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
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
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
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
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
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
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
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
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
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
