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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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Is a complete Noetherian local ring a homomorphic image of a Gorenstein local ring of the same dimension?

From the proof of Theorem 29.4 in Matsumura's book Commutative Ring Theory we can see that a complete Noetherian local ring is a homomorphic image of a regular local ring, but how can we prove that a complete Noetherian local ring $R$ is a quotient…
nick
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A question about a ring ext. $k \subset R$ where $k$ is a field, $R$ is not a field, $Spec(R)$ consists of only closed points and is finite

It is a well know fact that if $k \subset R$ is an extension of rings such that $R$ is a finite dimensional vector space over $k$, then every point of $Spec(R)$ is closed (i.e., equivalently every prime ideal of $R$ is maximal), and $Spec(R)$ is…
Rankeya
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$\operatorname{Hom}_\Lambda (B, \operatorname{Hom}_{\Bbb Z}(\Lambda, X)) = \operatorname{Hom}_\Bbb Z (B, X)$

Let $G$ be a group, $B$ a $G$-module and $X$ an abelian group. Let $\Lambda:=\Bbb Z[G]$. Serre states in his book local fields that we have the equality: $$\operatorname{Hom}_\Lambda (B, \operatorname{Hom}_{\Bbb Z}(\Lambda, X)) =…
Rodrigo
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Homomorphism in case of local ring

Let $A$ be a local ring and $\mathcal m$ the maximal ideal, considered as an $A$-module. Is then every $A$-module-homomorphism $\mathcal m \rightarrow A/\mathcal m$ equal to zero? Remark: I pose this question because I read that $Hom_A(A/\mathcal…
Veen
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Radical of an ideal after adjoining roots

Let $A$ be a Noetherian domain containing an algebraically closed field $k$. Let $x_1,\ldots,x_r\in A$ be irreducible elements generating a radical ideal $I=(x_1,\ldots,x_r)$. Set $B:=A[y_1,\ldots,y_r]$ where $y_i^n = x_i$. I am wondering whether…
Jesko Hüttenhain
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Square of tensor product

Let $A$ be an integral domain and $B$ be an $A$ algebra. Let $I$ be and ideal of $B$. Something has been bugging me : Is it true that $$(B\otimes I)/(B\otimes I)^2 \cong B \otimes (I/I^2)$$ We obviously have a surjective map $B\otimes I \to B…
schlomo
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an isomorphism of extension functors

Let $(R,m)$ be a Noetherian *local ring and suppose that $m$ is maximal in the ordinary sense. Then why is it true that $\operatorname{Ext}^i_R(R/m^j,M) \cong \operatorname{Ext}^i_{R_m}(R_m/m^jR_m,M_m)$ for every graded $R$-module $M$? Reference:…
Manos
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epimorphism in the category of commutative rings

Let $\phi:A\to B$ be an epimorphism in the category of commutative rings, we can find that the induced continuous map $\phi^*$ from Spec$B$ to Spec$A$ is injective as a map between sets, I want to know if $\phi^*$ is also an immersion of…
wxu
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Question about Going-Up

Let $k$ be a field, not necessarily algebraically closed. Then how would you show that the extension $k[x] \subset k[x,y]$ does or does not satisfy Going-Up?
Rankeya
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Uniqueness of representation of an element of an ideal in domains

Suppose $R$ is a domain and $I=aR$ be a non-zero principal ideal. Then, every element of $I$ has a unique representation, for if $ra=sa$ then $(r-s)a=0$. Since, $a\neq 0$ and $R$ is a domain, we have, $r-s=0$ and thus, $r=s$. Can we extend this to…
BMI
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Eisenbud 3.11(d) - A Uniform Bound on the Length of Certain Modules

I am trying to solve this exercise from Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. There is a hint or possibly a solution in the back, but I want to try to get some more organic feedback before peeking. 3.11(d): Let $k$…
pturner
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local Noetherian of zero depth implies Artinian?

Let $(R,m,k)$ be a local Noetherian ring such that $\operatorname{depth}R=0$. Question: Is it true that $R$ is Artinian? PS: If it is true then please only say so, as i am still attempting to prove it. If it is not true then please provide a…
Manos
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If A is a finitely generated algebra over the integers and m is a maximal ideal, then A/m is finite

I'm trying to prove the following: Suppose $A=\mathbb{Z}[x_1,\ldots,x_n]/I$ where $I$ is some ideal. Then for all $m \in Specm(A)$ we have $\mid A/m \mid$ is finite. I've seen some proofs of this on the site, but I have the following restriction…
user115697
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Preimage of Maximal ideals

It is a common fact that when F is a surjective homomorphism from a commutative ring A to another B (with 1), we have inverse images of the maximals in B maximal in A. Could anyone be so kind to me help solve the same problem without the condition…
karparvar
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What are some concrete examples of ideals and modules where $I M = M$?

I'm trying to get more of intuition for the cloud of ideas surrounding the abstract Cayley-Hamilton theorem, Nakayama's lemma, etc., so I'd like to see some concrete examples. The problem is that the main concrete applications I'm aware of are just…
Daniel McLaury
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