Abstract

In this talk I consider an old and exhaustively studied situation: A is a commutative ring, and \a is an ideal in it. (”\a” stands for gothic “a”.) We are interested in the \a-adic completion operation for A-modules, and in flatness of A-modules. The departure from the classical and familiar situation is this: the A-modules we care about are not finitely generated. The following was considered an open problem by commutative algebraists: if A is a noetherian ring and M is a flat A-module, is the \a-adic completion of M also flat? Partial positive answers were published in the literature over the years. A few months ago I found a proof of the general case. But then, a series of emails led me to prior proofs, that are embedded in pretty recent texts, and were unknown to the algebra community. Nonetheless, my methods gave more detailed variants of the general result mentioned above, that are actually new. One of them concerns the case of a ring A that is not noetherian, but where the ideal \a is weakly proregular. The latter is a condition discovered by Grothendieck a long time ago, but became prominent only very recently (in the context of derived completion). In the talk I will give the background, mention the new and not-so-new results (with some proofs), and give a few concrete examples. There will be no derived categories in this talk (!). The talk should be totally understandable to anyone with knowledge of \a-adic completion and flatness.