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

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Injectivity Unclear

Let $R=K[x_1,...,x_n]/I$ and $m$ be maximal ideal of $R.$ Let $(s_1,...,s_d)$ be a base of $m/m^2$ where $\dim(R_m)=\dim(m/m^2)=d.$ Then by Kunz Chapter V.5.10 the canonical epimorphism ...
1
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1answer
17 views

An exercise on tensor product in a local integral domain.

Let $M$ be a finite module over a local integral domain $(A,m)$. Let $k$ be its residue field and $Q$ its fraction field. Consider the $k$-vector space $M \otimes_A k$ and the $Q$-vector space $M ...
2
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2answers
33 views

What is an example of a homomorphism of rings that doesn't preserve gcd's?

Given a commutative ring $R$, we say that $x$ is a gcd of $(y,z)$ iff the following conditions hold: $x \mid y,z$ For all $x' \in R$, if $x' \mid y,z$, then $x' \mid x$. This gives a ternary ...
2
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2answers
45 views

Commuting of Hom and Tensor Product functors?

Let $V_i,W_i$ be finite dimensional vector spaces, for $i=1,2$. Assume we have homomorphisms $\phi_i:V_i\rightarrow W_i$. Then, there is an induced map $\widehat{\phi_1 \times \phi_2} \in Hom(V_1 ...
2
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1answer
47 views

Dickson's Lemma

I am doing a course in Commutative algebra and there is a lemma called Dickson's lemma which states the following: Let $\mathfrak{I} = \langle X^{u}: u \in A\rangle$ for some set $A \subset ...
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1answer
14 views

Why is the degree condition for a degree reverse lexicographic order necessary?

A degree reverse lexicographic order $\prec$ is defined as follows: Given the polynomial ring $R=K[x_1,...,x_n]$. Two monomials in $R$ have the order $x^u\prec x^v$, if $\deg(x^u)<\deg(x^v)$, or ...
1
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1answer
26 views

Subsheaf of a torsion-free sheaf

Let $X$ be a noetherian projective scheme, $\mathcal{F}$ a torsion free $\mathcal{O}_X$-module on $X$ and $\mathcal{G} \subset \mathcal{F}$ submodule. Is it possible that $\mathcal{G}$ is ...
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2answers
30 views

Isomorphism between affine varieties

I am working with a ring and I am trying to show it is not isomorphic (as $k$-algebra) to another ring: $k[x,y,z]/\langle xy-z^2\rangle$ and $k[u,w]$. What I tried so far was. I aim for a ...
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4answers
60 views

Are primary ideals always contained in unique maximal ideal?

Just wondering, is this a standard fact? I notice a couple Banach algebra texts define primary ideals in this way. Another question: does this property, i.e. being contained in a unique maximal ideal, ...
0
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1answer
32 views

Finding a bijective morphism

I am given two Varieties $Z=V(x^2+y^2+1) \subset C^2$ and $W=V(x^2-y^2-1) \subset C^2$. We need to find a bijective morphism f such is an isomorphism with the inverse of f. First how we defined ...
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0answers
26 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
2
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1answer
48 views

Isomorphism of Localizations

I believe, though a not sure, that any two ideals $A, B$ of a Dedekind domain $X$ are isomorphic as $X$-modules iff their localizations $A_p, B_p$ are isomorphic for any prime ideal $p$. Could anyone ...
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0answers
19 views

Not connected Zariski topology implies the existence of an idempotent element. [duplicate]

I am trying to prove that for a commutative ring $A$ (with the unit) the Zariski topology is not connected if and only if there is an idempotent element different from 0 and 1. I proved "Existence of ...
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1answer
22 views

Commutative Algebra and Monomial orders

So whenever we are doing any problem related to ideals in the polynomial ring $k[x_{1},x_{2},\dots x_{n}]$,(e.g. calculating a grobner basis for instance or doing the division algorithm for a set of ...
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0answers
20 views

Homogeneous prime ideal in $K[x_1,\ldots,x_n]$

Let $K$ be a field and $P$ a homogeneous prime ideal in $K[x_1,\ldots,x_n]$, with height $r$. I want to show that there is a chain of homogeneous primes $P_0\subsetneqq P_1\subsetneqq\cdots\subsetneqq ...
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1answer
21 views

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite.

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite. $P$ is a maximal ideal of $R/P$ is a field. I ...
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0answers
42 views

Class number of $\mathbb Q(\sqrt{10}) $

I am interested in knowing how to compute the class number of $\mathbb Q(\sqrt{10}) $. I am confused with these class number computations.
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0answers
36 views

Krull's height theorem in the non-Noetherian case

Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $\operatorname{ht} I\le n$. When $R$ is not Noetherian, this is not ...
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2answers
26 views

Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
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1answer
23 views

MaxSpec of the ring of continuous function on a compact topological space. [on hold]

Consider a compact topological space $X$ and let $A$ be the ring of continuous functions $f: X \to \mathbb{C}$. Let $\mathfrak m_x$ for a point $x \in X$ be the kernel of the evaluation map $f \mapsto ...
2
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0answers
27 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 ...
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1answer
28 views

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

$\require{AMScd}$ I recently read the chapter 10 on Completions in Atiyah-MacDonald. They describe the $\mathfrak{a}$-adic completion $\hat{M}$ of an $A$-module $M$ as the inverse limit of an inverse ...
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2answers
72 views

On the minimal set of generators of monomial 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 ...
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1answer
66 views

relation between vector space and torsion free module [on hold]

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. ...
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0answers
34 views
+50

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
40 views

tensor of two vector space [on hold]

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 ...
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1answer
20 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 ...
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2answers
38 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 ...
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0answers
38 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 ...
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0answers
27 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
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3answers
66 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
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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?
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2answers
81 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?
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1answer
44 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 ...
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2answers
40 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
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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 ...
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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: ...
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2answers
45 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 ...
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1answer
17 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 ...
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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 ...
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1answer
79 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 ...
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0answers
93 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 ...
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1answer
22 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 ...
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0answers
53 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 ...
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0answers
40 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 ...
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2answers
76 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
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1answer
52 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 ...
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0answers
19 views

theorem 31.6 Matsumura [on hold]

Let (A,m) be a formally equidimensional Noetherian local ring and I be an ideal of A If A/I is equidimensional Why A/I is formally equidimensional?
2
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1answer
27 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
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1answer
37 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 ...