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

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2
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18 views

Ramification group - do you know/can produce a simple proof to this?

Let $(K,v)$ be a valuation field, $L$ a finite extension of $K$, and $w$ a valuation of $L$ above $v$. The ramification group of $w$ in $L$ is the subgroup of ${\rm Gal}(L/K)$ of all automorphisms ...
2
votes
1answer
20 views

Understanding the $\mathfrak{a}$ adic completion of a $A$-module as a functor

$\require{AMScd}$ I recently read the chapter 10 on Completions in Aityah-MacDonald. They describe the $\mathfrak{a}$ completion $\hat{M}$ of an $A$-module $M$ as the inverse limit of an inverse ...
-1
votes
1answer
19 views

On the minimal set of generators of ideals in $\mathbb{C}[x,y]$.

I am trying to do exercise 2.6 of Hassett's "Introduction to algebraic geometry": i) Give an example of a monomial ideal $I\subseteq\mathbb{C}[x,y]$ with a minimal set of generators consisting of ...
2
votes
1answer
37 views

relation between vector space and torsion free module

Can you please help me to prove this. Let $R$ be a domain, $A$ be an $R$-module, and $Q=Frac(R)$. Then a module $A$ is a vector space over $Q$ if and only if it is torsion-free and divisible. Thanks ...
0
votes
0answers
15 views

Submodules $H$ satisfying: “if $ax \in H$ for some non-zero scalar $a$, then $x \in H$.”

Suppose $R$ is a commutative ring and that $X$ is an $R$-module. Question. Is there a term for those $R$-submodules $H$ of $X$ satisfying the following? For all $x \in X$, if $ax \in H$ ...
0
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0answers
24 views

tensor of two vector space

I don't know how to show this problem please help me. let $R$ be a domain and $Q=Frac(R)$ if either $C$ or $A$ is a vector space over $Q$,prove that both $C\otimes_RA$ and $Hom_R(C,A)$ are also vector ...
1
vote
1answer
19 views

Maximal ideals of finite algebra over a local ring

Let $R$ be a local ring with residue field $k$. Let $A$ be an $R$-algebra which is finitely generated as $R$-module. I want to show that the maximal ideals of $A$ are in one-to-one correspondence ...
1
vote
2answers
36 views

Prove Kähler Differential is always surjective using universal property.

Let $A$ be an $R$-Algebra. An $R$-linear derivation $d \colon A \to \Omega_{A/R}$ is called universal derivation or Kähler differential if for every $R$-linear derivation $D \colon A \to M$ there is a ...
1
vote
0answers
36 views

Algebraic closedness in residue field [on hold]

If $A\subseteq B$ are affine domains over an algebraically closed field of $k$ of characteristic zero such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
0
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0answers
37 views

Tangent Cone of a Complete Intersection

Can you give me an example of an affine variety $X \subseteq \mathbb{A}^n_{\mathbb{C}}$ over the complex numbers which is a complete intersection such that the reduced tangent cone at some point $p ...
0
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0answers
24 views

Relation between two definitions of primary modules

Let $A$ be a commutative ring, $M$ be an $A$-module and $N \leq M$. There are two definitions of primary modules: 1) $M/N$ is coprimary (i.e., every zero divisor is nilpotent); 2) $\text{Ann}_A(N)$ ...
9
votes
3answers
62 views

Definition of $\mathfrak{m}$-adic completion.

Let $V$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}$ and let $T$ be a prime element of $V$. Assume that we have a subfield $k\subseteq V$ such that the induced map $ k \to ...
4
votes
2answers
42 views

Is $m$ a projective $A$-module?

$A$ is a Noetherian local ring and $m$ be its maximal ideal. Then is $m$ a projective $A$-module? I got this problem while solving another problem. Can anyone please help me to figure it out?
1
vote
2answers
73 views

Cohen-Macaulay rings and Normal rings

is there an example that R is Cohen-Macaulay but not normal ring? what about the converse example?
3
votes
1answer
39 views

Prove that the following is a non zero tensor.

