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

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3
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1answer
33 views

Computing prime factorization of ideals?

I want to compute the prime factorizations of the ideals $\langle 4\sqrt{-14}\rangle$, $\langle 6\sqrt{-6} \rangle$ and $\langle 4\sqrt{-5} \rangle$ in the ring of algebraic integers of ...
1
vote
1answer
55 views

Associativity of the tensor product of bimodules

Let $A_0,\dots,A_n$ be algebras over some fixed commutative ring $k$ (you may assume $k=\mathbb{Z}$ for simplicity). Let $M_i$ be an $(A_{i-1},A_i)$-bimodule for $i=1,\dots,n$. A multilinear map from ...
1
vote
1answer
46 views

Krull dimension of affine $\Bbbk$-algebra

Given an ideal, $\mathfrak{a} = \langle x_2x_3 \rangle \subseteq \Bbbk[x_1, x_2,x_3]$, where $\Bbbk$ is a field. We have that the maximal set of indeterminates independent modulo the ideal ...
0
votes
0answers
22 views

A direct summand of a sequence, Rotman, Homological Algebra, ex. 10.15 [duplicate]

If $0 \rightarrow A' \xrightarrow{\delta} A \rightarrow A'' \rightarrow 0$ is a split short exact sequence in an abelian category $\mathcal{A}$ (if you like, let $\mathcal{A}$ be the category of ...
4
votes
1answer
65 views

Castelnuovo-Mumford regularity and exact sequence.

In a question on MathOverflow it is said that: It is known that given a short exact sequence of finitely generated graded modules over a polynomial ring over a field:$$0 \to M'' \to M \to M' \to ...
4
votes
1answer
31 views

(Finitely many minimal primes) + ($R_M$ domain for all maximal $M$) $\Rightarrow$ ($R$ = product of domains)

We are doing a homework problem for our commutative algebra class, which asks us to prove: Let $R$ be a commutative ring with $1$ containing finitely many minimal prime ideals $P_1, \dots, P_n$. ...
1
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2answers
39 views

Artinian - Noetherian rings and modules suggest study guide

What text or any document that has gathered this part of Algebra theory. Thanks. Pd: I seek on variety's book of commutative algebra but the subject is partially dealt
0
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1answer
45 views

Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
0
votes
1answer
44 views

Quick way to show that inclusion is a local property? [duplicate]

I have encountered a problem which requires me to prove that ideal inclusion is a local property. That is to say, suppose $S,T \subset R$. Show that $S \subset T $ if and only if $SR_P \subset SR_P$ ...
3
votes
0answers
33 views

On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finite generated case. Let $M$ be an free abelian group, $N$ a subgroup ...
1
vote
1answer
42 views

Let $A$ a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$.

Let $A$ be a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$, where $q^{(n)}= \lbrace a \in A \mid \exists d \in A \setminus q\text{ such that }da \in q^n \rbrace$ and ...
3
votes
1answer
36 views

Definition of primary ideal [duplicate]

I am confused with the definition of a primary ideal. The definition states that if $R$ is a commutative ring then $I$ is called a primary ideal of $R$ is the following condition holds. If $xy\in I$ ...
-1
votes
2answers
74 views

Notation in commutative algebra

I am doing some exercises on commutative algebra and came along the following expressions, which were not elaborated on. Is someone familiar with them? The first is for $p$ a prime number ...
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2answers
82 views

How does extension of restriction of $M$ relate to $M$?

Let $A,B$ be rings, $f:B\to A$ be a ring homomorphism, and $M$ be an $A$-module. We can view $M$ as a $B$-module via restriction, and we may then extend the restriction of $M$ to an $A$-module by ...
1
vote
1answer
16 views

Number of zero-solutions for two bivariate polynomials $p$ and $q$

If I consider two bivariate polynomials $p,q \in \mathbb{C}\left[ x,y \right]$ where $p$ has total degree $m$ and $q$ has total degree $n$. To keep things simple I'm not interested in special cases ...
1
vote
1answer
22 views

Question about proof of Krull principal ideal theorem

How can we explain the following step in the proof of Krull principal ideal theorem: $l\{ ((z):x^n)/(z) \}$ or $l\{ ((x^n):z)/(x^n) \}$ is finite? $l(M)$ - length of module.
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0answers
31 views

Definition of hypersurface singularity

I am really confused about this notion. Suppose $X$ is an arbitrary variety over an algebraically closed field $k$ (if you like, let the characteristic be $0$), and $p$ is a $k$-valued point. If $p$ ...
0
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0answers
28 views

if $F_{\bullet}$ is a complex and $r$ an integer, what is $F_{r-\bullet}$?

