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3:00 pm, Seminar Hall Reducedness of formally unramified algebras Alapan Mukhopadhyay University of Michigan. 20-08-19 Abstract An algebra over a field is called \textit{formally unramified} over the field, if the module of Kahler differentials with respect to the field is zero. It is well known that a formally unramified algebra over a field which is also finite type over the field is isomorphic to a product of finitely many finite separable field extensions of the field. So, it is natural to ask whether a formally unramified algebra over a field is always integral over the field or reduced. In positive characteristic, it is easy to find examples showing that the answers to the above questions are negative. In this talk, we shall first show that a formally unramified algebra over a characteristic zero field must be integral over the base field. Then we shall address the question of reducedness in characteristic zero. Although Ofer Gabber has constructed examples of formally unramified algebras over characteristic zero field which are not reduced, we shall show that in certain cases formally unramified algebras are reduced. In particular, we shall see that if the formally unramified algebra is either $\mathbb{N}$-graded or the localization at every maximal ideal is separated, the algebra is reduced. At the end, we shall spell out some open questions which will come up naturally during the talk. If time permits, we shall sketch Gabber's construction. These results were obtained in an ongoing joint work with either Karen Smith or Shubhodip Mondal.
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