Questions tagged [commutative-algebra]

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

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40 views

$p$-adic and mod $p$ relation on tensor

Let $G$ an abelian group. $\Bbb Z_{(p)} $ is the $p$-adic integers. What can one say about $\Bbb Z/p \otimes G$ given that $\Bbb Z_{(p)} \otimes G \not= 0$? Is it possible to conclude whether the ...
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1answer
66 views

Is the ideal $x^3-y^5 \subseteq \mathbb{C}[x,y]$ prime? Is it maximal? [closed]

I have some idea like I is a maximal ideal of a commutative ring R iff R/I is a field. but not able to formulate for this case. first, I thought about the irreducibility of ideal $x^3-y^5 \subseteq \...
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1answer
49 views

Base change of injective ring homomorphism

Let $f:A \to B$ be an injective ring homomorphism of $\mathbb{C}$-algebras and $A, B$ are integral domains. Suppose that $B$ is the integral closure of the ring $A$ (in the fraction field of $A$) i.e.,...
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57 views

Find the integral closure of a ring

Let $k$ be a field, let $A=k[x,y]/(y^{2n}+y^{2n+2}-x^{2n+2})$ with fraction field $K$. Let $u=x^{n+1}/y^{n}, v=y/x$, then $x=uv^n, y=uv^{n+1}$. It's then easy to see that $1-u^2+u^2v^{2n+2}=0$. Let $A'...
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1answer
37 views

On an isomorphism of rings

Let $k$ be a field, it's claimed in algebraic geometry textbook that $k[v^2, v^3]\cong k[t, u]/(t^2-u^3)$ via $v^2\mapsto u, v^3\mapsto t$. But I can't show it's well-defined, since an integer can ...
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2answers
94 views

Localization commutes with Hom for finitely presented modules

I am trying to solve an exercise given in Vakil's Algebraic Geometry notes. Suppose $M$ is a finitely presented $A$-module. The $M$ fits inside an exact sequence $A^q\rightarrow A^p\rightarrow M\...
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70 views

A problem on Jacobian matrices

Let $k$ be a field of characteristic $0$. Let $f_1, \dots, f_m \in k[x_1, \dots, x_n]$. Are $f_1, \dots , f_m$ algebraically independent over $k$ if and only if the rank of the Jacobian matrix $(\frac{...
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1answer
44 views

Proof of Quillen's patching theorem

The following are from Lam's book Serre's problem on projective modules, on page 163 and 164. For any ring $A$, the notation $m \in \mathscr{R}^A(A[t_1, \dots, t_n])$ means that there exists a $A-$ ...
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1answer
63 views

Showing surjectivity of the trace map $B\to A$ from faithful flatness

I've come across the claim if $A\hookrightarrow B$ is a finite etale extension of rings (commutative w/ $1$) with $A$ Noetherian then the trace map $\operatorname{Tr}_{B/A}:B\to A$ is surjective and ...
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1answer
33 views

Computing singular points of curves, exercise 5.1 (Hartshorne)

I am just trying to cross check my answer as it slightly differs from https://math.berkeley.edu/~reb/courses/256A/1.5.pdf to be sure of any mistake I am making. Here $k$ is an algebraically closed ...
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1answer
49 views

Exercise on finitely generated $A$-modules

Here is the exercise I'm trying to solve: Let $M$ be a finitely generated $A$-module (where $A$ is a commutative ring) and let $g:M\rightarrow A^n$ a surjective $A$-module morphism. Prove that $\text{...
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2answers
51 views

Constructing the inverse of a surjective homomorphism $g\otimes \operatorname{id}\colon B\otimes G \to C\otimes G$

Given an exact sequence of group homomorphisms on abelian groups $$A\xrightarrow{f} B \xrightarrow{g} C\to 0$$ I want to prove that the induced sequence $$A\otimes G \xrightarrow{f\otimes \...
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1answer
47 views

Completion of a polynomial ring over a complete ring

I'm learning about ring completions, and this question came to mind: If $R$ is a complete local ring with maximal ideal $\mathfrak{m}$ (e.g. $R = \mathbb{Z}_p$ or $R = k[[x]]$), is the completion of $...
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1answer
99 views

Is $\mathbb{C}[x,y]/(x^3+y^3−1)$ is a UFD or not?