While reading the paper Some results and questions on the Castelnuovo-Mumford regularity, by Marc Chardin, I encountered in the proof of Theorem 5.1 the notation $F^N_{r-\bullet}$. To provide some ...
0
votes
1answer
39 views

Commutativity of ring $R$ necessary for $\mathrm{Hom}_R(M,M')$ being an $R$-module

Why do we need $R$ to be commutative if we want $\mathrm{Hom}_R(M,M')$ (where $M$ and $M'$ are $R$-modules) to be an $R$-module itself? I tried to find out which axiom for modules does not hold if ...
4
votes
1answer
23 views

$G_{\mathfrak a}(A)$ integral domain and $\bigcap \mathfrak a^n = 0$ implies $A$ is integral domain

This is Lemma 11.23 in Atiyah: For an ideal $\mathfrak a \subseteq A$, define $G_{\mathfrak a} (A) = \bigoplus _{n=0} ^\infty \mathfrak a^n / \mathfrak a^{n+1}$. The statement of the Lemma: ...
2
votes
1answer
47 views

Showing the polynomials form a Gröbner basis

Let $A$ be an $m \times n$ real matrix in row echelon form and $I \subset \mathbb{R}[x_1,\dots,x_n]$ is an ideal generated by polynomials $p_i = \sum_{j = 1}^na_{ij}x_j$ with $1 \leq i \leq m$. ...
3
votes
1answer
62 views

Hilbert function and homogenous polynomials.

Let $\{[1:0:0],[0:1:0],[0:0:1],[1:1:1] \} = \{p_1,p_2,p_3,p_4\}$ be four points in the projective space $\mathbb{P}^2$. For every $p_i$, show there is a homogenous polynomial $f_i$ such that ...
4
votes
1answer
90 views

When is $\mathbb{Z} [x]/f(x) $ a Dedekind domain?

Given a monic separable irreducible polynomial $f$ with integer coefficients, when $\mathbb{Z} [x]/f(x)$ is a Dedekind domain? And when it happens to be a Dedekind domain, how to know its class ...
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0answers
74 views

Infinitely generated torsion free modules over PID

Let $R$ be a PID and $\mathbf{V}$ a torsion-free $R$-module, not necessarily finitely generated. If I understand it correctly, every rank 1 submodule of $\mathbf{V}$ is isomorphic to a submodule of ...
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vote
2answers
84 views

How do I find $\gcd(p,q)$ and $\mathrm{lcm}(p,q)$ by using syzygies?

Let us introduce the setting and recall some definitions. Setting: We are in a UFD polynomial ring $K[x]$ with $p,q \in K[x]$. Definition: Given a finitely generated $R$-module $M$ (where $R$ is a ...
2
votes
1answer
37 views

Arithmetically Cohen-Macaulay curve on a quadric

If $Y$ is a curve of bidegree $(a,b)$ on a smooth quadric surface $Q\subset \mathbb{P}^3$, how do we see that it is arithmetically Cohen-Macaulay (ACM, for short) iff $|a-b|\leq 1$? If (like me) ...
2
votes
1answer
47 views

About a short proof of Krull principal ideal theorem

How from this theorem I can get a proof of Krull principal ideal theorem? I understand that w.l.g. we can prove it for a Noetherian local ring. But why we can consider that $(x)$ is $M$-primary? ...
1
vote
1answer
37 views

Show that some monomial ideal is primary

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary. I noticed that ...
0
votes
2answers
39 views

Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated ...
0
votes
1answer
39 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
vote
1answer
69 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
vote
0answers
46 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
49 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
votes
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
vote
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
votes
0answers
53 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
29 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
votes
0answers
68 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
votes
2answers
62 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
37 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
75 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
votes
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 ...
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votes
0answers
23 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 ...
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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
votes
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
84 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
50 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
votes
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.