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 ...
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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 $\phi$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $K[X_1,...,X_n]$ ...
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1 answer
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$s\in {\frak{p}}\cap S$ 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 ...
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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 ...
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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 ...
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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 ...
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1 answer
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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 ...
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Set of all elements of a ring which semicommute with other elements of a ring

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 ...
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The set of regular functions on an open subset $U=X\setminus Y$ of an affine variety $X$ is the coordinate ring $A(X)$ implies codim($Y$) $\ge 2$.

This exercise came from Gathmann's algebraic geometry notes. And I've already found that there are similar questions asked here. But What I'd like to know is The set of regular functions $\mathcal{O}...
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Idempotent element [closed]

Let $A$ be a commutative ring. I've come across the following problem. Show that if $I$ and $J$ are two ideals of $A$ such that $I+J=A$ and $I\cap J=\def\Nil{\operatorname{Nil}}\Nil(A)$, the ...
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Meaning of dimension in module finiteness.

I am reading the book by Fulton on algebraic curves.In order to prove the weak version of Nullstellensatz they use some machinery.One of the definitions used in developing this machinery is that of ...
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Is there difference between support of function and support of module?

Let $A$ be a commutative ring, and $M =A$ can be treated as an $A$-module. Let $f\in A$ we have the submodule $(f) \subset A$ therefore we can consider the support of the module $(f)$, which is : $$\...
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Fiber product of local Artinian rings with fixed residue field

Let $\Lambda$ be a complete Noetherian local ring with residue field $k$ and suppose $A \to C$ and $B \to C$ are local homomorphisms of Artinian local $\Lambda-$algebras with residue field $k$. The ...
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Don't know where to start: $\oplus _{i\in I}(L_i\otimes M )\simeq (\oplus _{i\in I}L_i\otimes M)$

Both $L_i$ and $M$ are $R$-modules, where $R$ is a commutative ring with 1. We need to show there exists a $R$-linear isomorphism. My questions: We need to construct both a $R$-linear map with its ...
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relationship between `degrees' in Euclidean domain equations

Let $R$ be a Euclidean domain that admits a multiplicative Euclidean function $d$. Let $a,b$ be nonzero elements of $R$; assume $b \notin aR$. Then $b=aq+r$ for nonzero $q,r \in R$ with $d(r)<d(a)...
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Prove that $\frac{x^2}{x^2 + y^2}$ is irreducible in $\Bbb{Q}[x,y,\frac{1}{x^2+y^2}]_0$

I was doing Vakil's FOAG, in exercise 5.4 N needs to investigate an important example that $\Bbb{Q}[x,y,\frac{1}{x^2+y^2}]_0$ is not UFD (where we take localization of $\Bbb{Q}[x,y]$ at $x^2+y^2$ and ...
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Example of a non-Noetherian Laskerian Ring using property (L1) and (L2) from Atiyah Macdonald Exercise 4.17&4.18

In Exercise 4.17 and 4.18 of Atiyah and Macdonald's book Introduction to Commutative Algebra we proof the following statement. In a commutative Ring $A$ with $1$ every ideal has a primary ...
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Projectivity of exterior powers/algebras?

Suppose that $R$ is a commutative ring and that $P$ is a projective $R$-module. Is it then true that as an $R$-module, $\bigwedge_R(P)$ and/or each exterior power $\bigwedge_R^n(P)$ is projective?
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$A\otimes_k l$ is faithful flat extension of $A$

Let $A$ be $k$- algebra and $l$ be finite extension of field $k$. Prove $B = A\otimes_k l$ is a faithful flat extension of $A$. My attempt: let $M$ be arbitary $A$- module, therefore $M\otimes_AB = M\...
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1 vote
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Proposition 3.3.3 in Bruns and Herzog

The proposition says that If $(R,m,k)$ is a Cohen-Macaulay (CM) local ring of dimension $d$ and $C$ is a maximal Cohen-Macaulay (MCM) $R$-Module, then a) Suppose $M$ is a MCM $R$-module with $Ext^j_R(...
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prove $K(A)\bigotimes_k l$ is an integral domain if $A\bigotimes_k l$ is a integral domain

I am trying to solve Ex. 5.4.M in Vakil's notes (which has been discussed here) my question is different than that, I want to solve: If $A$ is a integral domain which is also $k-$algebra, and $l$ is a ...
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Atiyah-Macdonald Exercise 5.1: Doubt about detail in final step

I'm working on Atiyah-Macdonald Exercise 5.1: Let $f:A\to B$ be an integral homomorphism of rings. Show that $f^*:\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ maps closed sets to closed sets. Here $f^*$ ...
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Set of polynomials with coefficients in $\mathbb{Z}_n$ [closed]

Is it correct that any polynomial from the set of polynomials with coefficients in $\mathbb{Z}_n$ can be seen as an integer in base $n$ ? I'm asking this seemingly trivial question because I am seeing ...
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3 votes
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Embedding of a torsion free module into a free module

Let $A$ be a commutative ring with the unit element (not necessarily being an integral domain) and $M$ be a finitely generated $A$-module. The following proposition is well-known: If $A$ is an ...
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Higher Direct Images of Fiber Products

Let F, G be coherent sheaves on schemes X, Y, respectively, over $\mathbb{C}$. We have the natural maps $X \to \mathbb{C}$ and $Y \to \mathbb{C}$ and so we have the natural fiber product $X \times_{\...
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Proof that open set of a variety has the same dimension [duplicate]

I have been trying to understand this proof: So the first part I am unsure of for the proof of the first inequality: is it possible that when we take the closure of $U_i$ that two of the $X_i$ are ...
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the irreducible component of an affine algebraic set given by a nonconstant polynomial is the vanishing set of the irreducible factorization

I am reading this set of notes about dimension theory for algebraic geometry:https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c4.pdf For the proof of iv), I am just wondering ...
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5 votes
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Formal Differentiation and Evaluation of Polynomials

Let $R$ be a commutative ring and consider the ring of polynomials $R[x]$. Let $D:R[x]\to R[x]$ be the formal derivative. For any element $a\in R$ we also have an evaluation homomorphism $\varphi_a:R[...
1 vote
1 answer
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Confusion in a step of the proof of Strong Hilbert's Nullstellensatz.

