# Questions tagged [commutative-algebra]

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

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### Inclusion of Rings after Localization

Let $\phi:A \to B$ an injective ring map between noetherian integral domains $A,B$.. Let $C \subset B$ a subring of $B$ and assume that there exist a prime ideal $\mathfrak{p} \subset A$ , such that ...
• 683
1 vote
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### Corollary from Theorem on Acyclic Carriers in Hilton and Wylie's Homology Theory

Let $C=(C_p, \partial_p)$ and $D=(D_p, \partial_p)$ two free chain complexes in category of $\mathbb{Z}$-modules, ie every $C_q, D_p$ is freely generated. A carrier (function) $E$ of a chain map $\phi$...
• 683
1 vote
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### Vakil's FOAG exercise 6.5 M, prove generic point of irreducible component of some $\text{supp }M$ is associated prime.

I was doing exercise 6.5 M (2022 version) in professor Vakil's FOAG. which asked me to show that : Suppose $M$ is a finitely generated module over Noetherian $A$, and $\mathfrak{p} \subset A$ is a ...
• 4,090
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### Lemma from galois representation book

Let $\mathcal{O}$ be a complete Noetherian local rings with residue field $k$ and suppose $A \to B$ is a surjective morphism of Noetherian local $\mathcal{O}$-algebras (with residue field $k$) with ...
• 9,056
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### Dimension of $k[x_1,...,x_d]$ localized at arbitrary maximal ideal

In Atiyah-Macdonald's Introduction to Commutative Algebra, when proving the equivalence between the local dimension at any point of a variety and the dimension of the variety (Proposition 11.25), he ...
• 111
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### Testing injectivity/projectivity of specific objects in an abelian category

I have a valuation ring $A$, a full abelian subcategory $\mathcal{C}$ of $\text{Mod}_A$ and an object $M$ in $\mathcal{C}$. I know that $M$ is not a projective $A$-module, but that it can still be a ...
• 2,992
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### The stalks at embedded points are non-reduced.

I was trying to prove the claim that: Given locally Noetherian scheme $X$, the stalks at embedded points are non-reduced. (where embedded points means those associated point that not coming from ...
• 4,090
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### Proof of Zariski lemma.

I am studying algebraic curves but I have no background of commutative algebra.An important theorem in this topic is the weak Nullstellensatz which states that: Any maximal ideal of $K[X_1,...,X_n]$ ...
• 2,849
1 vote
20 views

### $s\in {\frak{p}}\cap S$ is nilpotent for $A_{\mathfrak{p}}/\operatorname{Ann}_A(\frac{m}{s})_{\mathfrak{p}}$

This question is motivated by About weakly associated primes . I have some detail in the solution that can not work out, I can rephrase the problem as : Let $M$ be a $A$-module, we can taking ...
• 4,090
1 vote
27 views

### The equality between the local dimension and the dimension of a variety

When I am reading Atiyah-Macdonald's Introduction to Commutative Algebra, the author wants to establish the equality of the Krull dimension of the localized coordinate ring at a point and the ...
• 111
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### When is $R\cong \Pi_{Q\in \max(R)}R_Q$?

Problem: Let $A$ be a semilocal Noetherian ring and let $\text{max}(A)$ be the finite set of maximal ideals of $A$. I am trying to prove the following statement (which I hope to be true but am not ...
• 1,000
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### Computing the length of a module

I have some trouble to compute the length of the $\mathbb{R}[x]$-module $M=\mathbb{C}[x]/\mathbb{R}[x]$. Usually, I would try to find a composition series of the module, but I am not sure how to find ...
• 31
1 vote
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### About the integral closure of a DVR inside a galois extension of function field with two variables

Edit: I see my mistake now (murphy), w(X + Y) = min(w(X), w(Y)) only when the valuations are not the same. I have the following algebraic problem, which I encountred after thinking about blowups. My ...
1 vote
I am wondering that as we define $Z(R)$, center of a ring is a subring of $R$. Can we define a subset which is collection of all those elements which semicommute with other elements. Will this set ...