## completion of finitely generated module over non-Noetherian ring

Let $A$ be a commutative ring with unity and fix $f \in A$. Any $A$-module $M$ has its $f$-adic completion, the $\hat{A}$-module $\hat{M} = \underset{n}{\lim} M/f^nM$. There is a canonical map $\hat{A} \otimes_A M \to \hat{M}$, which is surjective if $M$ is finitely generated. My question is about additional conditions which imply that this map … Read more

## A question related to bousfield localization and nilpotent completion

I am reading Bousfield’s paper entitled “The localization of spectra with respect to homology” (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum E has countable homotopy and satisfies some vanishing conditions in the associated Adams spectral sequence, then the localization is equivalent to nilpotent completion. So, my question is the following: … Read more

## Does every Lindelof uniform space have a Lindelof completion?

Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces? Note it is well known to be true for metric spaces, since a metric space is separable if and only if it is Lindelof (see e.g., Engelking Gen … Read more

## Mapping cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules For the setting: Let A be a ring and let I be some finitely generated ideal in A. Let f:C→D be a map of chain complexes of derived I-complete A-modules. I am trying to apply the “derived Nakayama” to … Read more

## Enriched Cauchy completions and underlying categories

The ordinary Cauchy completion ¯C of a small category C satisfies a number of conditions: Every idempotent in ¯C splits, there’s an equivalence of categories [Cop,Set]≃[¯Cop,Set], etc… There’s also a notion of Cauchy completion for enriched categories, my questions are about it: 1 – Let X be a V-enriched category (where V is a closed … Read more

## Is completion of isolated singularity isolated?

Let K be an algebraically closed field and let A=K[x1,…,xn]/I be a K-algebra of finite type which has only an isolated singularity at the origin. Let m=(x1,…,xn) and consider the localization Am and its m-adic completion B. Is B also an isolated singularity? The localization at m should not be a problem, but completion might. … Read more