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

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0
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1answer
38 views

Prove that a monomial ideal $I$ is determined by the set of monomials it contains. [closed]

For an ideal $I \subseteq k[X_1, \dots ,X_n]$ prove that the following are equivalent: $I$ is generated by monomials. If $f =\sum \limits _a c_a X^a \in I$, and $c_a \ne 0$, then $X^a \in I$, where ...
1
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1answer
66 views

Chern class of ideal sheaf

Let $X$ be a smooth projective surface. Let $Z$ be a dimensional $0$ subscheme of length $l$. Suppose $I_Z$ is the ideal sheaf of $Z$. Then it claimed that $c_1(I_Z) = 0$ and $c_2(I_Z) = l$. (1)Why ...
1
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0answers
45 views

The same algebraic variety defined by different sets of polynomials

Let $\emptyset\neq X\subset\mathbb{P}^{n}$ be an algebraic variety such that $$ X=V(F_{1},\ldots,F_{m}) $$ for certain linearly independent homogeneous polynomials $F_{1},\ldots,F_{m}\in ...
0
votes
1answer
48 views

Determine the integral closure of a ring.

Let $R=F[X,Y]/(Y^2-X^3)$. Determine the integral closure of $R$ in its quotient field. I guess I should reduce the problem to some statement related to $F[X]$. For $F$ of characteristic not equal ...
2
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1answer
33 views

Atiyah-Macdonald, Exercise 4.6 [duplicate]

Let $X$ be an infinite compact Hausdorff space and let $C(X)$ be the ring of real-valued continuous functions on $X$. Does $(0)$ have a primary decomposition in this ring? I feel like the answer ...
1
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1answer
35 views

Noether's normalization lemma in practice (example)

I would like to know how to use the Noether's normalization lemma in practice. Noether's normalization lemma Let $k$ an infinite field, and $k[a_1,\dots ,a_n]$ be a finite $k$-algebra. There ...
2
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0answers
52 views

Is this result about the defining ideal true?

I am trying to generalize a result whose precise statement is the following: Let $X$ denote a set of $d+1$ points of $\mathbb P^{d}$ and the points are in linearly general position. Then $I_X$, ...
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0answers
21 views

Commutative rings and PI algebras

Any commutative ring $R$ with unity is a PI ring (polynomial identity ring). When could one take $R$ as a PI algebra? Essentially, what is the relation between an arbitrary commutative ring and a ...
0
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0answers
27 views

Prove that over an infinite field, a finite set of points in $\mathbb{ A}^n$ can be obtained as vanishing set of n polynomials [duplicate]

While reading Ernst Kunz's commutative algebra book, I came across this problem: Let $K$ be an infinite field, and $V \in\mathbb{ A}^n (K)$ (the affine n-space) be a finite set of points. Show ...
4
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0answers
66 views

Completion of Power Series

Let $k$ be field, char. not equal to two. Let $A = k[X,Y]/(Y^2 - X^2(X+1))$ with $\mathfrak{m}=(X,Y)$-adic topology. I want to show that $A'$, the completion of $A$, is isomorphic to ...
5
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2answers
61 views

If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring?

Suppose I have a graded polynomial ring $k[x_1,\ldots,x_n]$ on homogeneous generators, where $k$ is a field and the $x_i$ indeterminates, and further that I have a homogeneous graded subring $A$ such ...
2
votes
0answers
35 views

When does the equality $\mathrm{ht}\:\mathfrak{p}+\mathrm{coht}\:\mathfrak{p}=\dim R$ happen?

In the context of Krull dimension, given any commutative ring $R$ and $\mathfrak{p}\subset R$ a prime ideal, we have (almost by definition) $$ \mathrm{ht}\:\mathfrak{p}+\mathrm{coht}\:\mathfrak{p} ...
3
votes
1answer
74 views

Exercise on radical ideal and formal derivatives

I need some help for solving the following exercise, because at the moment I'm a little bit lost and don't know where to start. Given a field $k$ with $\mathrm{char}(k)=0$ and a polynomial $f\in ...
2
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0answers
20 views

Show that there is no coefficient field containing $k(X+Y^p)$.

Let $k$ be a field of characteristic $p$, let $R = k(Y)[[X]]$ be the power series ring with coefficients in $k(Y)$. Now $R$ is a local ring whose unique maximal ideal $\mathfrak M$ consists of power ...
3
votes
1answer
58 views

Blow-up and resolving a singularity

Given a variety $X=\{F:=x_0^3+t(x_1^3+x_2^3+x_3^3+x_4^3+1)=0\}\subset \mathbb{C}^6$, where $(x_0,...,x_4,t)$ are coordinates of $\mathbb{C}^6$. How to resolve the singularity by only blow-up smooth ...
0
votes
0answers
22 views

projective resolution for an $I$-torsion $R$-module

Let $R$ be a commutative Noetherian ring with non-zero identity, $I$ be an ideal of $R$ and $M$ be an $I$-torsion $R$-module. We know that there exists an injective resolution of $M$ in which each ...
1
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0answers
34 views

