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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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Definition of split exact sequence and splitness of some exact complex

First recall the definition of splitness of the short exact sequence : Def. (Split short exact sequence) A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category, ${\displaystyle…
Plantation
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tensor product and localization of module

$M$ be $\mathfrak m^{-1}R$ module where $\mathfrak m$ be maximal ideal in a ring $R$ I want to show that: $\mathfrak m^{-1}R \otimes_R M$ is isomorphic to $M$ First: I define a map $f$: $\mathfrak m^{-1}R \otimes_R M$ $\to$ $M$ by …
lee
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Unmixedness theorem and associated primes

Let $R$ be a commutative Noetherian ring. $I=(x_1,...,x_n)$ is an ideal generated by $n$ elements such that $\operatorname{height}I=n$. If $R$ is Cohen-Macaulay, then every associated prime of $I$ is minimal over $I$. This is the statement of…
user782932
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Under a derivation on a ring, is the image of a nilpotent element still nilpotent?

Let $k$ be a field. Let $A$ be a commutative $k$-algebra. A $k$-derivation is $k$-linear map $\partial:A\to A$ such that $\partial(fg)=f\partial g+g\partial f$. Let $G$ be an affine algebraic group over $k$. If there is a dual action of $G$ on $A$,…
Display Name
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Clarifiying lemma on dimension of quotient of regular local ring

https://stacks.math.columbia.edu/tag/00KW proves that if $R$ is a regular local ring and $x\in\mathfrak{m}$ is not a zero divisor, then $\dim(R) = \dim (R/(x)) + 1.$ I have a question about one part of their proof. They write that if $x$ is not a…
user960774
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Construction on the equivalent class for localization

I have a question about the following example about the localization : If $x$ is an element of a commutative ring $R$ and $ S = \left\{1, x,x^{2},...\right\}$ then $S^{-1}R$ can be identified (is canonically isomorphic to) $…
Hanwoong Cho
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$R/I$ is finitely generated iff $I$ contains a monic polynomial

Problem: Let $R=\mathbb{Z}[t]$ and $I$ an ideal of $R$. Then $R/I$ is finitely generated as a $\mathbb{Z}$-module if and only if $I$ contains a monic polynomial. Suppose $I$ contains a monic polynomial $g$, then for each $f\in R$ there are $h,r\in…
defacto
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Abuse of notation to write injective maps as subset inclusion

I have one basic question in the accepted answer given here In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal and I don't have enough reputation to comment so I'm posting my question here: Given…
User
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Non-injective induced maps on spectra

So this is a very basic question and I am sure that I am just missing something obvious. Given a commutative ring homomorphism $f: A \rightarrow B$, we have an induced map on spectra $f^*: Spec(B) \rightarrow Spec(A)$ via the formula…
szantag
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Suppose $x^2$ is homogeneous of degree 2. Does it follow that $x$ is homogeneous of degree 1?

Say $k$ is a field. Consider the standard graded $k$-algebra $R = k[A,B]$, where $A$ and $B$ are indeterminates. Suppose, for some $x \in R$, that $x^2$ is homogeneous of degree $2$. Does it follow that $x$ is homogeneous of degree $1$? My attempt…
beanshadow
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Zero-divisors vs nilpotents (in Noetherian rings without idempotents)

I've looked at earlier similar questions and, as far as I could see, the examples of zero divisors that are not nilpotent are idempotents. I tried to prove that those are the only examples, at least in some cases, but could not. So: Let $k$ be a…
Boogie
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Proof verification( Localization from Maximal ideals)

This question is from a lecture notes from which I am studying commutative algebra and this question was left as an exercise for students. For two ideals I , J in A, prove that $I \subset J$ iff $I_M \subset J_M$ in $A_M$ for all maximal ideal…
user775699
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Is finitely generated algebra over a field the same as polynomial ring over a field? (Proposition $7.9$ of Atiyah-Mcdonald)

Let $k$ be a field, $E$ be a finitely generated $k$-algebra (suppose it is generated by $x_1, x_2,...,x_n$). This means that every element of K can be written as a polynomial in $x_1, x_2,...,x_n$ with coefficients in $k$, so the evaluation…
JBuck
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Rees algebra of a monomial ideal

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\geq 1$ for all $i$, then prove that $$ R[It]/\mathfrak m R[It]\simeq K[f_1t,\ldots, f_q t]\simeq…
Fbakhshi
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Show that $(A / m) \otimes_A A^{n}$ is a vector space of dimension $n$.

Let $A$ be a ring $\neq 0$, and $\mathcal{m}$ a maximal ideal of $A$. Then both of the field $A / m$ and $A^{n}$ (n-tuple direct sum) are $A$-modules. Question: How to show that $(A / m) \otimes_A A^{n}$ is a $A / m$-vector space of dimension…
Lelong Wang
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