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Questions tagged [commutative-algebra]

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

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23 views

Find an explicit Noether normalization

I am trying to solve the following exercise: Let $k$ be an infinite field and let $f \in k[x_1,...,x_n]\setminus\{0\}$. Define $A = k[x_1,...,x_n, f^{-1}]$ as subring of $k(x_1,...,x_n)$. Find a ...
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21 views

Quotient of free module is still free

Let $f\colon A\longrightarrow B$ be a finite extension of local rings and suppose $f$ flat. We know that under these assumptions (Thm. 2.16 Algebraic Geometry and Arithmetic Curves, Qing Liu ) $B$ is ...
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21 views

Show that this domain is integrally closed

The following is stated on the Wikipedia entry for integrally closed domains as an example: Let $k$ be a field of characteristic not $2$ and $S=k[x_1,...,x_n]$ a polynomial ring over it. If $f$ is ...
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24 views

Question regarding books for commutative algebra

So I was searching about books for commutative algebra. I have read most of the algebra namely galois theory and field theory and basic algebra from dummit Foote. So I was thinking about studying ...
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2answers
21 views

Contraction of quotient ideal is quotient of contractions?

Let $\mathfrak a,\mathfrak b$ be ideals in a ring $A.$ The quotient of $\mathfrak a$ and $\mathfrak b$ is $(\mathfrak a:\mathfrak b)=\{x\in A:x\mathfrak b\subseteq \mathfrak a\}$ and if $f:B\to A$ is ...
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37 views

The maximal ideals $\mathrm{Maxspec}(\Bbb Z[X])$ of $\Bbb Z[X]$

In this post, we will try to find all the maximal ideals of $\Bbb Z[X]$, that is $\mathrm{Maxspec}(\Bbb Z[X])$. Of course, there are some posts in MSE or out, but nowhere I found a complete proof. So, ...
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48 views

What is needed to arrive at Bézout's theorem in Algebraic Geometry?

I'm writing a paper that deals with Bézout's theorem, and I'd like to do something from scratch, showing everything I need to get to prove Bézout's theorem and give some examples of this theorem. My ...
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1answer
32 views

Dominant projection of affine schemes

Let $A$ be a $k$-algebra of Krull dimension $1$, where $k$ is a field. Let $n\geq 2$ a natural number and $\frak{p}$ a prime ideal of $A^{\otimes n}$ which is not maximal. Consider the ring ...
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1answer
28 views

About the concept of a valuation ring

I got a little confused by the different definitions of valuation rings while reading Atiyah's introduction to commutative algebra. Let $A$ be an integral domain and $K$ its field of fractions. We ...
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1answer
43 views

Is $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ an integral domain and is $(A/\mathfrak{q})_\mathfrak{p}$ a ring?

Let $A$ be a commutative ring with identity and $\mathfrak{q}\subset\mathfrak{p}$ two prime ideals of $A$. I am trying to determine whether $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ is an integral ...
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15 views

Quotient of units in the formal power series ring

Let $k[[x,y]]$ be the ring of formal power series in two variables over a field $k$. A unit in $k[[x,y]]$ is of the form $a_0+f$ where $f\in k[[x,y]]$ and $a_0$ is a unit in $k$. I heard that the ...
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12 views

Graded endomorphisms factoring through nilpotent ones

Let $M$ be a finitely generated 2-graded $k[x,y]$-module, concentrated in nonnegative degrees, and $f$ be a degree $(n,m)$-endomorphism of $M$. We can consider $f$ as a morphism $M\langle n,m\rangle\...
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1answer
26 views

What is the kernel of the induced map on symmetric algebras?

Let $$0\to M\to N\to P\to 0$$ be an exact sequence of $R$-modules over a commutative ring $R$.Then consider the induced map on the $i^{th}$ graded piece of the symmetric algebras: $$S^i(N)\to S^i(P).$...
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1answer
35 views

Cancelling finitely generated projective modules from a tensor product of finitely generated projective modules

Let $R$ be a commutative Noetherian ring (with unity) and $M,N,P$ be finitely generated projective modules over $R$ such that for some $n\ge 1$, we have $M\otimes_R N \cong M \otimes_R P \cong R^n$. ...
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39 views

Radical of the ideal generated by $x^3 - y^6$ and $xy-y^3$

I am following Andreas Gathmann's notes on algebraic geometry. He asks, right after he shows that $V$ and $I$ are bijection between algebraic varieties and radical ideals (so I suppose I must use ...
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41 views

A strictly henselian ring is final in the category of etale neighborhood of its closed point.

In Etale cohomology theory, by Lei Fu, Page, 195, it is stated that given a strictly Henselian ring $A$ with closed point $s$, then $\mathrm{Spec}A$ is a final object in the (opposite) category of ...
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31 views

Characterization: Automorphisms of $\mathbb{A}_{\mathbb{C}}^n$

Is there any sort of characterization(/research?) of automorphisms (of varieties) on $\mathbb{A}_{\mathbb{C}}^n$? (The question came up, as we had to find a "non-trivial" automorphism on $\mathbb{A}_{...
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1answer
26 views

Mori domain if and only if every v-ideal is of finite type?

