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

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5
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If the module of Kahler differentials $\Omega_R$ is free than $\operatorname{rk} \Omega_R = \operatorname{dim} R$

Let $k$ be an algebraically closed field of characteristic zero, and $R$ is a (local as ring with maximal ideal $m$) algebra over $k$ of essentially finite type, such that $$ k \cong R/m. $$ It is ...
4
votes
0answers
66 views

Finite extensions of $\mathbb{Q}((t))$

Let $K$ be a number field. Is every finite extension of $K((t))$ of the form $L((\pi^{1/e}))$, where $L/K$ is finite and $\pi = a_1t + a_2t^2 + \cdots$ for some $a_i\in L$? Is every finite flat ...
3
votes
1answer
28 views

Why do we pass to fraction fields of domains of residue?

To look at an arbitrary commutative ring $R$ as a ring of continuous functions on its spectrum, we define the evaluation map $R\rightarrow R/\mathfrak p\rightarrow \kappa(x)$ for $x=\mathfrak ...
3
votes
2answers
40 views

Prime ideals in $R= \left\{ \frac ab\in \mathbb Q\mid a,b\in \mathbb Z,p\nmid b \right\}$

Let $p\in \mathbb Z$ be a prime. Define $R= \left\{ \frac ab\in \mathbb Q\mid a,b\in \mathbb Z,p\nmid b \right\}$. I'm supposed to prove $pR$ is the only prime ideal in $R$, and the $R/pR\cong \mathbb ...
2
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0answers
40 views

does every affine closed subscheme have an open affine neighborhood?

Let $X$ be a scheme and $D\subset X$ be a closed affine subscheme. Under what conditions must there exist an open affine $U\subset X$ containing $D$? The situation I'm imagining is this: $f : ...
4
votes
1answer
47 views

In what generality does every module show up over some residue field?

Let $R$ be a commutative, unital ring and $M$ an $R$-module. It is well-known that if $M$ is nonzero, there is some prime $\mathfrak{p}\in \operatorname{Spec} R$ with $M\otimes_R R_\mathfrak{p}$ ...
0
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0answers
21 views

Units in Semiperfect Skew Group Rings

Let $k$ be a field and $S$ the ring $k[[x_1,\ldots, x_n]]$. Let $G$ be a finite subgroup of $GL_n(k)$ that does not contain any nontrvial pseudo-reflections and such that $|G|$ is invertible in $k$. ...
0
votes
1answer
34 views

Finding all prime ideals in $\mathbb Z[\frac 1n]$ [duplicate]

As homework, I need to find all the prime ideals of $\mathbb Z[\frac 1n]$. For starters, I needed to prove that given any two rational numbers $q_1,q_2$, there exists a natural number satisfying ...
2
votes
2answers
35 views

Minimal polynomial of integral elements [duplicate]

Let $R$ be an integrally closed domain and let $K$ be its fraction field. Let $L\supseteq K$ be a field. If $\alpha\in L$ is integral over $R$ (i.e. if it satisfies a monic polynomial in ...
1
vote
1answer
44 views

Example for Integral closure

Let $R=k[x^4,x^3y,xy^3,y^4]$ be a polynomial ring. We can see $R$ is not integrally closed (since $x^2y^2 \in Q(R)$ is integral over $R$ but $x^2y^2 \notin R$ ). Therefore ...
0
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0answers
35 views

Radical ideals and prime decompositions for nonprincipal ideals

Fact. Let $R$ be a UFD. A principal ideal $ \left\langle a \right\rangle$ is radical iff every element in its prime decomposition has multiplicity one. Does this statement generalize in any way to ...
1
vote
1answer
26 views

Height of $I$ = Height of $I K[X]$

In the proof of Theorem 23 in Matsumura's Commutative Algebra on page 85, he wrote Since $A$ is a subring of $B = A[X]/I$, we have $A \cap I = (0)$. Therefore, if $K$ denotes the quotient field of ...
0
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1answer
68 views

