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Questions tagged [commutative-algebra]

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

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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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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-Resiner ideal $I$ (i.e. the face ideal https://en.wikipedia.org/wiki/Stanley%E2%80%93Reisner_ring ) of the Simplicial Complex ...
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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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28 views

Question which is similar to sheaf property

In Hartshorne chapter 3, exe3.7(a): $A$ is Noetherian, $\mathfrak{a}$ is an ideal, $U=\operatorname{Spec} A-V(\mathfrak{a})$. For any $A$-module M, $\Gamma(U,\tilde{M})\cong \varinjlim\operatorname{...
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1answer
31 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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$\text{Max Spec}\:A$ 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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30 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
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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
25 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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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
58 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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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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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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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
37 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
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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
51 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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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
29 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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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 ...
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1answer
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Finitely generated projective resolution of a module over a regular local ring

Let $R$ be a regular local ring and let $M$ be an $R$-module. Then there exists a finite projective resolution $P_\bullet\to M\to 0$. However, need there exist a finite projective resolution ...
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1answer
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How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules?

Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the ...
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Torsion-less module over commutative ring whose injective hull is Hopfian

Let $M$ be a module over a commutative ring (with unity) $R$. Let $E_R(M)$ denote the injective Hull of $M$ . If $M$ is torsion-less (i.e. $\cap_{f\in M^*=Hom_R(M,R)} \ker f=(0)$ ) and $E_R(M)$ is ...
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1answer
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Elements of the spectrum of complex numbers

I recently learned that the elements in the spectrum of $\mathbb{C}[x]$ are in the form $x-a$. I understand that a spectrum consists of all prime ideals of a ring, but I'm a little confused as to why ...
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2answers
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Reduced and integral rings

Let $R$ be a commutative ring with unit. Are the following true? If $\operatorname{Spec}(R)$ is irreducible i.e, cannot be written as union of two proper closed subsets, and $R$ is reduced, then $R$ ...
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A trivialization of line bundle is same as nonvanishing section

I am reading Lemma 17.22.10. My fundamental confusion is how is Nakayama Lemma applied. Claim: Let $X$ be a ringed space. Assume that each $O_{X,x}$ is a local ring with maximal ideal $m_x$. Let $ ...
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Are the normalizations of $k[X,Y] / (Y^3 - X^5)$ and $k[X,Y] / (Y^5 - X^{19})$ just $k[y/x]$?

In Reid's Undergraduate Commutative Algebra he shows that the normalization of $A = k[X,Y]/(Y^2 - X^3)$ is $k[t]$, where $t = y/x$, because the field of fractions of $A$ is $k(t)$, $t \in k(t)$ is ...
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1answer
35 views

Map induced on fraction fields is finite

Let $\phi:R\rightarrow R'$ be an injective, finite map of integral domains. Is it true that induced map $\phi_1:\operatorname{Frac}R \rightarrow \operatorname{Frac}R'$ is also finite? Note: Finite ...
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Free resolutions in $\mathbb{R}[x,y,z]$ of ideals generated by 4 elements

working in $\mathbb{R}[x,y,z]$ I would like to know free resolutions of ideals generated by 4 elements. If $I$ is generated by 3 elements and these do not have a common zero then they are a regular ...
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On Hopfian modules over commutative Noetherian rings

Let $R$ be a commutative Noetherian ring with unity. Let us call an $R$-module $M$ to be Hopfian if every surjective endomorphism $M \to M $ is injective. 1) If $M_1$ and $M_2$ are Hopfian modules, ...
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Map between localizations induces map on underlying modules for Zariski covering

While working through a proof of this paper,1 at the middle of page 45, the author's claim of a short exact sequence seems to amount to the following problem: Let $A$ be a commutative ring and let $...
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Equivalent conditions under Nakayama Lemma

I am sorry for the vague title and sorry if the question is trivial but I am new to this kind of mathematics. In the book "Introduction to Singularity Theory and Deformations" I found a condition for ...
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Why can't we use different approach on the last problem of Exercise 2.11 of Atiyah-Macdonal?