I'm asked to prove that the ideal $I=(x,y)$ in $R=k[x,y]$ is not a flat R-module. My approach was to use the exact sequence $$0\rightarrow I \to R \to R/I \to 0$$ to induce a non injective map ...
0
votes
2answers
39 views

Jacobian of n linearly independent forms in n variables

Let $k$ be a field of characteristic zero and let $f_1, \ldots, f_n \in k[x_1, \ldots, x_n]_d$ be linearly independent forms of degree $d$ in $n$ variables. Is there a nice algebraic argument for ...
1
vote
1answer
25 views

Is torsion-free equivalent to free for non-finitely generated modules over a PID?

Maybe this is a trivial question. If $A$ is a PID and $M$ is a finitely generated $A$-module, it's well known that $M$ is torsion-free iff $M$ is free. However, if $M$ is not finitely generated, does ...
2
votes
2answers
54 views

Why is it called the category of representations?

Let $A$ be a (Hopf) algebra. Let $C_A$ be a category whose objects are $A$-modules and whose morphisms are $A$-linear maps. This category is called "the category of representations". My question is: ...
0
votes
2answers
40 views

How do I find an isomorphism between varieties

Our book defines an isomorphism between varieties when there exist two maps say $\phi: V \rightarrow W$ and $\psi: W \rightarrow V$ both morphisms and $\psi \circ \phi =id_V$ and $\phi \circ \psi =id ...
1
vote
1answer
13 views

Representation matrix for modules map

Here $S=\mathbb Q[x,y]$, and we define $\oplus Se_i$ to be a $S$-free module with basis $\{e_1,e_2,e_3\}$. Define a map from $\oplus Se_i$ to $S$ by $e_1\to x^2$, $e_2\to xy+y^2$, $e_3\to y^3$. Is the ...
0
votes
1answer
20 views

Showing $\hat{A} \otimes_{A} M \cong \hat{M}$ when $M$ is a finitely generated free $A$-module.

I had a reading question on Proposition 10.13 from Atiyah-MacDonald. The proposition is the following PROPOSITION. For any ring $A$, if $M$ is finitely-generated, $\hat{A} \otimes_{A} M \rightarrow ...
0
votes
1answer
68 views

Short exact sequence of modules over a Noetherian local ring of depth $1$.

I am reading an article in algebraic geometry and am having trouble understanding a particular point that reduces to a problem in commutative algebra. I'm not familiar with the concepts involved so am ...
1
vote
0answers
91 views

How to prove that an ideal can not be generated by 2 elements

In Kunz's "Introduction to commutative algebra and algebraic geometry", page 137-139, particular monomial affine curves are described. Here is the link. In case the curve is not an ideal ...
1
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1answer
21 views

Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is ...
4
votes
0answers
52 views

Is the preimage of the non-normal locus a divisor?

Let $X$ be a complex, affine variety. Let $\nu:\tilde X\to X$ be the normalization of $X$ and denote by $D\subseteq X$ the closed set of points where $\nu$ fails to be an isomorphism, i.e. $D$ is the ...
0
votes
1answer
32 views

What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
2
votes
2answers
74 views

One-dimensional Noetherian UFD is a PID

I am looking for a reference which has a self-contained (elementary, that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does anyone ...
3
votes
1answer
48 views

Maximal ideal in a polynomial ring over a field that is not algebraically closed

I want to prove that although $K$ is a field that IS NOT algebraically closed, every maximal ideal in $K[x_1, \ldots, x_n]$ can be generated by $n$ elements. To prove this, I am following the next ...
2
votes
1answer
26 views

How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
1
vote
1answer
36 views

Contraction of a maximal ideal in a polynomial ring

I have two questions: If $K$ is a field, $R=K[x_1,\ldots,x_n]$, the ring of polynomials over $K$ with $n$ indeterminates, and $M$ is a maximal ideal of $R$ why is the contraction $N$ of $M$ to ...
3
votes
0answers
74 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
1
vote
1answer
58 views

Question on a property of $\mathrm{Ass}(M)$ for modules over noetherian rings

I got stuck reading a proof of the following lemma (Lemma 0.19 in this file): Lemma Suppose that $M$ is a module over a commutative noetherian ring $R$ and let $m\neq 0 \in M$. Let $S$ be a ...
2
votes
2answers
59 views