I'm wondering if $\mathbb{C}[x,y]/(x^3+y^3−1)$ is a UFD or not. I know that a Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal, and I know that the krull ...
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1answer
41 views

Let $A$ be a torsion abelian group. Then $A$ has no $p$-torsion iff $A \otimes \Bbb Z_{(p)} = 0$.

Let $A$ be a torsion abelian group, $p$ a prime. Then $A$ has no $p$-torsion iff $A \otimes \Bbb Z_{(p)} = 0$. I could prove one directioin $\Rightarrow$. But how does one prove the converse?
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2answers
66 views

Dimension of union of two varieties

Suppose $X$ and $Y$ are two varieties. By varieties, I mean affine varieties or quasi-affine varieties or projective varieties or quasi-projective varieties. Suppose Krull dimension of $X$ is $n$ and ...
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0answers
50 views

On transfer and base change map on Grothendieck groups induced from injective ring homomorphism

For a commutative Noetherian ring $R$, let $G_0(R)$ denote the Grothendieck group (an abelian group) of the abelian category of finitely generated $R$-modules (Note that I'm Not talking about $K_0(R)$ ...
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1answer
79 views

A question to the exercise I.5.4(c) Hartshorne

Much before Bezout's theorem the following exercise is given: If $Y$ is a curve of degree $d$ in ${\mathbb{P}}^2$ and if $L$ is a line in ${\mathbb{P}}^2$, $L \neq Y$, show that $(L \cdot Y) = d$. The ...
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24 views

Determining the subfield of rational function field with a given subset

Let $n$ to be a positive integer. Let $L$ to be the rational function field $\mathbb{C}(X_1,...,X_n)$ of $n$ variables over $\mathbb{C}$, and $S\subset L$ to be a given finite subset of $L$. Consider ...
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42 views

field of fractions of ring of fractions = ring of fractions of field of fractions?

Suppose $A$ is an integral domain and $K$ its field of fractions. Let $S$ be a multiplicatively closed subset of $A$. I was wondering if $S^{-1}K$ is the field of fractions of $S^{-1}A$. I assume yes ...
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1answer
41 views

Dimension of product of affine varieties is the sum of dimensions of each variety

How do I prove that the dimension of the product of affine varieties is the sum of dimensions of each affine variety? I am aware that similar questions had been asked in Dimension of product of affine ...
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3answers
82 views

Why is $\operatorname{Hom}_{\mathbb{Z}}(-,\mathbb{Q})$ right exact functor?

I'm looking for an example of a ring $R$ and an $R$-module $M$ such that the functor $\operatorname{Hom}_R(-,M)$ is exact. I've seen that this is equivalent to saying that $M$ is an injective module, ...
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4answers
521 views

Why do we need prime ideals in the spectrum of a ring?

I'm reading Atiyah Macdonald, where they introduce in the exercises of chapter one a topological space $\operatorname{Spec}(A)$ associated to a ring $A$, which is defined as $\operatorname{Spec}(A) \...
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22 views

Exceptional Set vs Tangent Cone

Let $k$ be an algebraically closed field, $R=k[x_1,\ldots,x_r]/J$ for some ideal $J$, $X=Z(J)\subseteq\mathbb{A}^r$, and $I=(x_1,\ldots,x_r)$. I'm following Eisenbud's Commutative Algebra with a View ...
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46 views

Inverse Image of Ideal under morphism is Ideal extension

Let $\varphi: X \rightarrow Y$ be a morphism of affine varieties. Then for any subvariety $Z \subseteq Y$ with ideal $\mathcal{I}(Z) = J \subseteq K[Y]$ the equation $\mathcal{V} (J \cdot K[X]) = \...
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1answer
115 views

How can I show that the inclusion $\mathbb{C}[x^2, xy, y^2] \to \mathbb{C}[x, y]$ is not flat?