I already read solutions for injectivity at MathOverflow and MathStackexchange and understand some of the solution. (Those are brilliant!) What I didn't understand is the reason we have to find ...
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1answer
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Proof explanation: Every module is the quotient of free module

Let $M$ be a left $R-$ module, it is said (here for example) that $M \approx F(M)/\sim$ The proof is apparently $F(M) \stackrel{\pi}\to M$ where $(0,\dots,1_m,\dots0) \stackrel{\pi}\to m$ and appeal ...
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Affine $k$-domain of dimension$1$ can always be embedded in a polynomial ring?

Let $k$ be an algebraically closed field. Let $R$ be a UFD which is a finitely generated$k$-algebra. If $\dim R=1$, then is it true that there exists an injective $k$-algebra homomorphism from $R$ ...
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1answer
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Proof about length of quotient modules

Let $M$ be an $R$-module and let $x, y \in R$ such that $y$ is not a zero divisor of $M$ and $M / xyM$ has finite length. Show that $l(M/xyM)=l(M/xM)+l(M/yM)$. In the above, $l$ denotes the lenght ...
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1answer
60 views

Is the Evaluation map of an R-Module of rank 1 and hom injective

This is in context of a larger problem of showing that the dual of an invertible sheaf is invertible on a scheme. I want to show that given a free R-module A of rank 1, the standard evaluation map ...
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Proving that $\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a complete intersection ring

I have found in some sources that the ring $R=\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a local complete intersection ring. I need this result in order to apply a related theorem and I ...
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Flat $\mathbb{Z}$-modules and algebraic independence

I know that this question is naive, but I really want to find out if this observation is correct. First, let $\alpha_1, \alpha_2, \dotsc, \alpha_k$ be real numbers. Consider the $\mathbb{Z}$-module ...
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Ideal generated by its uniformiser

Given $A$ a Dedekind ring, with $X=\operatorname{Spec}(A)$. I want to understant why $\operatorname{Cl}(X)=\operatorname{Cl}(A)$ (Hartshorne page 132 example 6.3.1). For that I need to verify that ...
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1answer
34 views

Finitely generated projective modules, over commutative Noetherian ring, of constant rank $1$, if stably isomorphic, then isomorphic?

Let $M,N$ be finitely generated projective modules over a commutative Noetherian ring $R$ such that $M_P \cong N_P \cong R_P, \forall P \in Spec(R)$. If $\exists n\ge 1$ such that $M \oplus R^n \...
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1answer
30 views

if $A$ is a finite dimensional commutative $\mathbb{C}$ algebra with dimension $n$, must $A$ have $n$ simple modules up to isomorphism?

I'm trying to prove the question above. I'm not sure whether this is true or not, but I'm trying to figure it out. If $A$ is semi-simple I don't think it's too hard to see that it's true, but ...
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35 views

Blow up at point is finite?

Let $X$ be an affine algebraic curve with $0 \in X$ and $\tilde{X}$ the strict transform of $X$ w.r.t the blowup of $X$ at $0$. How to prove that $\pi \colon \tilde{X} \to X$ is finite? Is it even ...
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0answers
17 views

depth $R_{\mathfrak p}=0$ implies $\mathfrak pR_{\mathfrak p}$ is associated to zero

Let $R$ be a commutative Noetherian ring with unity. I want to prove the fact that depth $R_{\mathfrak p}=0$ implies $\mathfrak pR_{\mathfrak p}$ is associated to zero. Since depth $R_{\mathfrak p}=...
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32 views

How much commutative algebra needed for studying Hartshorne's algebraic geometry book? [duplicate]

I am studying commutative algebra from Atiyah's ' Introduction to Commutative Algebra' book but I am interested for algebraic geometry. My question is how much commutative algebra is needed for ...
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1answer
22 views

tensor product of modules over algebra and isomorphism

I have always wondered this statement is true. Let $S$ be commutative algebra over $R$ and $M,N$ be modules over $S$. Then, $M\otimes_R N\simeq M\otimes_S N$ as $S$-modules. (Or can be as $R$-...