Reflexive Graded Module

Let $R=k[x_1,\dots,x_d]$ be a polynomial ring and $M=M_0\oplus M_1\oplus M_2\oplus\cdots$ be a graded $R$-module. Is it true that $M$ is reflexive as an $R$-module if and only if $M_i$ is reflexive as ...
0
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0answers
48 views

Finding genus of projective curve

Can anyone help me in finding the genus of the curves a) $x^2y^2-z^2(x^2+y^2)$ b) $(x^3+y^3)z^2+x^3y^2-x^2y^3$ c) $y^4+z^4-2x^2(y-z)^2$ d) $y^2z^2-x^4-Y^4$ e) $(x^2-z^2)^2-2y^3z-3y^2z^2$ Here ...
4
votes
3answers
82 views

Motivation for rings of fractions?

I'm learning about rings of fractions and localization. I like the material a lot and feel engaged with it, but I do lack a broader perspective on things. For example, I'm aware of things such as ...
0
votes
1answer
46 views

Support and Annihilator of Tensor Product of Modules

Let $M$ and $N$ be $R$-modules. Let $\mathrm{Supp}(M)$ be the set of primes $P$ such that $M_P\neq 0$, and let $\mathrm{Ann}(M)$ be the ideal of elements $r\in R$ such that $rm=0$ for all $m\in M$. ...
2
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1answer
54 views

Is R/m a flat R-module?

Let $(R,\frak m)$ be a commutative Noetherian local ring. Is $R/\frak m$ a flat $R$-module? Thanks.
2
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0answers
35 views

Intersection of hypersurfaces in the projective space

Fix an integer $n>0$. Is it true that for any $k>0$ and a closed point $x \in \mathbb{P}^n$, there exists hypersurfaces sections of degree $k$ (i.e., global sections of ...
0
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1answer
46 views

$I=(f_1, \ldots, f_n)\subset k[x_1, \ldots, x_n]$ with $f_i\in k[x_i]$ irreducible polynomials

Let $A=k[x_1,\ldots, x_n]$ and $I=(f_1, \ldots, f_n)\subset A$ with $f_i\in k[x_i]$ irreducible polynomials. Is it true that $I$ is a maximal ideal in $A$? $I$ is a maximal ideal $\iff$ $1\in ...
2
votes
1answer
66 views

$P$ prime implies $V(P)$ irreducible?

Let $P\subset k[x_1, \ldots, x_n]$ be a prime ideal. Is it true that the variety $V(P)$ is irreducible? This is easy to show when $k$ is algebraically closed. Is it also true in general?
0
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1answer
28 views

Module Completion and Right Exactness

This question is based on Atiyah Macdonald, Ex. 1, in Ch. 10. Let $A = \mathbb{Z}$ be the ring of integers with $p$-adic topology. The topological module $\mathbb{Z}/p\mathbb{Z}$ is then discrete. ...
-1
votes
1answer
53 views

Valuation rings of dim 1,2

I am studying valuation rings (beginner). I have read some theorems but still don't know a nontrivial example. Please give me an example which is not field. Also Need help to have examples of Krull ...
1
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2answers
39 views

Ring with spectrum homeomorphic to a given topological space

I would like to ask whether given a topological space $X$, we can find a commutative ring with unity $R$ such that $\operatorname{Spec} R$ (together with the Zariski topology) is homeomorphic to $X$. ...
0
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1answer
69 views

Compute Ext with Macaulay2

I want to compute Ext with Macaulay2. I see in the website they write how to do but I can not do. Can anyone help me with an example? For example, let $S=k[x,y,z,t]$. How compute ...
2
votes
1answer
87 views

Height of a contraction of prime ideal

Let $P\neq 0$ be a prime ideal of $\mathbb{Z}[X_1,\ldots,X_m]$ of height $n$, i.e. the longest chain of prime ideals contained in $P$ has length $n$, with $P \cap \mathbb{Z} = 0$. I want to show ...
1
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1answer
49 views

Need some suggestion for an introductory talk on 'Local Cohomology'?

Next week i am to give a talk on 'Local Cohomology' and i am writing to request suggestions for some basic interesting results for the talk.The relevant information is as follows: (1) The audience ...
0
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2answers
46 views

Noether normalization lemma proof

I would like to prove the following statement without using Noether normalization lemma (cause it is actually the base case in the induction process of the proof of this lemma). Let $k$ a field ...
2
votes
2answers
44 views

Zero module of differentials implies finite extension?

I am in the middle of a proof where the author asserts that for a field extension $K/E$, $\Omega_{K/E} = 0$ implies that $K/E$ is a finite extension. I am aware of results in the other direction with ...
3
votes
1answer
38 views

If $\phi: R \rightarrow S$ satisfies lying over, then $\textrm{ht } IS \leq \textrm{ht } I$.

