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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.

16857 questions
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rank function on Spec (help with definition)

one definition of the line bundle over a ring is: a finitely generated projective A-module such that the rank function Spec A → N (positive integers) is constant with value 1. We call A itself the trivial line bundle. so here i think that spec is…
El Moro
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Picard group for dummies

a picard group is the set of isomorphism classes of invertible R-modules. I just read that phrase in the CRing project notes without further explanations: Here are my questions: 1-under which law (I am guessing it's a restriction of a tensor…
El Moro
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Non-decreasing Hilbert Function

In commutative algebra, people care about the Hilbert function of the associated graded ring is non-decreasing. As far as I know, motivation came from Sally's question in 1978 which is one dimensional Cohen-Macaulay ring with a small embedding…
user619499
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Smallest cardinality of generating set of maximal ideal in a ring $R$

I wonder the smallest cardinality of generating set of maximal ideal in a ring $R$ in it coincide with the Krull dimension? i.e. Let $J=\{$Maximal ideal in $R\}$, $S_{I}$ is the generating set of maximal ideal $I$,Krull dimension of $R=n$. Is…
Ken.Wong
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deciding if a chain is a composition series (sanity check)

A small sanity check related to Question 2 from here: proof of the Krull-Akizuki theorem (Matsumura) Let $C$ be an $A$-module, with $A$ commutative ring and suppose that there exists a chain of submodules $C=C_0 \supset C_1 \supset \cdots \supset…
Manos
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Finitely generated $k$-algebra with vanishing module of differentials has Krull dimension zero

I have come across this question reading the answer to this question: An algebraic result corresponding to etale morphism. How can I prove: Let $k$ be a field, R a finitely generated $k$-algebra. If the module of differentials $\Omega_{R/k}=0$,…
Linda
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Integral extension of a DVR which is of finite type over a field $k$

Let $(R,\mathfrak{m})$ be a DVR over an algebraically closed field $k$ and $K=\operatorname{Frac}R$. Assume that a field $k^{\prime}$ of finite type over $k$ and a finite field extension $K\subset K^{\prime}=K\otimes k^{\prime}$ are given. Then I…
Aoki
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Two $R$-module homomorphisms induce a bilinear map

Say $A,B,C$ are three $R$-algebras. And given two homomorphisms of $R$-algebras, $f:A\to C$ and $g:B\to C$. We could induce the "product homomorphism" $f\times g$ which is an $R$-bilinear map $A\times B\to C,(a,b)\mapsto f(a)g(b)$ by forgetting the…
CO2
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Showing explicitly by using the valuation that DVRs are Dedekind domains.

We know that DVRs are Dedekind domains by the characterization of Dedekind domains in terms of localizations at maximal ideals. There are many nice abstract ways to prove that DVRs are Dedekind domains. This is not what this question is about. I…
Mark Murray
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$B\otimes_A A/\mathfrak{m}$ a domain with $B$ not a domain

I want to find an example of where $A$ is a dedekind domain, $B$ is a flat algebra over $A$, $m$ is some maximal ideal of $A$ with $B\otimes_A A/\mathfrak{m}$ a domain but $B$ is not a domain. Specifically could you hint me towards how you would…
samlanader
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Is the morphism $A^G\rightarrow A$ faithifully flat?

Let $A$ be a ring with a an action of a group $G$. Is the morphism $A^G\rightarrow A$ faithfully flat? If not, is it true under some reasonable conditions? My motivation for this problem is that the morphism on topological spaces $Spec(A)\rightarrow…
xlord
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Ring of Invariant

Let $G \subset SL_2(\mathbb{C})$ be a finite subgroup acting linearly on $\mathbb{C}[X, Y]$. Then it is claimed that the ring of invariants $\mathbb{C}[X, Y]^G$ is always a hypersurface. I am not able to see its proof by myself. Please help.
A.G
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Annihilators of Modules

I'm stuck trying to prove that for two $R$-modules $M,N$ ($R$ commutative with a 1), then $$Ann(M+N)=Ann(M) \cap Ann (N)$$ I was trying to do double inclusion, and I can prove the RHS is contained in the LHS, but im stuck in the other direction…
Chris Birkbeck
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Let $R \subset S$ be an integral ring extension and $I$ an ideal in $S$ then $R/(I \cap R) \subset S/I$ is an integral extension

I am working on the following exercise: Let $S$ be a commutative ring and let $R \subset S$ be an integral ring extension and $I \vartriangleleft$ an ideal in $S$. Then $R/(I \cap R) \subset S/I$ is an integral extension. I do recognize that $R/(I…
3nondatur
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An example of a non-coherent ring

What's the easiest example of a non-coherent ring ? I know that the ring of polynomials in an infinite number of variables over a Noetherian ring A is an example of a coherent ring, so possibly if we take A non-noetherian we can find a non-coehrent…
Laassila souhayl
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