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Presented By: Department of Mathematics

Commutative Algebra Seminar

Triviality of jet closures and the local isomorphism problem

Jet schemes are a higher order analogue of the tangent scheme of a variety; algebraically, just as the tangent scheme of a variety corresponds to derivations on its local rings, jet schemes corresponds to higher order (Hasse--Schmidt) derivations. It's a natural question to ask whether a morphism inducing an isomorphism on jet schemes must itself be an isomorphism. Towards this question, de Fernex, Ein, and Ishii introduced an ideal closure operation for local k-algebras (R,m,L), the "jet closure". The triviality of this closure operation detects a positive answer to the preceding question, and so it's natural to ask whether this closure operation is always trivial. In this talk, we give an affirmative answer to this question under mild hypothesis (i.e., when L is separable over k). Our methods are elementary, commutative-algebraic, and are inspired by similar techniques in tight closure theory. Speaker(s): Devlin Mallory (University of Michigan)

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