I am reading the book by Fulton on Algebraic Curves.There is a very important theorem and in fact a milestone theorem which is called Hilbert's Nullstellensatz.Let for a subset $S$ of $K[x_1,x_2,...,...
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Is every modular lattice the ideals of a commutative ring?

Fix a (unital) commutative ring $R$ and let $L(R)$ denote the lattice of ideals of $R$, partially ordered by inclusion. In this answer, rschwieb notes that "the problem of representing lattices ...
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Chinese Remainder Theorem for commutative rings [closed]

Let $R$ be a commutative ring with identity, $I_1 , I_2 , \ldots I_n \subset R$ be ideals of $R$ and $y_1 , y_2 , \ldots , y_n \in R$. Consider the system of congruences below. $x \equiv y_1 \text{ ...
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polynomial $x_1^2+\cdots+x_n^2$ is irreducible for $n\ge 3$ [duplicate]

I was doing Ravi's FOAG exercise, in 5.4I (see related questionhere) there is a question ask me to show that $$\operatorname{Spec}\left(k[x_1, \ldots, x_n]/(x_1^2 + \cdots + x_m^2)\right) $$ is normal ...
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prove $K(B)\cong K(A)[z]/(z^2-a)$

Let $A$ be a UFD, and $z^2 - a$ be a irreducible polynomial in $A[z]$ such that $a$ has no repeated prime factor, let $B = A[z]/(z^2-a)$ then it's an integral domain (since $z-a^2$ prime in UFD). I ...
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Term for an integral domain $R$ with a "norm" satisfying $N(a) < N(ab)$ for all non-units $a,b \in R \setminus \{0\}$?

Is there a standard term for an integral domain $R$ under a map $N \colon R\setminus \{0\} \to \mathbb{Z}_{\ge 0}$ such that $N(a) < N(ab)$ for all non-unit elements $a, b \in R\setminus \{0\}$? I ...
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$A[[X]]$ is a regular ring. power series over regular ring is regular

Assume $A$ is a com. ring with unity. I am trying to find a more elementary proof on the following statement, than what I have read so far (see the list at the end). If $A$ is a regular ring, so is $A[...
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$R$ is a commutive ring, and $A, B, C$ are ideals in $R$. If $A + B = A + C$ and $A ∩ B = A ∩ C$, then it is necessary that $B = C$?

My question is: $R$ is a commutive ring, and $A, B, C$ are ideals in $R$. If $A + B = A + C$ and $A ∩ B = A ∩ C$, then it is necessary that $B = C$? I come up with this question because when $ R $ is ...
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Check my proof that $\operatorname{Proj}\ k[x_0,...,x_n]/I$ is a reduced projective $k$ scheme for $I$ radical

I was doing Ravi's FOAG book exercise 5.3 E which asks me to prove that : $\operatorname{Proj}\ k[x_0,...,x_n]/I$ is reduced projective $k$ scheme for $I$ is radical homogeneous ideal To show it's ...
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What is the motivation behind the definition of topological dimension and Krull dimension.

Usually in algebraic geometry,we define the dimension of a topological space as follows: $\dim(X)=\sup\{ m:\text{ there exists a descending chain of irreducible closed sets of length } m\}$ and in a ...
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The set of prime ideals containing any non-zero proper ideal in a commutative Noetherian ring is finite, by consideration of the topology on $Spec(R)$

In the notes by J.P. May on ‘Associated Prime Ideals’, for $R$ a commutative Noetherian ring and $M$ a non-zero finitely generated $R$-module, I do not understand the first statement in the proof of ...
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Transcendence degree over arbitrary point given information over generic point

This question came up while attempting to do an exercise (12.5.C) in Vakil's FOAG (Dec 31, 2022 draft)—specifically, an exercise related to the proof of the following statement (questions about a ...
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1 vote
2 answers
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Hom of tensor products and change of algebra

Motivated by the question in this post here, I'm wondering if a similar statement is true in the following sense: Let $A$ be a commutative algebra over a field $\mathbb{K}$, and let $0\not=I\subset A$...
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1 answer
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Submodules of a direct sum of ideals in a valuation ring

Let $K$ be the field of Hahn series in an indeterminate $t$ with exponents in $\mathbb{R}$, coefficients in $\mathbb{F}_2$ and valuation $\nu$. Set $$A:=\{a\in K:\nu(a)\geq 0\},$$ which has unique ...
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Hilbert Function of Ideal in Polynomial Ring

I came across the notion of Hilbert function in a book about commutative algebra. I have some trouble understanding the meaning of the definition and how to apply it to practical examples. The author ...
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Definition of presheaf, given in Basic Algebraic geometry 2 Shafarevich

I was reading Basic Algebraic geometry 2 by Shafarevich, In the definition of presheaf, $\mathscr F(\emptyset)=1$ To illustrate this definition they have given an following example: suppose first ...

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