Let $\phi: R \rightarrow S$ be a homomorphism of Noetherian rings with prime spectra $X, Y$, and suppose the contraction map $\phi^{\ast}: Y \rightarrow X$ is surjective. I'm trying to show that for ...
0
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1answer
29 views

Field of fractions of integral extension is an algebraic extension [duplicate]

Let $A\subset B$ be an integral extension. If $F$ and $E$ are the fields of fractions of $A$ and $B$, respectively, I want to show that $E$ is an algebraic extension of $F$. I know that since $A ...
0
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1answer
45 views

Compute intersection of ideals in a polynomial ring

Consider the ideals $\mathfrak{p}_1=(x,y)$, $\mathfrak{p}_2=(x,z)$ and $\mathfrak{m}=(x,y,z)$ in $k[x,y,z]$. How to show that ...
0
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0answers
39 views

Integral extension of a complete Noetherian local ring is local.

I try to prove the following result: Let $R \subset S$ be an integral extension of rings where R is a complete Noetherian local ring and S is a domain. Show that S is local. To start with, ...
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0answers
19 views

A field $F$ that is finitely generated as a ring is finite.

I have a proof but I doubt that it is correct. We assume that the characteristic of the field is positive, say prime $p$ (can we do this?). Now let $x_1,...,x_n$ generate $F$ as a ring. Let ...
0
votes
1answer
52 views

Why is this vanishing set nowhere dense? [closed]

Let $A$ be a commutative ring and $f\in A$ be a nonzerodivisor. Why is $\mathrm{V}(f)$ nowhere dense in $\mathrm{Spec(A)}$?
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0answers
23 views

Ring is Artinian iff nilpotent maximal ideal in Noetherian rings

I am browsing through some old lecture notes, and I am trying to prove the following: Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak m$. Show that the following are equivalent: (a) ...
2
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0answers
36 views

Principal local Artinian ring is a quotient of discrete valuation ring.

I have seen here the following statement: Let $R$ be a principal local Artinian ring. Clearly the quotient of a discrete valuation ring is such a ring; conversely it is not difficult to show that ...
0
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3answers
93 views

Finitely many complex solutions imply all solutions algebraic

I came across this rather interesting problem: Suppose that $f_1(x_1, \dots , x_n), \dots , f_m(x_1, \dots , x_n)$ are $m$ polynomials with integer coefficients, and that these have a finite number ...
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0answers
37 views

Proof of commutative Artinian ring is Noetherian

I think that I have a proof, but it seems much simpler than all proofs that I can find on the internet. Hence I suppose that there must be a mistake in my proof. The commutative ring $R$ is ...
1
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0answers
25 views

Module of differentials of a tensor of algebras

Fix a base ring $A$. I am looking at Lemma 6.1.11 in Liu. Let $B_1, B_2$ be $A$-algebras, $R = B_1 \otimes_A B_2$. Then there is a canonical isomorphism $$ \varphi: (\Omega_{B_1 / A} ...
0
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1answer
44 views

Localization of $\mathbb{Z}_6$ with respect to the powers of $2$

I want to localize $\mathbb Z_6$ with respect to the powers of $2$. Now if $\frac{2}{2} = \frac{a}{b}$, by $c(2a-2b)= 0$ and letting $c=3$ we conclude if $a \neq 0$ then $\frac{a}{b} =1$. Thus ...
1
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1answer
20 views

determinant annihilates ring vector

Let $R$ be a ring, and let $A$ be an $n\times n$ matrix with coefficients from $R$. Suppose for $r\in R^n$ we have $Ar=r$. Prove that $\det (A-I)\cdot r=0$. It is actually part of a bigger problem ...
8
votes
5answers
176 views

$K[[X]]$ is not a finitely generated $K[X]$-module.

How can I prove that $K[[X]]$ is not finitely generated over $K[X]$ as a module, where $K$ is a field. What I tried: if above is not true then $K[[X]]$ is integral extension over $K[X]$. But I ...
1
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1answer
25 views

What is the Hilbert Series of $R/I$ for a regular sequence? [closed]

What is the Hilbert series of $R/I$ for $I = (F,G)$ where $F,G$ is a regular sequence on $R = k[x,y]$ with $\deg F \leq \deg G?$ Definition: A sequence $F,G$ is regular on $R$ if $F$ is a nonzero ...
2
votes
1answer
45 views

Hilbert function of $k[x_1, x_2, x_3]/ (x_1^2,x_2^2x_3,x_2^3)$

If $R = k[x_1, x_2, x_3]$ is a polynomial ring and $I = (x_1^2,x_2^2x_3,x_2^3)$ how do you see that the Hilbert function $H(R/I,i) = 4$ for $i \geq 4$? So the free resolution of $R/I$ is $$0 \to ...
1
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0answers
30 views

Any characterization for commutative rings over which “projective modules” equal “free modules”?

As far as I know, over any PID, an polynomial rings over a field, or an local ring, projective modules are always free. This kind of results make me curious about if there are any overall ...
1
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1answer
72 views

Hilbert function and series.

If $f$ is a homogeneous polynomial of degree $d$ in a polynomial ring in $t$ variables over a field, and it generates an ideal $I$. Then the Hilbert function of $R/I$ is $$H(R/I,n) = ...