I want to show that $\mathbb{C}[s, t, u] \to \mathbb{C}[x, y] : s \mapsto x^2, t \mapsto xy, u \mapsto y^2$ is not flat. If $s \otimes y \neq t \otimes x$ in $(x^2, xy, y^2) \otimes_{\mathbb{C}[s, t, ...
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1answer
63 views

Classifying all commutative $\mathbb{R}$-algebras of matrices over $\mathbb{R}$?

I initially thought they were all isomorphic to some subring of the $n \times n$ diagonal matrices $\mathcal{D} \cong \mathbb{R} \times \dots \times \mathbb{R}$, but this was wrong: Every commutative ...
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1answer
40 views

quotient ideal & primary decomposition

A 'quotient ideal' associated to a pair of ideals $\frak{a}, \frak{b} $ $\subset R$ of a commutative ring with $1_R$ is a new ideal defined as $(\frak{a}:\frak{b})$ $= \{r \in R \mid r\frak{b} \...
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0answers
67 views

When are quotients of the polynomial ring local?

Let $\mathbb{Z}[x_1,...,x_n]$ denote the polynomial ring in $n$-variables with coefficients in $\mathbb{Z}$. This ring is not local, because (for example) any ideal of the form $(x_1 - a_1,...,x_n-a_n)...
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2answers
41 views

Check Noetherian-Artinian for $\mathbb{Q}[x]$ as a $\mathbb{Q}$-module and $\mathbb{Q}[x]$-module

I have to check if these modules are Artinian or/and Noetherian. $\mathbb{Q}[x]$ as a $\mathbb{Q}$-module $\mathbb{Q}[x]$ as a $\mathbb{Q}[x]$-module For the second one I know that $\mathbb{Q}$ is a ...
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1answer
109 views

Equicharacteristic Noetherian local domain of embedding dimension $3$ and krull dimension $2$ [closed]

Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $2$ and assume $\mu(\mathfrak m)=3$. Also assume $R$ contains a field. Then, is it true that $R$ is Cohen-Macaulay ? I can prove this ...
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2answers
69 views

If $R$ is a reduced Noetherian ring, then every prime ideal in the total quotient ring $K(R)$ is maximal.

I know that in $K(R)$, the set of maximal ideals is the set of associated primes of $K(R)$ and that an ideal is maximal if and only if it is the localization of a maximal associated prime of $R$. So, ...
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1answer
47 views

Kernel of map of Kahler differentials

This is lemma 10.130.6, stacks project I understand those objects describe lie in the middle. I have trouble understanding how this "diagram" chase is done. Especially when we are dealing ...
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1answer
49 views

Which of the following are Dedekind domains? [closed]

Could you please advise me which of the following is a Dedekind domain, which not and why? $a) \; \mathbb Z[1/3]$ $b) \; \mathbb Z[\sqrt{-5}]$ $c) \; \mathbb Z[x]$ $d) \; \mathbb C[x,y]/(y^2 - x^3 +...
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1answer
42 views

Integrality and normality of ideals

Consider a ring $R$, an idela $I$. An element$z\in R$ is integral over $I$ if $z$ satisfies the equation $$z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n=0,$$ where $a_i\in I^i$. We define the integral closure $\...
2
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1answer
39 views

Standard etale maps are etale

We define a finitely presented $R$-algebra $A$ to be etale if for every $R$-algebra $S$, every ideal $I\subset S$ such that $I^2=0$ and every homomorphism $$\phi: A\to S/I$$ there exists a unique ...
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36 views

Minimal Prime Ideal is an Associated Prime

So given a Noetherian Ring $R$, an ideal $I \subseteq R$, I want to show that if $J \supseteq I$ is a minimal prime ideal of $I$, then $J \in \operatorname{Ass}(R/I)$. I have managed to prove two ...
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1answer
36 views

A complete DVR $A$ is a $\Bbb Z_p$ module, Serre's local field

I am trouble understanding how one obtains a $\Bbb Z_p$ action in the last line in this statement in pg. 36 of Serre's Local fields In particular Observe that $\Bbb Z$ injects into $A$ and by ...
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65 views

The complete regular ring in Cohen's structure theorem can be chosen to have dimension equal to the embedding dimension of the starting ring?

