Commutative Algebra Seminar -- (derived) jets and arcs and an imperfect curve selection lemma

Fix k a field and X an affine k-variety. For each non-negative integer r, associated to X are r-th jet spaces which parameterize polynomial differential equations of order at most r, and their limit J^infty X the arc space of X, with a natural projection map pi : J^\infty X \to X. A fundamental problem introduced by Nash relates essential valuations to those that appear in the irreducible decomposition of the inverse image under pi of the singular locus of X. Critical to all known cases of determining when these valuations are the same is the `curve selection' lemma, which comes down to proving a noetherianity statement of a local ring of the arc space at so called stable points. This is known when k is perfect. Extending techniques of Docampo-de Fernex, we employ derived techniques to address the case when k is not perfect. We will review background on jet/arc spaces. This is joint work with R. Docampo, C. Eric Overton-Walker.