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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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Hilbert Serre Theorem and Atiyah-MacDonald 11.3

I have some difficulty with a detail in the treatments of the proof of this theorem, most of which follow that of Theorem 11.1 in A-M (many more or less word for word). This is not really significant till Proposition 11.3 when it seems to me some…
Noel Robinson
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Question on local property

In commutative algebra, sometimes people consider the polynomial ring over a field as a local ring and they uses the Nakayama's lemma to get some informations about the generators of a finitely generated module over that polynomial ring. My question…
Arsenaler
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Intersection of Subcomodules is a Subcomodule?

Let $(C, \Delta, \epsilon)$ be a coalgebra over a commutative ring $k$. Let $M$ be a right comodule over $C$, that is a $k$-module $M$ together with a $k$-linear map $\delta \colon M \to M \otimes C$ such that $(1 \otimes \epsilon)\delta = 1$ and…
Paul Slevin
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Associated Primes and Hom modules

Let $R,\mathfrak{m}$ be a Cohen Macaulay local ring and $M$ be an $R$ module such that $\mathfrak{m}\in Ass(M)$. i.e., the maximal ideal $\mathfrak{m}$ is an associated prime of $M$. Now suppose $Hom(R/\mathfrak{m},M)=0$ then does it necessarily…
dongrugose
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Precursor to Serre's criterion in Eisenbud

Theorem $11.2$ in Eisenbud states the following: A Noetherian domain $R$ is normal iff for every prime $P$ of $R$ associated to a principal ideal, $P_P$ is principal. Since $R$ is an integral domain, then for any principal ideal $Q=(r)$, the…
user7090
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References needed for Spectrum R

This is my frist course in commutative algebra. I took two weeks ago Zariski topology which is interesting. , but my text book (Sharp) doesn't have this topic. I want to know how I can find spectrum for some ring like Z[x] and others. I found them…
Team
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$\text{dim}_{\mathbb{C}}Hom_{\mathbb{C}[x,y]}(I , \mathbb{C}[x,y]/ I)$

Given an ideal $I \subset \mathbb{C}[x,y]$ such that $\text{dim}_{\mathbb{C}}\mathbb{C}[x,y]/I = n$, can the dimension of the space $Hom_{\mathbb{C}[x,y]}(I , \mathbb{C}[x,y]/ I)$ be determined in some elementary…
baltazar
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Proove that in regular rings, some module length grows slower than some polynomial (read description)

I'm working on the F-signature of triples $(R, \Delta, f^t)$ in the case $\Delta = 0$. More precisely, I need to show that for a regular ring R with maximal ideal m, and for any principal ideal f of R, and any $0 \leq t \leq 1$, $$\lim\limits_{e \to…
user402952
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Proving that a ring is von Neumann regular

This last week I've been trying to prove that a certain ring is von Neumann regular. Here come the details. Let $A$ be a commutative ring with unity. Put $A'=A[X_a]_{a\in A}/(a^2X_a-a,aX_a^2-X_a)_{a\in A}$ and let $\psi:A\to A'$ be the canonical…
Tobias Shuxue Laoshi
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Proposition 2.14 iii) Atiyah Macdonald introduction to commutative algebra

Hi I am refering to proposition 2.14 in Atiyah-MacDonald introduction to commutative algebra and I can't find the bilinear maps that will induce $A$-module homomorphisms $f,g$ where $f:(M \oplus N) \otimes P \to (M \otimes P) \oplus (N \otimes P)$…
Oliver
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Number of Generators and Localization

I thought I had a fairly good understanding of finite projective modules over commutative rings, but I recently asked myself a few questions about them and this exposed how little I actually know. My first question was: can a finite projective…
Noah Olander
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About completion over a noetherian local ring.

Let $R$ a noetherian local ring with maximal ideal $\mathfrak{m}$. Let $M$ and $N$ $R-$modules finitely generate and $\hat{M} \cong \hat{N}$ (the completion over the ideal $\mathfrak{m}$) . I have that $\widehat{\mbox{Hom}_R(M,N)} \cong…
ÝTAN
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In what generality does the second argument of Hom distribute over tensor?

Let $R$ be a commutative ring, and let $M,N,P$ be $R$-modules. In what generality can we say that $Hom_R(M,N\otimes_R P)\cong Hom_R(M,N)\otimes_R Hom_R(M,P)$. This is true in a cartesian monoidal category, and from that, it seems like we might be…
dtripleez
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Blow-Up Algebra definition

Suppose $R$ is a ring and $I$ is an ideal of that ring, then the blow-up algebra of $I$ in $R$, as defined in Eisenbud, is: $B_IR := R \oplus I \oplus I^2 \oplus \dots \cong R[tI]$ Why is this direct sum isomorphic to $R[tI]$?
user7090
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What does it mean when elements act as units on a set?

I'm reading Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry, and I am confused by one of his proofs. The setup is that $R$ is a commutative ring, $U$ is a multiplicatively closed subset, and $\varphi$ is the natural map…
Mehta
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