Vinteuil
  • Member for 8 years, 6 months
  • Last seen more than 5 years ago
Math books for advanced high school students
15 votes

Two excellent old books (but still very interesting) by great mathematicians. They are translated to several languajes: Courant-Robbins, What is mathematics? Rademacher-Toeplitz: Von Zahlen und ...

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Is a Gorenstein ring a quotient of a local complete intersection
12 votes

No. A quotient of a local complete intersection ring has complete intersection formal fibers and so its complete intersection locus is open. In "Greco-Marinari, Nagata's Criterion and Openness of Loci ...

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Hochschild cohomology and formal smoothness
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6 votes

Yes, the Hochschild-Konstant-Rosenberg theorem has a converse. More generally you have vanishing characterizations of smoothness in terms of Hochschild homology (one of them is e.g. Avramov, Luchezar ...

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Does regular field extension preserve regularity?
6 votes

If $k \rightarrow K$ is formally smooth for the discrete topology (i.e. separable), by flat base change $A \rightarrow A\otimes_kK$ is formally smooth for any $k$-algebra $A$ essentially of finite ...

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Descending étaleness
5 votes

The "Main theorem" in "Mark S. McCormick, Etaleness and Normality, Journal of Algebra 219, 1999, 437-465" almost answers your question. Sorry, this should be a comment not an answer, but I have few ...

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When a smooth algebra is regular?
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4 votes

Yes. More generally, let $A \to B$ be a homomorphism of noetherian rings satisfying the condition (1) of your question (that is, B is formally smooth in the sense of [EGA IV.17.1.1]). Let $\mathfrak q$...

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if $R$ is Noetherian local with a finite module of finite injective dimension and if "?" , then $R$ is "Gorenstein"
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4 votes

If you admit $M$ cyclic as additional assumtion, then $R$ is Gorenstein by a theorem in Peskine-Szpiro paper "Dimension projective finie et cohomologie locale", Theorem II.5.5.

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Is the embedding dimension minus the dimension upper semicontinuous?
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2 votes

Edit. Finally the proof was not so long, so I include it complete: Question 3. Embedding codimension (sometimes simply codimension). Question 1. I don't have access here to "Lech, Inequalities ...

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A question about Complete Intersections
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2 votes

If $n=\text{dim}_{K}(H_1(\mathbf{x},A))$ then $\text{dim}_{K}(H_2(\mathbf{x},A))=\frac{n(n-1)}{2}$. This was proved by Assmus in 1958. In general, $H_*(\mathbf{x},A)$ is the exterior algebra over $H_1(...

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Regularity of a tensor product
2 votes

Another well known example: it is clear that the tensor product of two field extensions need not be regular. So this answers your original question. Regarding the one raised in the comments, if $A$ ...

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Formally smooth map from a regular ring
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2 votes

As I wrote in the comments, the answer is usually yes, depending on the topology you are considering. The two more common cases are (it should be direct references for them, maybe in EGA $0_{IV}$): ...

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Projective dimension of a quotient ring
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1 votes

I will try to answer to both questions together but the second one only in a few very particular cases. I'm sorry for not having complete answers. If $\phi :A \to B$ is flat and the rings $A$, $B$ ...

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When every module is a scalar extension?
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1 votes

An example: if $B$ is the henselization of a local ring $A$, then for any finite type $B$-module $N$ there exists a finite type $A$-module $M$ such that $N$ is a direct summand of $M\otimes_AB$. You ...

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formally etale deformations of algebras
1 votes

Question 1 can be deduced from EGA $0_{IV}.19.7.1.5$

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Can André–Quillen homology detect the property of being Gorenstein?
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1 votes

"No" if you want a standard vanishing result, and "no up to now" if you are thinking in another kind of characterization (see however for a related result: Garcia-Soto, Ascent and descent of ...

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local cohomology mayer-vietoris sequence
1 votes

In line -3, first isomorphism.

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zero homology of augmented Koszul complex implies the sequence is regular?
0 votes

In my second-hand copy of Matsumura's, that is pointed as a mistake, but actually I do not know the answer to your question. Maybe you are interested in http://arxiv.org/pdf/math/0406566 (since Koszul ...

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