In commutative alebra, I proved that in a Mori domain, every v-ideal(divisorial ideal) is of finite type. But converse is hard to me..I don't know it's correct. Someone help me plz
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1answer
36 views

Non zerodivisors in ideals of polynomial rings [duplicate]

The question is the following: Let $f$ be a polynomial of the ring $R[x_1, \ldots, x_n]$, with $R$ any ring, and let $\mathrm{cont}(f)$ be the ideal generated by the coefficients of $f$. Why if $\...
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1answer
23 views

Does Noetherian + finite dimenasional imply essentially of finite type?

Let $R$ be a Noetherian (commutative) algebra over a field $k$. If $\dim R<\infty$ (Krull dimension), does it follow that $R$ is essentially of finite type over $k$? (Meaning: $R=S^{-1}(k[x_1,\...
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1answer
44 views

Localization at annihilators of an ideal

I was reading this post and on line +10-11 of the proof of lemma 27.25.1, it seems to claim the following: Let $A$ be a ring, $I \subseteq A$ an ideal, and $M$ an $A$-module. Let $M_I:=\{ x \in M\...
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30 views

Equivalence relation on field extensions, explanation

I am confused with lemma 25.13.3. Especially with the equivalence. At -2 lines of lemma 25.13.3 , the author writes, Given any set of extensions $\kappa \subseteq K_i$ there exists some field ...
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1answer
46 views

coprime ideals in a ring

Suppose $R$ is a ring ($R$ may not have a unit and can be non-commutative), $I,J$ are two nonzero proper ideals in $R$ such that $I+J=R$ and $I\cap J\neq 0$. I wonder if there exists a possibility ...
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1answer
20 views

Flatness ascends up a inverse system

Let $R$ be a ring with $f \in R$ a nonzero divisor. Then, the claim is that specifying an f-adically complete and f-torsion free $R$-algebra $S$ with $R/f\to S/f$ flat is the same as specifying a ...
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1answer
31 views

Tensor product of exact complexes is exact

Let $M_\circ = \dots \to M_n \dots \to M_0 \to 0$ and $N_\circ = \dots \to N_n \dots \to N_0 \to 0$ be exact complexes of modules over a ring $A$ such that each module is flat. Is it then true that $(...
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1answer
43 views

Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?

Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module. Question: Is $M$ also a finitely generated $R$-module? I know that $M$ is an $R$-module ...
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1answer
57 views

Question about associated primes and annihilator

I am trying to solve the following exercise: Let $R$ be noetherian and $M$ a finitely genererated $R$-module. Show that $\mathrm{Ass}(R/\mathrm{Ann}(M)) \subseteq \mathrm{Ass}(M)$ and both sets ...
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1answer
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On the depths of symbolic powers of the Stanley-Reisner ideal of a bow-tie complex

Consider the polynomial ring $S=k[x_1,...,x_5]$. Consider the Stanley-Reisner ideal $I$ (i.e. the face ideal) of the simplicial complex which is a bow-tie $\Delta:=\left<x_1x_2x_3,x_3x_4x_5\right&...
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34 views

Annihilator and maximal ideal in finite ring

I have this proposition Let $R$ be a finite commutative ring with unity. If $M$ is a maximal ideal in $R$ then $\exists m\in M: M=Ann(m)$ I do not know how to give this $m$ and why the ...
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1answer
46 views

Question which is similar to sheaf property

Hartshorne, Chapter 3, Exercise 3.7(a): $A$ is Noetherian, $\mathfrak{a}$ is an ideal, $U=\operatorname{Spec} A\setminus V(\mathfrak{a})$. For any $A$-module $M$, $\Gamma(U,\tilde{M})\cong \varinjlim\...
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1answer
46 views

On covariant linear functors $T: R$-Mod $\to R$-Mod which preserves direct-limits or inverse-limits

Let $R$ be a commutative ring with unity. Let $R$-Mod denote the category of $R$-modules, and $Ab$ denote the category of Abelian groups. Now, it is known that a covariant additive functor $T: R$-...
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25 views

$\text{Max Spec}\:A$ is Hausdorff for an algebra $A$ of finite type over a field.

Let $\text{Max Spec}\:A$ be the subspace of maximal ideals of $\text{Spec} \:A $ for the $F$-algebra $A$ that is of finite type over a finite field $F$. Show that $\text{Max Spec}\:A$ is Hausdorff. I ...
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33 views

Special case of isomorphism $S\otimes_R \mathrm{Hom}_R(M,N) \simeq \mathrm{Hom}_S(S\otimes_R M, S\otimes_R N)$

$R$ is commutative ring, $S$ is commutative $R$-algebra and $M$ is $R$-module. So we have the S-module isomorphism $$S\otimes_R \mathrm{Hom}_R(M,N) \simeq \mathrm{Hom}_S(S\otimes_R M, S\otimes_R N)$$ ...
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1answer
27 views

Element of $T(R)$ that projects to element of $R/P$ for every minimal prime $P$ of $R$?

