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

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5
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1answer
52 views

Is this result on the bound of regularity of an ideal true?

I am solving a problem in which i need to use the following result but i am not sure whether the result is true on not: If the ideals $I_0,...,I_n$ are generated by linear polynomials in ...
10
votes
3answers
87 views

Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
4
votes
0answers
88 views

Testing if a submodule is free

This is hopefully a very simple question. In Gröbner Bases in Commutative Algebra by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, $S=K[x_1\ldots ...
1
vote
1answer
50 views

Integral extension and s.o.p.

Let $R\subset S$ be an integral extension. Is a system of parameters of $R$ a system of parameters of $S$? I think so, since there is good behavior in dimensions. Many thanks.
0
votes
0answers
50 views

Generic fiber to the Frobenius morphism

Let $k$ be an algebraically closed (perfect) field of characteristic $p>0$ and $f:\mathrm{Spec}\, k[t] \to \mathrm{Spec}\, k[X]$ be the Frobenius morphism induced by $X \mapsto t^p$ and identity on ...
1
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0answers
25 views

A Question about the Intersection Multiplicity

In a recent lecture the lecturer defined the local ring of an irreducible affine variety $V$ at $P\in V$ as $$ \mathcal{O}_{V,P}=\{\phi\in K(V)\mid\phi\text{ is defined at }P\}. $$ Then he defined ...
0
votes
1answer
33 views

The Archimedean place of $\mathbb{Q}$

Is there a way to extract the Archimedean absolute value of $\mathbb{Q}$ from its field structure in a way analogous to its non-archimedean absolute values? Here is some context: Given a valuation ...
2
votes
0answers
20 views

K-theory of a classifying space (part two)

Continuing my previous question, given a compact, connected Lie group $G$, there is a sequence of maps $$R(G) \to \hat{R}(G) \overset\sim\to K^*(BG) \to \hat{H}^*(BG;\mathbb Q)$$ apparently first ...
0
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0answers
42 views

Krull dimension localization with coefficients [duplicate]

Let $A[x_1, \ldots, x_n]$ be a polynomial ring over an integral domain, $A$. Let $s = a\prod_{1\leq i \leq n}{x_i}^{\alpha_i}$, $a \in A$. What is the Krull dimension of $A[x_1, \ldots, x_n]_s$?
3
votes
1answer
55 views

Krull dimension on localization

Given $A$ an integral domain and $A[x_1, \ldots, x_n]$ a polynomial ring over $A$. Let $s = \prod_{1\leq i \leq n}{x_i}^{\alpha_i}$. What is the Krull dimension of $A[x_1, \ldots, x_n]_s$? Will it be ...
9
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1answer
89 views

Commutativity of a ring from idempotents.

In a ring $R$ with unity, every element can be written as product of finitely many idempotents. Can one show that the ring is commutative?
3
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1answer
22 views

Primary decomposition in a Dedekind Domain

I was a little bit puzzled with the following problem that I have recently come across: Let $R$ be a Dedekind domain and let $P$ be a prime ideal in $R$. Is it true that $P^k$ is an irreducible ...
1
vote
2answers
27 views

Why is $\varphi(X_i) = X_i + b_i$ an automorphism of $K[X_1,\dots,X_n]$?

I'm trying to justify to myself the assertion (used here) that given a field $K$ and elements $b_1,\dots,b_n\in K$, the map $\varphi(X_i) = X_i + b_i$ is a $K$-automorphism of $K[X_1,\dots,X_n]$. ...
1
vote
2answers
66 views

Corollary to Lemma of Nakayama

In Matsumura's Commutative Algebra there is the following Corollary to the Lemma of Nakayama: Let $A$ be a ring, $M$ an $A$-module, $N$ and $N'$ submodules of $M$, and $I$ an ideal of $A$. Suppose ...
2
votes
1answer
75 views

Possible Inaccuracy at classic paper by Bayer and Stillman

In reading the paper Bayer and Stillman, "A criterion for detecting $m$-regularity", i believe i have encountered what may be a little inaccuracy, which i describe next. Let $I$ be a homogeneous ...
1
vote
1answer
30 views

Height of prime ideal containing the variable of a polynomial ring

I have a ring $R$ and a prime ideal $P$ of $S=R[t]$ with $t \in P$. I'm trying to prove that if $\mathrm{ht}(P/tS)$ is finite then $\mathrm{ht}(P) > \mathrm{ht}(P/tS)$. Here ...
1
vote
1answer
35 views

