of Buchweitz) the authors prove the triviality of all evolutions of reduced algebras which are defined by perfect ideals in a regular local ring and are linked to a complete intersection. These include all perfect ideals of codimension 2 and all Gorenstein ideals of codimension 3. Similar results are true for monomial ideals in polynomial rings, for quasi-homogeneous ideals in polynomial rings over fields of characteristic 0 and (due to Kunz) for almost complete intersections. On the other hand the authors give an example of a 1-dimensional quasi-homogeneous \(k\)-algebra in characteristic \(p\) that admits nontrivial evolutions.

Using a result of H. Lenstra the authors show: Let \(\Lambda\) be a regular ring, let \((P,{\mathfrak M})\) be the localization of a polynomial ring in finitely many variables over \(\Lambda\) and let \(I\) be an ideal in \(P\) such that \(T:=P/I\) is reduced and generically separable over \(\Lambda\). Then every evolution of \(T\) is trivial iff the symbolic square \(I^{(2) }\) is contained in \({\mathfrak M}I\). For generically complete intersections \(I\) the elements of \(I^{(2)}\) (and more generally those of \(I^{(d)})\) are characterized with the help of certain Fitting ideals. – Applying these results (and a theorem of Buchweitz) the authors prove the triviality of all evolutions of reduced algebras which are defined by perfect ideals in a regular local ring and are linked to a complete intersection. These include all perfect ideals of codimension 2 and all Gorenstein ideals of codimension 3. Similar results are true for monomial ideals in polynomial rings, for quasi-homogeneous ideals in polynomial rings over fields of characteristic 0 and (due to Kunz) for almost complete intersections. On the other hand the authors give an example of a 1-dimensional quasi-homogeneous \(k\)-algebra in characteristic \(p\) that admits nontrivial evolutions.

Reviewer: H.Wiebe (Bochum)

### MSC:

13N05 | Modules of differentials |

13C40 | Linkage, complete intersections and determinantal ideals |

13H05 | Regular local rings |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |