Evolutions, symbolic squares and Fitting ideals

Given a reduced local algebra T over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections R → T which induce isomorphisms Ω R/k ⊗ T → Ω T /k. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions of an algebra T can be characterized in terms of the symbolic square of an ideal defining T. We give a characterization of the symbolic square in terms of Fitting ideals. Using this and other techniques we show that certain classes of reduced algebras — codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras in general, and some others — admit no nontrivial evolutions. On the other hand we give examples showing that non-trivial evolutions of reduced Cohen-Macaulay algebras of codimension 3 do exist in every positive characteristic.