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 ...

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 ...

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 ...

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 ...

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 ...

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 $\mathfrak q$...

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.

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 ...

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 $H_1(... View answer 2 votes Another well known example: it is clear that the tensor product of two field extensions need not be regular. So this answers your original question. Regarding the one raised in the comments, if$A$... View answer Accepted answer 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$0_{IV}$): ... View answer Accepted answer 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$A$,$B$... View answer Accepted answer 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 ... View answer 1 votes Question 1 can be deduced from EGA$0_{IV}.19.7.1.5\$

"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 ...