If the localizations of two submodules with respect to any prime ideal are equal then the submodules are equal [closed]

I want to prove the following: Let R be a commutative ring with 1 and let N and L be two submodules of an R-module M. If the localizations of N and L with respect to any prime ideal of R are ...
0
votes
0answers
62 views

Localization of a regular local ring is regular

Quoting Hartshorne's Algebraic Geometry Definition. We say a scheme $X$ is regular in codimension one if every local ring $\mathcal{O}_x$ of $X$ of dimension one is regular. The most ...
4
votes
1answer
69 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
0
votes
0answers
21 views

Module length and connection between $\varphi,\det\varphi$ [closed]

$e_A(\varphi,M) = l_A(\mathrm{coker}\varphi) - l_A(\ker\varphi)$ Let $A$ be domain. I want to prove that $e_{A}(\varphi,A^n) < \infty \iff e_{A}(\det\varphi,A) <\infty$ using the fact that ...
2
votes
1answer
35 views

Relatively prime ideals in Dedekind Domains

I am currently working through Lang's Algebra and have come across an exercise I can not solve (Chapter II, Exercise $19$). Any help would be greatly appreciated. Let $R$ be a Dedekind domain. ...
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votes
0answers
25 views

Talking about varieties

hi I was recently reading ideals varieties and algorithms. I ah having problems showing things are not affine varieties. Previously with problems like. $V= \{ (a,a) | a \in R^* \}$ it was much easier ...
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votes
0answers
75 views

Overrings of holomorphy rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$, where $O_P$ is the valuation ring associated to the place $P$. ...
2
votes
0answers
51 views

Partial derivatives with respect to algebraically independent polynomials

Suppose that $\{f_1, \ldots, f_n\}, \{g_1, \ldots, g_n\}$ and $\{h_1, \ldots, h_n\}$ are algebraically independent polynomials that generates the same algebra of $\mathbb{R}[x_1, \ldots, x_n]$. Then I ...
2
votes
0answers
43 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...
0
votes
1answer
44 views

Let I be an unmixed radical ideal of R. then (I:x) is unmixed

Let $R$ be commutative ring with $1$. One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors (in other words, if the associated prime ideals of $R/I$ are the minimal prime ideals ...
3
votes
1answer
45 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
2
votes
1answer
65 views

How can I verify that the ideal $(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$ in $\mathbb Q[x,y,z,w]$

I want to show that the ideal $$(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$$ in the ring $\mathbb{Q}[x,y,z,w]$ is prime, how can I?
0
votes
1answer
55 views

If every maximal ideal is finitely generated is the ring Noetherian? [duplicate]

$R$ is a commutative ring with $1$. Suppose every maximal ideal is finitely generated. Is this ring Noetherian? Equivalently, is every prime ideal finitely generated?
0
votes
0answers
30 views

Subvarieties and finding ideals

Hi guys I am stuck working on this problem. I have a surface $W= V(xz-y^2)$ and we are trying to find an ideal $J \in K[W]$ so that the $V_w(J)=V(y-x^2,z-x^3)$ I showed that the second thing which is ...
2
votes
1answer
77 views

Functorial construction with two integral domains

Motivated by this question: Let $\mathsf{Int}$ be the category of integral domains with ring homomorphisms (perhaps only injective ring homomorphisms, if you need this). Is there a functor ...
1
vote
2answers
38 views

Residue class field of coordinate ring

If $X$ is an irreducible affine curve over an algebraically closed field $k$, then its coordinate ring $O(X)$ is a Dedekind domain. Suppose $\mathfrak{p}$ is a prime (hence maximal) ideal in $O(X)$ ...
0
votes
0answers
38 views

Property of free submodules for a module over a PID [duplicate]

This question was asked here and remains without solution. It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is ...
0
votes
1answer
14 views

Residue class ring of Dedekind domain

Zariski and Samuel Commutative Algebra Ch V para 7 makes the following statement: If $R$ is a Dedekind domain with an ideal $\mathfrak{a}=\prod_i\mathfrak{p}_i^{n(i)}$ factored into prime ideals, ...