Every non-zero prime ideal of $K[x,y]/(y^2-x^3)$ is maximal

Let $R=K[x,y]/(f)$ where $f(x,y)=y^2-x^3$. I can show that $R$ is an integral domain and Noetherian. I have to show that every non-zero prime ideal of $R$ is maximal, but I can not realize the form ...
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0answers
38 views

Prove that the upper half plane of Cartesian plane is not an affine variety

The question is stated as follows, Q. Let $R=\{(x,y):y>0\}$ be the upper half plane. Prove that $R$ is not an affine variety. My attempt was simply first assuming that it is an affine variety ...
1
vote
1answer
32 views

Tangency of varieties - example from Gathman

In general $\mathbf I(Y_1 \cap Y_2)=\sqrt{\mathbf I(Y_1)+\mathbf I(Y_2)}$. In remark 1.27 of these notes by Gathman, an example is given to illustrate the RHS is not radical if the varieties $Y_1,Y_2$ ...
2
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1answer
71 views

Zariski topology questions from Atiyah and Macdonald's Introduction to Commutative Algebra

Exercise 17 in Chapter 1 in Atiyah and Macdonald's Introduction to Commutative Algebra introduces the Zariski topology. There are 7 subquestions, of which 4 I've solved on my own, but the last couple ...
1
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0answers
28 views

Intersection of height one prime ideals

Let $A$ be a commutative, integrally closed, noetherian ring and as $\mathfrak p$ ranges over height one prime ideals, we have: $$A = \bigcap_\mathfrak p A_{\mathfrak p}.$$ The proof I have seen ...
1
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2answers
42 views

When is the set of prime ideals that don't intersect a monoid zariski open?

The $R$ be a commutative ring and $S$ be a multiplicative subset. When is the set $ \left\{ \mathfrak p\in \operatorname{Spec}R:\mathfrak p\cap S=\emptyset \right\}$ Zariski open in ...
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2answers
59 views

Proof of Proposition 2.12 in Neukirch ANT

I'd like a reference or a direct proof of the following statement: Let $K|\mathbb Q$ be a finite extension and consider the ring of algebraic integers $\mathcal O_K$. Let $\mathfrak ...
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0answers
9 views

Exactness and conjunction of finite character properties for modules

From Commutative Algebra - Constructive Methods by Lombardi and Quitte: Definition 2.9. A property $\mathsf P$ concerning commutative rings and modules is called a finite character property if it is ...
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2answers
65 views

Is the localization at a prime ideal of a non-zero ring non-zero again?

Let $R$ be a non-zero ring. Is it true that for any prime ideal $p$ in $R$, the localization $R_p$ is non-zero again? Stated in another way: Is it true that $R \neq 0 \iff \forall p \in Spec(R): R_p ...
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1answer
30 views

inclusion of f.g submodules and localization [closed]

Suppose $M,N$ are submodules of some other module and that $M$ is f.g. Suppose $S^{-1}M\subset S^{-1}N$. How can I prove $M[\frac 1s]\subset N[\frac 1s]$ for some $s\in S$?
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1answer
26 views

Is the localization of a f.g module at a f.g multiplicative monoid a f.g module?

Let $M$ be a f.g $R$-module for $R$ commutative. Let $S$ be a multiplicative submonoid of $R$ which is f.g as a monoid. Is it true that $S^{-1}M$ is a f.g $S^{-1}R$-module?
2
votes
3answers
149 views

Let $f:A\rightarrow B$ be a homomorphism of integral domains. Is it possible to extend $f$ to the integral closures of $A$ and $B$?

I am working on question II.3.8 in Hartshorne's Algebraic Geometry. Our professor mentioned a very useful algebraic result for the problem but I cannot find a reference anywhere. Let ...
2
votes
0answers
28 views

Why does the singular locus of a random rational curve always have the same Hilbert function?

