# Tag Info

### Completion of a local ring of a curve

My answer is most likely going to be rephrasing Francesco's answer above. Here's how I think about your question. IMHO, the Cohen Structure Theorem is too big a thermonuclear weapon to invoke, because ...
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### Does Grothendieck's algebraization imply existence of colimits of schemes?

Here's one way to see what's going on. I will use the Tannakian duality theorem of Hall and Rydh (see Theorem 1.1 here). It is stated for algebraic stacks, but if you replace the word "algebraic ...
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### Why is $K_{\upsilon}|K$ separable for a global field $K$?

By definition an extension of fields $K'/K$ is separable when $K' \otimes_K F$ is reduced for all field extensions $F/K$, and by limit considerations it is the same to say that all finitely generated ...
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### link between completion of the universal enveloping algebra and an endomorphism of functor

It sounds like you want to prove that $\text{End}(F)$ is the profinite completion of the universal enveloping algebra $U(\mathfrak{g})$. I don't understand your strategy for proving this (in ...
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### Is completion of isolated singularity isolated?

I believe that in your situation, $B$ indeed has an isolated singularity at the maximal ideal $\mathfrak{n} \subseteq B$. Let me first give two possible definitions for “isolated singularity”; please ...
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### Is Cauchy completion the largest extension with the same free cocompletion?

The answer is positive. I found a published account with details to be chapter 6 and 7 of Handbook of Categorical Algebra 1 by Francis Borceux. Thanks to the comments, useful links that summarize how ...
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### Completion of a local ring of a curve

One possible reference is Mumford, The red book of varieties and schemes, chap. III, § 6.
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### Is there a complete local analogue of the Artin-Tate lemma?

Yes. In fact, if $B \subset C$ is a module-finite extension of noetherian rings with $C$ local and complete then $B$ is local and complete. Indeed, by standard prime-lifting stuff with module-finite ...
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### Does CZF prove there is a minimal cauchy completion of the rationals?

You can prove this from the regular extension axiom $\mathbf{REA}$ using the general theory of inductive definitions. See e.g. Theorem 5.11 in Aczel & Rathjen, Notes on Constructive Set Theory. I ...
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### Doesn't completion of a representation ring preserve its indecomposables?

In fact $\text{dim }I(G)/I(G)^2=2$. Let $x=[V_{3L_1}]$, $y=[V_{2L_1+L_2}]$ and $z=[V_{3L_1+3L_2}]$. Then \begin{eqnarray}R(PSU(3))\cong\mathbb{Z}[x, y, z]/(y^3-y^2-xz-2y(x+z)-x-y-z).\end{eqnarray} ...
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### Is a filtered colimit of complete module complete?

The answer should be negative. For example, just take $R=k[[t]]$ in your case. We can choose a series of indeterminates $x_i$, and let $k_i=k(x_1,x_2,...,x_i)$, $M_n=k_n[[t]]$, and take the filtered ...
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### Completed stalks of the pushforward of the structure sheaf

This has little to do with morphisms, and follows immediately from the following commutative algebra lemma: Lemma. Let $R$ be a Noetherian ring, and $f \colon M \to N$ a morphism of finite $R$-modules....
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### Completion and extension by scalars

This follows by adapting the proof of Tag 00MA, even without assumption 5. We also don't need $S$ to be an algebra; a complete $R$-module suffices. Finally, we never use that $R$ is $I$-adically ...
The comment by skd basically answers your question. I am writing to flesh it out with references, so your question doesn't stay open forever. The derived category of $I$-complete $A$-modules has ...