Direct Product of Completions

This question is regarding Theorem 8.15, page 62 of Matsumura's Commutative Ring Theory. It says that if $A$ is a semi-local ring and $I=m_1\cdots m_r$ be the Jacobson radical of $A$. Then ...
1
vote
1answer
42 views

Deduce that there are short exact sequences

Show that for $n>0$ there is a short exact sequence of chain complexes $0\rightarrow C_i(X;\mathbb{Z})\stackrel{f}{\rightarrow} C_i(X;\mathbb{Z})\stackrel{g}{\rightarrow} ...
2
votes
0answers
21 views

Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?

In a lecture notes on 'Cohomology modules' i read the following remark: Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where ...
2
votes
1answer
56 views

Is there an adjective for rings whose every non-zero prime ideal is maximal?

(All my rings are commutative and unital.) Question. Is there an adjective for rings whose every non-zero prime ideal is maximal? Remarks: Every PID has this property; more generally, every ...
2
votes
1answer
57 views

a little “paradox” in local cohomology of zero-dimensional ideals

Let $S = k[x_1,x_2,x_3]$ be a polynomial ring of dimension $3$ over an infinite field, and let $I$ be a homogeneous ideal of height $3$. Since $S$ has no zero divisors, the Krull dimension of $I$ is ...
2
votes
0answers
18 views

About the discriminant ideal

Let $E/K$ be a separable field extension of degree $n$, let $A$ be a Dedekind Domain which quotient field is $K$, and let $B$ be the integral closure of $A$ in $E$. Then we have that the ideal ...
1
vote
1answer
35 views

Minimal free resolution of ideal generated by three homogeneous polynomials

I am trying to solve the following exercise; Let $R=k[x_0,x_1,x_2]$ and $f_i$ homogeneous polynomials of degree $d_i, 0\leq i \leq 2$. Suppose $f_0,f_1,f_2$ have no common roots in $\mathbb P^2$. ...
0
votes
1answer
32 views

Krull dimension of an algebra

Given the ring, $\mathbb{Z}_6[x,y]/\langle x \rangle$. What is the Krull dimension of the ring? Isn't the following a chain of prime ideals in the ring, $\langle \overline{2}\rangle \subsetneq ...
1
vote
1answer
39 views

Book recommendation on Primary decomposition of ideals [closed]

I'm trying to prepare a presentation on "Primary Decomposition of Ideals" which is the title of my project. But I'm new for the subject so I need help on the following points How to outline my ...
0
votes
0answers
11 views

Inverse limit of ideals equal to expected ideal of inverse limit?

Suppose we have a map $(A_n \to B_n)_{n \in \mathbb N}$ of inverse systems of unital rings and a system $\mathfrak a_n \lhd A_n$ of ideals, one sent into the next under the maps $A_n \to A_{n-1}$. ...
2
votes
1answer
62 views

Exercise II-11 from Eisenbud-Harris, subscheme of dimension $0$, degree $3$, supported at origin isomorphic to what?

Suppose that $K$ is algebraically closed, and let $Z = \text{Spec}\,K[x_1, \ldots, x_n]/I \subset \mathbb{A}_K^n$ be any subscheme of dimension $0$ and degree $3$, supported at the origin. How do I ...
0
votes
0answers
29 views

Proving “up to unique isomorphism”: Universal property of cokernel

This is a homework question. I have already searched the web, but the proofs I have found are very generalized using category theory. Problem is the following: Suppose you have homomorphism $f: M ...
0
votes
1answer
74 views

Krull dimension of a $\Bbbk$-algebra

Given an ideal, $\mathfrak{a}\subseteq \Bbbk[x_1,\ldots,x_n]$, where $\Bbbk$ is a field. Let the maximal set of indeterminates independent modulo the ideal $\mathfrak{a}$ be of cardinality $k$. ...
3
votes
1answer
32 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
44 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
21 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
64 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
votes
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
42 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
32 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
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1answer
41 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 ...
1
vote
1answer
30 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 $xy\in I$ ...
0
votes
2answers
63 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
81 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
20 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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votes
0answers
30 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 $1 \leq i \leq m$. ...