As background, let's first note that a random rational curve of degree $d$ in $\mathbb P^n$ is smooth when $n\ge 3$. Indeed, picking a random rational curve of degree $d$ in $\mathbb P^n$ corresponds ...
2
votes
1answer
41 views

Faithfulness of Galois extensions of commutative rings

I need some help to understanding why Galois extensions of commutative rings are faithful. The definitions I'm using for Galois extensions is the one below. Let $R \rightarrow T$ be a ring ...
1
vote
1answer
33 views

Symbolic power of a prime ideal is primary

Let $A$ be a commutative ring, $S$ a multiplicatively closed subset of $A$. For any ideal $\mathfrak a$, let $S(\mathfrak a)$ denote the inverse image of $S^{−1}\mathfrak a$ under the localization map ...
2
votes
1answer
21 views

Do ideals with the same saturation have the same Hilbert polynomial?

Suppose $I$ and $J$ are homogeneous ideals in $k[x_0,\dots, x_n]$ (with the usual grading) that have the same saturation with respect to the irrelevant ideal. Do they have the same Hilbert polynomial? ...
1
vote
1answer
31 views

Finite character properties and localization

From Commutative Algebra - Constructive Methods by Lombardi and Quitte: Definition 2.9. A property $\mathsf P$ concerning commutative rings and modules is called a finite character property if it is ...
0
votes
0answers
33 views

Inverse of an isomorphism involving flat and finitely presented modules

I want to find the inverse of the map that appears in the black box. I have searched a lot and I could not find it. The books that I have come across display this proof technique without finding the ...
2
votes
1answer
42 views

Applications of Nagata's Lemma

In the spirit of this MO question I would like to ask for applications of a (somewhat lesser known?) lemma. Lemma. (Nagata) Let $R$ be an atomic domain. TFAE: $R$ is a UFD There exists a ...
7
votes
0answers
66 views

$\mathbb C[X_1, \ldots, X_n]$ is a free module over $\mathbb C[X_1, \ldots, X_n]^G$

Let $G$ be finite subgroup of $GL_n( \mathbb C )$. Let $\mathbb C[X_1, \ldots, X_n]^G$ be the set of all G-invariant polynomials of $\mathbb C[X_1, \ldots, X_n]$. Is there any rule by which we can ...
1
vote
1answer
54 views

Every nonzero prime ideal of $\mathcal {O_K}$ is maximal.

Theorem 1. If ring $B$ is an integral extension of ring $A$ and $P$ is prime ideal of $B$, then $P$ is maximal ideal of $B$ $\Leftrightarrow$ $A \cap P$ is maximal ideal of $A$. Theorem 2. If an ...
3
votes
2answers
34 views

$K$-homomorphisms from an étale $K$-algebra to a field

I am somewhat confused by the proof of Theorem 1.5.4 (p.22) (Grothendieck's version of the main theorem of Galois Theory) in Szamuely's Galois Groups and Fundamental Groups. The theorem establishes an ...
0
votes
0answers
28 views

When identifying $X$ with $\operatorname{mspec}k[X]$, how do we identify $\phi\colon X\to Y$?

Suppose $\phi\colon X\to Y$ is a morphism of affine varieties. It's common place to identify $X$ with the maximal ideals of $k[X]$, by $x\leftrightarrow M_x=\{f\in k[X]:f(x)=0\}$. What does this ...
2
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0answers
25 views

If $\varphi\colon X\to Y$ is a dominant morphism of affine varieties, $X$ irreducible, then $\varphi X$ contains nonempty open of closure.

