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Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in …

Here is a summary of what I learned from a nice expository account by Eisenbud (written in French), can be found as number 27 here.1 First, one studies a more general problem: Let $A$ be a Noetherian …

I will argue that the examples you gave are "simplest" in some strong sense, so although they look unnatural, if Martians study commutative algebra they will have to come up with them at some point. …

Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the intersectio …

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not rem …

To add to David Speyer's answer, since this story continues with a rather interesting and illustrious history: When $A$ is regular, the Tor functor satisfies the following property: (1) $\text{Tor}_ …

I have heard during some seminar talks that there are applications of the theory of matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any s …

(To supplement Alberto's example) If $V$ is projective, then the gap between being locally c.i and c.i is quite big. In particular, any smooth $V$ would be locally c.i., but they are not c.i. typicall …

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 …

Recently I dug up some biographical details of Lindsay Burch, of Hilbert-Burch Theorem fame, whose few papers have had quite an impact on commutative algebra. This made me curious about the first wome …

ADDED: There is an account written by Buchsbaum (see page 1 and 2 of number 23 here) which described in more details what they wrote in [1]. So the localization problem for regular rings was definitel …

Hi Pete, this sounds like a lot of fun! I wish I could be there (-: Here is a concrete and useful property of flatness, you can explain it without using Tor. Suppose $R\to S$ is a flat extension. T …

In a paper I need to make reference to two conjectures by Gabber, from Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37 …

Dear Andrea: Hartshorne was right, but we need to do some work. Let $\mu(I)$ be the minimal number of generators of $I$, and $\mu_h(I)$ be the minimal number of a homogenous system of generators of $I …

Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open propertie …