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It is worth noting the following result by S. Katz: If the singularity $X$ defined by $xy-g(z,t)$ is isolated (so $g=0$ is reduced) and $cA_n$ (meaning a general hyperplane section is a type $A_n$ s …

I do not know a reference, but here is my guess. I will use the notion of the wikipedia article. The order is: type, class group, the generator ideals. ($i^2=-1$, and I assume char. 0 for simplicity) …

Recently I noticed an intriguing talk by Kollár at the MAGIC conference. The abstract says: Title: Cohomology groups of structure sheaves Abstract: I will discuss the behavior of cohomology groups o …

This new wonderful note, F-singularities: a commutative algebra approach, written by Linquan Ma and Thomas Polstra, two card-carrying commutative algebraists, is perhaps what you need. From the Introd …

That would be number 11 on her paper site "Linear equivalence of topologies". As for Problem 0.9, it is known for regular local rings over fields by Ein-Lazarsfeld-Smith and Hochster-Huneke. The most …

To expand on Emerton's answer: Using the excision sequence, Cartan's result in the algebraic case boils down to showing the following: Let $R$ be a regular local ring, and $I$ and ideal of height at l …

This question is motivated by this (highly recommended) comment by Emerton on Terry Tao's post "Learn and relearn your field". In particular, the following paragraphs: In particular, the first couple …

This is kind of embarrassing, but I never figured out how to cite journal names in the bibliography, especially when to abbreviate and how. For example, do we write "Adv. in Math." or "Advances in Mat …

For a reference on Cohen-Macaulay and Gorenstein rings, you can try "Cohen-Macaulay rings" by Bruns-Herzog. Also, Huneke's lecture note "Hyman Bass and Ubiquity: Gorenstein Rings" is a great introdu …

I was reading Auslander's talk at the 1962 ICM (beginning of Section 2 on this page). At the end, the reference began: [1] M. Auslander, Modules over unramified regular local rings, Illinois. J. Math …

You don't get more algebraic geometry than understanding the equations defining a (affine or projective) variety, and there are still tons of questions being actively studied. For example: 1) Google …

This question arises from my discussion with a Master student. It concerns with the following situation: let $\phi: R \to S$ be a homomorphism between Noetherian commutative rings. Suppose the $R$-mod …

Let me answer and accept this in CW so that it will not be bumped periodically as not answered by the software. It was hoped that the case of $\phi$ surjective can be generalized, but as Laurent poin …

There are many open problems which are fairly easy to state, also one might need some basic definitions, for example derived functors. I will provide mostly pointers to the ones I know, you can google …

I think this question is quite important. For a commutative algebraist, some knowledge of algebraic geometry is extremely useful (and vice versa). Not only that, and more important to me personally, i …