There is a well known theorem that if $\varphi\colon X\to Y$ is a dominant map of affine varieties, and $X$ is irreducible, then $\varphi X$ contains a nonempty open set over $\overline{\varphi X}$. ...
4
votes
0answers
42 views

Finitely generated torsion module over a Dedekind domain

Let $M$ be a finitely generated torsion module over a Dedekind domain $R$. Show that there exist nonzero ideals $I_1 \supseteq \cdots \supseteq I_n$ of $R$ such that $M \cong ...
2
votes
2answers
57 views

Sheafyness and relative chinese remainder theorem

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let's take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ ...
1
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1answer
51 views

Associated primes of the symbolic power and ordinary power

I am struggling to understand the following quote from a paper of Arsie and Vatne "A note on symbolic powers and ordinary powers of homogeneous ideals": Our interest in the symbolic power stems ...
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2answers
62 views

A more technical definition of an Affine Variety

My textbook states that an affine variety is Definition. Let $k$ be a field and let $f_1,...,f_s$ be polynomials in $k[x_1,...,x_n]$. Then we set $$V(f_1,...,f_s)=\{(a_1,...,a_n) \in k^n : ...
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0answers
32 views

Calculating an endomorphism ring of a module over $R = \Bbbk[x,y,z]/(xy-z^2)$

Let $R = \Bbbk[x,y,z]/(xy-z^2)$, and let $M = (x,z)$. I want to determine $\text{End}_R(M)$. Since $R$ is a domain, I think that we're able to identify $\text{End}_R(M)$ with $ \{ q \in Q(R) \mid qM ...
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votes
0answers
37 views

Projective Resolution of $C^{\infty}(V)$ by Connes

In his article Noncommutative differential geometry (Inst. Hautes Études Sci. Publ. Math. No. 62 (1985), 257–360) A. Connes gives in Lemma 44 (p. 343f) a projective topological resolution of the ...
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2answers
48 views

Ideal of affine algebraic variety is radical

In remark 1.15(c) of these notes by Gathman, it is said that for a ring $R$, an ideal of an affine variety is radical, for if $f\in \sqrt I$ then $f^k|_X=0$ and hence $f|_X=0$ meaning $f\in I$. I'm ...
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1answer
78 views

The ring of real convergent sequences is Noetherian [closed]

Is it true that the ring of real convergent sequences is a Noetherian ring?
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2answers
58 views

Can we give an ring $R$ such that every prime ideal of $R$ be maximal with $|\operatorname{Max}(R)|=\infty?$

It is well known that in commutative rings, maximal ideals are prime. Can we give an example of a ring $R$ such that every prime ideal of $R$ is maximal with $|\operatorname{Max}(R)|=\infty?$
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51 views

Field extension is étale implies polynomial is separable

Following Johnstone (Exercise 0.11), a ring homomorphism $f: A\rightarrow B$ is étale if for every nilpotent ideal $N\subseteq R$ of a ring $R$ and every diagram of ring homomorphisms there is a ...
0
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2answers
60 views

Ring of polynomial functions on unit hyperbola is PID

Let $R=\mathbb{R}[X,Y]/(XY-1)$ be the ring of polynomial functions pn the unit hyperbola. How do I prove that $R$ is a principal ideal domain with unit group ...
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2answers
44 views

Let A a commutative ring with unity in which every element is idempotent ($x^n=x$ for some n>1 dependent on x), then every prime ideal is maximal. [duplicate]

Let $A$ a commutative ring with unity in which every element is idempotent ($x^n=x$ for some n>1 dependent on $x$), then every prime ideal is maximal. I came across this question revisiting Atiyah's ...
2
votes
0answers
41 views

Restriction of closed immersion to closed subset is a closed immersion

Let $f \colon X \to Y$ be a closed immersion in the category of algebraic sets. I think it's true that for $Z \subseteq X$ closed the induced map $Z \to f(Z)$ is an isomorphism, i.e. the restriction ...
1
vote
0answers
47 views

Isomorphism between ring of polynomial functions on unit hyperbola and Laurent polynomials [duplicate]

I want to prove that the ring $\mathbb{R}[X,Y]/(XY-1)$ of polynomial functions on the 'unit hyperbola' is isomorphic with the ring $\mathbb{R}[T,T^{-1}]\subset \mathbb{R}(T)$ of Laurent polynomials. ...