Questions tagged [commutative-algebra]
Questions about commutative rings, their ideals, and their modules.
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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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0answers
19 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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29 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 $$S\otimes_R \mathrm{Hom}_R(M,N) \simeq \mathrm{Hom}_S(S\otimes_R M, S\otimes_R N)$$ ...
2
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1answer
24 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 $a/b \in T(R)$ ...
1
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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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0answers
24 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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0answers
93 views
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 ...
2
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1answer
47 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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31 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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0answers
53 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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39 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
33 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
39 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
11 views
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
50 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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0answers
35 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
27 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
12 views
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 ...
2
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1answer
30 views
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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0answers
22 views
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
26 views
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
40 views
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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31 views
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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20 views
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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19 views
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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1answer
53 views
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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34 views
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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42 views
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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0answers
55 views
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
44 views
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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0answers
41 views
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
26 views
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
59 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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24 views
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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0answers
97 views
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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0answers
23 views
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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0answers
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 $R$-...
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1answer
22 views
The elements of the extended ideal $I^e\trianglelefteq S^{-1}R$
Let $R$ be a commutative ring with $1_R$, $S\subseteq R$ a multiplicative set, $I\trianglelefteq R$ and ideal of $R$ and $\nu:R\longrightarrow S^{-1}R,\ a\longmapsto \nu(a):=\frac{a}{1_R}$ the natural ...
0
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1answer
29 views
Valuation theory
what is the the natural, non-trivial valuation $\mathbb{v}$ in a field F? This term appeared in some articles about $\mathbb{R}$-places that I was studying. I do have some small background in ...
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0answers
13 views
Localization of Smooth Functions on R
Consider the ring of functions $C^{k}(\mathbb{R}^d;\mathbb{R}^d)$ from $\mathbb{R}^d$ to itself with $k$-continuous derivatives. Is the localization of $C^{k}(\mathbb{R}^d;\mathbb{R}^d)$ at the ideal ...
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1answer
33 views
Galois group of residue field over $\mathbb Q$
Assume $L$ is a number field and is Galois over $\mathbb Q$,$\mathcal O_L$ is algebraic integer in $L$,$k_L=\mathcal O_L/p\mathcal O_L$ is the residue field,for a prime ideal $(p) \subset \mathbb Z$ ...
