Let $R$ be a Noetherian domain, and let $\mathfrak{p}$ be a prime ideal; consider the completion $\hat{R_{\mathfrak{p}}}$ of $R$ at $\mathfrak{p}$ (the inverse limit of the system …

Suppose $X$ and $Y$ are two (smooth, affine) algebraic varieties. Let $\mathcal{F}$ be a locally free coherent sheaf over $X \times Y$, and let $\mathcal{G}$ be the pushforward of …

Let $K$ be a field and $L$ an extension of $K$. I wonder how much larger the multiplicative group $L^\times$ of $L$ is than the multitplicative group $K^\times$ of $K$. I know tha …

Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-pro …

Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y …

k is an alegraically closed field and A is a commutative k-algebra. We also know that A is a Noetherian domain and its Krull dimension is one. Are there any necessary and sufficien …

A student asked me why $\mathcal{O}_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him th …

I'm writing some commutative algebra notes, but I'm facing a difficulty in organizing the order of the topics. I'd like to have the topics about factorization before speaking of in …

All rings are assumed to be commutative and unital, with all homomorphisms unital as well. On last week's homework, there was a mistake in one of the questions: (2.5) Let …

If I have a flat family $f \colon X \to T$ such that some fiber is (locally) a complete intersection, does that imply that there is an open set $U$ in $T$ such that the fibers abov …

First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$. I want to find an example of an GCD-domain $R …

Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$? I am interested in the answer to …

In his answer to a question about simple proofs of the Nullstellensatz (http://mathoverflow.net/questions/15226/elementary-interesting-proofs-of-the-nullstellensatz), Qiaochu Yuan …

I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be: -More comprehensive than Atiyah-MacDonald -More readable than Matsumur …

Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques? One proof I know proves first the 'weak' Nullstellensatz which ensures th …