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

Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ and $r_n+Rx^{n}=r_{n+1}+Rx^n$). Denote the image of $r\in R$ under the canonical ring-homomorphism by $\hat r$. Localize $\hat R$ using the multiplicative subset $\{1,\hat y,\hat … Read more

Artin approximation theorems over non-regular rings/non-Noetherian rings

In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers $\underline{polynomial}$ equations. Is there some version like this: “Let $R$ be a local Noetherian Henselian ring(not necessarily regular), over a normed field. Given an arbitrary (possibly countable) system of analytic equations over $R$, … Read more

Can we recover the completion of a local ring $R$ if its associated graded is the coordinate ring of a Veronese variety?

Suppose $R$ is a localization of a normal closed point of a variety of dimension $n$ over an algebraically closed field $k$ with maximal ideal $\mathfrak{m}$. Suppose also that the associated graded $\operatorname{Gr}_{\mathfrak{m}} R$ is isomorphic to the degree $d$ Veronese subring $S_d$ of $k[X_1,\ldots,X_n]$. Is this enough to conclude that the completion of $R$ … Read more

Completion of local rings in the exceptional divisor of a blow-up

Let $X=\mathrm{Spec}(A)$ be an affine variety, $Z\subseteq X$ a closed, reduced subscheme. Let $$\beta:Y=\mathrm{Bl}_Z(X)\to X$$ be the blow-up of $X$ in $Z$. In other words, $Y=\mathrm{Proj}(A[IT])$ for $I:=I(Z)$. Let $E:=\beta^{-1}(Z)$ be the exceptional divisor. For a point $Q\in E$, I am now wondering how the completion $\hat{\mathcal{O}}_{Y,Q}$ looks like. I would like to understand its … 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