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Given a reduced local algebra $T$ over a suitable ring or field $k$ we studythe question of whether there are nontrivial algebra surjections $R\to T$ whichinduce isomorphisms $\Omega_{R/k}\otimes T \to \Omega_{T/k}$. Such maps, calledevolutions, arise naturally in the study of Hecke algebras, as they implicitlydo in the recent work of Wiles, Taylor-Wiles, and Flach. We show that theexistence of non-trivial evolutions of an algebra $T$ can be characterized interms of the symbolic square of an ideal defining $T$. We give acharacterization of the symbolic square in terms of Fitting ideals. Using thisand other techniques we show that certain classes of reduced algebras --codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras ingeneral, and some others -- admit no nontrivial evolutions. On the other handwe give examples showing that non-trivial evolutions of reduced Cohen-Macaulayalgebras of codimension 3 do exist in every positive characteristic.

Evolutions, symbolic squares, and Fitting ideals.