Let $(R, \mathfrak m)$ be a Noetherian complete local ring. Then by Cohen structure theorem, we have that $R$ is a homomorphic image of a complete regular local ring $(S, \mathfrak n)$ (https://stacks....
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0answers
18 views

Bound on the number of generators of monomial ideals

I'm studying integral closure of monomial ideals, and I learnt that if $I$ monomial ideal of $K[x_1,\ldots,x_n]$ is generated by monomials of degree at most $k$, then $\overline{I}$ is generated by ...
4
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2answers
49 views

If $I\lhd A$ and $P$ is a prime ideal in $I$, prove $P\lhd A$.

Q: Let $I$ be an ideal in $A$. If $P$ is a prime ideal in $I$, prove that $P$ is an ideal in $A$. First of all, if $p \in P$ and $q \in P$, then $p - q \in P$, because $P$ is an ideal in $I$. So, i ...
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1answer
28 views

If $\{N_i\}_{i\in I}$ is a totally ordered set of prime submodules then ${\bigcup}_{i\in I} N_i$ is a prime submodule

In a finitely generated $R$-module $M$, let $\{N_i\}, i\in I$ be a totally ordered prime submodules family. I have tried that definition $\Rightarrow$(There is a prime submodule that includes all ...
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1answer
28 views

Easy question about codimension of points of a variety

Let's say $X$ and $Y$ are varieties over $\mathbb{C}$. Suppose there is birational morphism $\pi: Y \rightarrow X$. In particular this is dominant. For example's sake let's say this is the blowup of ...
4
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1answer
95 views

$R$ Gorenstein implies $\operatorname{Proj}(R)$ Gorenstein

Let $k$ be a field and let R be a Gorenstein $k$-algebra which has a non-negative grading $R=\oplus_{k\geq 0} R_k.$ Assume further that $R_0=k$ and that $R$ is generated in degree one. I've seen it ...
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0answers
29 views

Tensor Product of Fractional Ideals

Reading Neukirch's book on algebraic number theory and on chapter III he calls the class group by the name of "Picard Group". This group, for me, had to do with invertible ideals modulo ...
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1answer
36 views

How to prove this sufficient condition for when a monomial ideal is primary.

This answer does a good job at explaining that if $I$ is primary monomial ideal in $k[x_1, \dots, x_n]$, then $I = (x_{i_1}^{a_1}, \ldots, x_{i_m}^{a_m}, m_1, \ldots, m_k)$ where $m_1, \ldots, m_k$ ...
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1answer
54 views

When does a morphism of varieties preserve codimension of points?

The context of this question is my attempt to generalise the Hurwitz theorem to varieties that aren't necessarily curves. In the case of curves, we have that a closed point always maps to a closed ...
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1answer
54 views

What is wrong with this proof that every ideal whose radical is prime is a primary ideal?

In Dummit & Foote, the definition of primary ideal says: A proper ideal of a commutative ring is called primary if whenever $ab \in Q$ and $a \notin Q$, then $b \in {\rm rad}(Q)$. Suppose $I$ is ...
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1answer
51 views

$M_x$ is free $\Rightarrow \widetilde{M}$ is locally free at $x$ [duplicate]

Let $X=\text{Spec}(A)$ where $A$ is noetherian. Suppose $M$ is a finitelly generated $A$-module and that $M_x$ is a free $A_x$-module with finite rank for some $x\in X$. Show that there exists an open ...
2
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1answer
28 views

If $S$ is simple a module over a ring $R$ which is noetherian, hereditary and every simple module is injective, then $S$ is finitely presentated.

Let $R$ be a left noetherian and left hereditary ring , also suppose every simple left module $M$ over $R$ is injective. Prove that a simple left module $S$ over $R$ is finitely presentated. So Im ...