Let $R$ be a reduced ring and $T(R)$ the total ring of fractions of $R$ (i.e. localizing $R$ at nonzerodivisors). Any element of $T(R)$ maps naturally to an element of $T(R/P)$ since $a/b \in T(R)$ ...
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1answer
26 views

Does every covariant, additive, faithfully-exact functor $T:R$-Mod $\to Ab$ preserve either direct sum or direct product?

Let $R$ be a commutative Noetherian ring. Let $Ab$ denote the category of abelian groups. Let $T:R$-Mod $\to Ab$ be a covariant, additive functor such that for any sequence of $R$-modules, $A \...
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28 views

On faithfully flat and faithfully projective modules

Let $R$ be a commutative Noetherian ring. Let $P,Q$ be some $R$-modules such that $-\otimes_R P $ and $ Hom_R(Q,-) $ are faithfully exact functors i.e., for any sequence of modules $A \xrightarrow{f}...
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1answer
32 views

$A$ algebraic over $B$, $B$ algebraic over $C$. $A$ algebraic over $C$?

Let $A/B/C$ be field extensions, with $A$ algebraic over $B$, $B$ algebraic over $C$. Must $A$ be algebraic over $C$? I think the answer is yes, but I don't know how to prove it.
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show $\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain

I've got to show that $A:=\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain I know that $A$ is isomorphic to $\mathbb{C}[t,t^{-1}]$ and that this a subfield of $\mathbb{C}[t]$ which is a PID. So ...
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1answer
64 views

Find the algebraic dimension of space of matrices

From A. Gathmann's Algebraic Geometry (2014): Exercise 2.31. Let $X$ be the set of all $2 \times 3$ matrices over a field $K$ that have rank at most $1$, considered as a subset of $\mathbb A^6 = \...
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36 views

construction of Cohen Macaulay graph

The graph can be seen in the image Let H be a graph with vertex set V(H)={x_1, x_2,...,x_k, z, w} and J is its edge ideal. Assume that z is adjacent to with deg(z) greater or equal to 2 and deg(w)=1. ...
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61 views

References on Kähler Differentials

I'm working on an independent study project on Kähler differentials for my commutative algebra class. I'm looking for any references on these that might help me out. Any references to their use in ...
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49 views

An example of an argument using generic points to prove a “closed” condition

Every irreducible affine scheme $\mathrm{Spec}(R)$ contains a generic point, namely $\eta:=\mathrm{Nil}(R)$. If $R$ is a domain then $\eta=(0)$. This is a point which is Zariski dense in $\mathrm{Spec}...
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1answer
47 views

Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
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1answer
40 views

Is this a maximal ideal of the ring of formal power series?

Let $ k $ be an algebraically closed field, and $ k[[T]] $ be the ring of formal power series in variables $ (T_{1},\dots,T_{n}) = T. $ Let $ \mathfrak{m}^{l} $ be the ideal of $ k[[T]] $ consisting ...
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1answer
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Support of localization of a module at a minimal prime over the support of the original module

If $M$ is a non-zero module over a commutative ring $R$ (not necessarily Noetherian), and $P$ is a minimal prime in $\mathrm{Supp}(M)$, then is it true that $\mathrm{Supp}(M_P)=\{PR_P\}$ (where $M_P$ ...
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1answer
53 views

The form of prime ideals in $S^{-1}R$

Theorem. Let $R$ be a commutative ring with $1_R$, $P\in \mathrm{Spec}(R)$ a prime ideal of $R$, $S\subseteq R$ an multiplicative subset of $R$ and $\nu:R\longrightarrow S^{-1}R,\ a \longmapsto \...
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37 views

Given $\phi \in \mathrm{End}(M)$, when does $\phi$ injective imply $\phi^*$ surjective and $\phi^*$ injective imply $\phi$ surjective?

Let $M$ be a module over a commutative ring (with unity) $R$. Let $\phi : M \to M$ be an $R$-module homomorphism. Then we have a dual map $\phi^* : M^* \to M^*$ given by $\phi^*(f)=f\circ \phi, \...
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1answer
33 views

The generating set of ideals in regular local ring

Let $R$ be a regular local ring. It is well-known that $R$ is a Cohen-Macaulay ring. Hence $grade(I,R)=ht(I)$ if $I$ is an ideal. If $I$ is a proper ideal, suppose $ht(I)=d$, is there exists $x_1,.....
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1answer
31 views

Equivalence for rings with localization property

I feel like I need a hint for the following exercise: Let $R$ be some commutative unitary ring. If $M$ is a $R$-module, let $M[f^{-1}]$ denote the localization of $M$ with respect to the set $\{ f^...
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0answers
64 views

Given a split exact sequence $0 \to N \to M \to M \to 0$, when can we say $N=0$?

Let $M$ be a module over a commutative ring $R$. Let $N$ be a submodule of $M$ such that there is a split exact sequence $0 \to N \to M \to M \to 0$. So, in particular, $M \cong M \oplus N$. Under ...