## 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.
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

Zbl 0193.00502
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