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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

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Proposition 11.3 in Atiyah MacDonald

Let $A=\bigoplus_{n=0}^{\infty} A_n$ be a Noetherian graded ring, in which case $A$ is generated as an algebra over $A_0$ by elements $x_1,\dots,x_s$ of degrees $k_1,\dots,k_s$. Let $\lambda$ be an additive (with respect to exact sequences)…
Manos
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Question about regular local rings

Let $A$ be a commutative regular local ring of dimension $d$ with maximal ideal $\mathcal m$ and $a \in A$ an element of the ring. Suppose that $\mathcal m \cdot a \subset \mathcal m^2$, i.e. if I multiply the element $a$ by an arbitrary element of…
Cyril
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Can a the variety associated to a finitely generated $K$-subalgebra of $K[X]$ be embedded into $\mathbb{A}^3$?

Let $K$ be a field. Is there an example of a finitely generated $K$-subalgebra $$ A\subseteq K[X] $$ which is not isomorphic to $K[T_1,T_2,T_3]/I$ for some ideal $I$? As $A$ is finitely generated, we may write $A\cong K[X_1,X_2,\ldots, X_n]/I$…
user8463524
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Dimension of Tensor Product for Flat Extensions

Suppose that $A,B,$ and $C$ are commutative unital rings, $A\to B$ is flat, and $A\to C$ is any map. I am trying to determine whether $$ \dim B\otimes_AC=\dim B+\dim C-\dim A $$ Any counterexamples or references? I am taking the Krull dimension…
xavier17
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Is this module finitely generated?

Suppose $M$ is a $A$-module, $A$ is a commutative ring with 1, such that for every countably generated submodule $N$ of $M$, there exists a finitely generated submodule $L$ which contains $N$. Must $M$ be finitely generated? (Maybe it should be…
wxu
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Finite injective dimension of the residue field implies that the ring is regular

Let $(R,\mathfrak m,k)$ be a noetherian local ring. If $\operatorname{inj dim}_R k$ is finite, then $R$ is regular. This is exercise 3.1.26 from Bruns and Herzog, Cohen-Macaulay Rings. I don't see how I can use the results from this chapter to…
Andrei
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Why is $\hat{I}$ contained in the Jacobson radical $J(\hat{R})$?

Suppose $I$ is an ideal of a commutative ring $R$, and $\hat{R}$ is the $I$-adic completion. I don't follow why $\hat{I}$ is in $J(\hat{R})$. I know $\hat{R}$ is complete wrt the $\hat{I}$-adic topology. Let $a\in\hat{I}$. Since…
Carrie Kawashima
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Is the localization of a ring $R$ at a prime ideal a finitely generated algebra over $R$?

Let $R$ be a ring and let $S=\{1,s,s^2,s^3,\dots\}$ be a multiplicative system of $R$. Consider the canonical map $R\rightarrow S^{-1}R$. Is $S^{-1}R$ a finitely generated algebra over $R$? It looks like $\frac{1}{s}$ will generate $S^{-1}R$ over…
Babai
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A step in showing that $\oplus_{i\in\mathbb Z}\mathbb Z$ is reflexive

I'm working on an assignment, in which in the end I'm trying to show that the countable direct product $\prod_{i\in \mathbb Z}\mathbb Z$ is not reflexive. I've already made some progress on the subject, and reduced the problem to showing…
kneidell
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Do localization and dual commute for locally free modules of rank $1$?

Let $A$ be an integral domain and $M$, $N$ be finitely generated $A$-modules. I know from this topic that one cannot expect $\hom_A(M,N)_P \cong \hom_{A_P}(M_P,N_P)$ to be true in the general case (although I lack the background to fully grasp the…
tommy
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If $A\subseteq B$ where $A,B$ are commutative domains, and $B$ is a finitely generated $A$-module is Frac$(A)\subseteq$ Frac$(B)$ finite?

If $A\subseteq B$ where $A,B$ are commutative domains and $B$ is a finitely generated $A$ module, is $\operatorname{Frac}(A)\subseteq \operatorname{Frac}(B)$ a finite field extension? I know this extension is algebraic and every element of…
birju
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On a localized ring tensor with a module

Let $A$ be a commutative ring, $S$ be a multiplicative subset of $A$ and $M$ be an $A$-module. The questions says to "describe a natural isomorphism $(S^{-1}A) \otimes_A M \cong S^{-1}M $ as $A$-modules". I manage to show these two are isomorphic…
Tom Mosher
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Integral closure under completion

Suppose $(R,\mathfrak{m})$ is a commutative local ring with identity and $I$ an ideal in $R$. If $I$ is integrally closed, does it follow that $I\hat{R}$ is integrally closed? If not, is this true with certain assumptions on $R$? EDIT: OK, I read…
Vince Cairns
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Atiyah-Macdonald, Chapter 10, Proposition 10.15 clarifications

In Proposition 10.15 in Atiyah-Macdonlad, what does the equality $\hat{\mathfrak a}=\hat A\mathfrak a$ mean? I know that there is an isomorphism $\hat A\otimes_A\mathfrak a\cong\hat{\mathfrak a}$ and I understand what it does (what is sent to what),…
aytio
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algebra homomorphism $k^S \to k$

Let $k$ be a field and $S$ be an infinite set. Assume $|S| \leq |k|$. Why is then every $k$-algebra homomorphism $k^S \to k$ equal to a projection $\mathrm{pr}_s$ for some $s \in S$? I don't know how to use the cardinality assumption here. The…
Martin Brandenburg
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