Questions tagged [commutative-algebra]

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

Filter by
Sorted by
Tagged with
0
votes
3answers
66 views

Example of $A \subseteq B$, $\dim(A)=\dim(B)$, $J$ non-maximal ideal of $B$ and $J \cap A$ a maximal ideal of $A$

Let $A \subseteq B$ be two $k$-algebras, $k$ is a field of characteristic zero. Assume that $\dim(A)=\dim(B) < \infty$. Is it possible to find a non-maximal ideal $J$ of $B$ such that $J \cap ...
0
votes
0answers
21 views

$u,v \in A \subseteq B$ satisfying $Au+Av=(Bu+Bv) \cap A$

Let $A \subseteq B$ be two $k$-algebras, $k$ is a field of characteristic zero. Let $u,v \in A$. Let $I=Au+Av$ be the ideal of $A$ generated by $u$ and $v$ and let $J=Bu+Bv$ be the ideal of $B$ ...
3
votes
1answer
78 views

Describe prime ideals and Krull dimension of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$

I want to describe the prime ideals of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$, where $\overline{\mathbb{Q}}$ denotes the integral closure of $\mathbb{Q}$ in $\mathbb{C}$, ...
0
votes
0answers
69 views

Notion of simple hypersurface singularity depends on the presentation?

Let $(S, \mathfrak n)$ be a regular local ring. For $0\ne f\in \mathfrak n^2$ define $c(f, S):=\{\text{ideals } I \text{ of } S : f\in I^2\}$ . Now let $(S_1, \mathfrak n_2)$ and $(S_2,\mathfrak n_2)...
0
votes
0answers
26 views

How to show that $\hat{R} \cong \prod_{i=1}^{r}\hat{R_{m_i}}$.

Let $\mathfrak m_1,\ldots,\mathfrak m_r$ be distinct maximal ideals of a Noetherian ring $R$ and $I=\bigcap_{i=1}^{r}\mathfrak m_{i}$. Let $\widehat{R}$ be the $I$-adic completion of $R$ and $\widehat{...
0
votes
1answer
32 views

Elementary Proof that every Number Ring is integrally closed in Number field

Is there an elementary way to see that a number ring $\mathcal{O}_K$ of a number field $K$ is integrally closed in $K$? In other words, let $\alpha \in \mathbb{C}$ and $a_i$ are algebraic integers. ...
2
votes
1answer
30 views

Localization and nilradical

I am trying to answer a question that has already been posted in here (About Nilradical and Localization). I did not have much success with the first two answers, and the other two mention sheafs, ...
1
vote
1answer
40 views

complex non algebraic manifold local ring of holomorphic functions is noetherian?

Consider $X$ a complex manifold. Denote $x\in X$ a point and $O_x$ as the local holomorphic function ring at $x$. Assume $X$ is not algebraic. $\textbf{Q1:}$ Is $O_x$ Noetherian? If it is ...
1
vote
0answers
31 views

Find the composition of finitely generated module over Dedekind domain

I'm taking a course on commutative algebra and we learn this theorem: Every finitely generated module M over Dedekind domain A is direct sum of projective module P and torsion module T. T is direct ...
0
votes
1answer
56 views

Flat extension of local rings with a specified extension of residue field [closed]

Let $(R, \mathfrak m_R, k)$ be a Noetherian local ring and $K$ be a field containing $k$. Then is it true that there is a Noetherian local ring $(S, \mathfrak m_S)$ and a flat ring homomorphism $f: ...
1
vote
1answer
88 views

For a projective cover $(\sigma, P)$ of a module $M$, $P$ is indecomposable implies $M$ is indecomposable

We say that a module $M$ is indecomposable if for $M=M_{1} +M_{2}$ (not direct sum) we have that $M_{1}=M$ or $M_{2}=M$. Let $\sigma:P \to M$ a projective cover of $M$, this means that $\sigma$ is an ...
1
vote
0answers
71 views

direct product decompositions of semi-rings

(With Hilbert's Basis Theorem in mind.) Is it true that every finitely presented commutative semi-ring with unit is a finite direct product of directly indecomposable factors?
1
vote
0answers
63 views

A puzzling computation by Mumford of a module of Kähler differentials.

Mumford in his Red Book gives on page 144 an example of computation of a module of Kähler differential forms. Namely, he lets $k$ be a field and considers the quotient algebra $B=k[X,Y]/(XY)$. He ...
2
votes
1answer
51 views

Integral closure of $k[x^3,x^2y,y^3]$ in field of fractions

Let $A = k[x^3,x^2y,y^3] \subset k [x,y] $. I want to find the integral closure of $A$ in its field of fractions. To do so, I first want to find the field of fractions $\mathrm{Frac}(A)$. I think ...
1
vote
1answer
64 views

A bundle pull-back along itself

Let $X$ be a scheme, $\mathcal{E}$ a locally free $\mathcal{O}_X$-module of finite rank and $p: E\to X$ the corresponding geometric vector bundle (with global sections $\mathcal{E}$). Do we have an ...
0
votes
1answer
57 views

Punctured spectrum of a (reduced) Noetherian local ring of dimension $1$ is an affine- scheme?

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Then the affine- scheme $X=\operatorname {Spec}(R)$ can be written as a set-theoretic union $\operatorname{Spec}(R)=Min(R)\cup \{\...
1
vote
0answers
43 views

Base change along a separable extension.

Let $L/K$ be a separable field extension of degree $n>1$. Is it true that $L\otimes_{K}L=L^{n}$ as $K$-algebras? My initial guess is yes, but I have no idea how to prove it.
4
votes
1answer
110 views

Can finitely generated reflexive module have strictly larger depth than the depth of the ring?

Let $M$ be a non-zero finitely generated module over a Noetherian local ring $(R, \mathfrak m)$. Then $\operatorname {depth}(M)\le \dim M\le \dim R$. So if $R$ is Cohen-Macaulay, then $\operatorname ...
1
vote
1answer
67 views

Localizations of $k[x,y]/(f)$ UFD

Let $k[x,y]$ be a polynomial ring in two indeterminants and $f \in k[x,y] \backslash k$ a non constant poynomial. I want to know if there exist any nice criteria to answer the question when the ...
1
vote
1answer
44 views

Base change of nilradical is nilradical

Let $B \to A, B \to B'$ be injective, finite ring homomorphisms (finite means that $A$ and $B'$ are finite $B$-modules). Suppose that $A$ and $B$ are integral domains. Denote by $N$ the nilradical of ...
2
votes
1answer
30 views

finiteness of Koszul groups

A basic question about Koszul homology from Matsumura's Commutative Ring Theory In Theorem 16.5(ii) it is assumed that $(A,m)$ is a local ring and $x_1,\ldots,x_n \in m$, and $M$ is a finite $A$-...
1
vote
0answers
55 views

Blow-up and regular sequence

I'd like how to deduce, if it's possible that: the blowup of an affine variety $X$ along $V(g_1,\ldots,g_k)$ is $V(t_i g_j-t_j g_i)_{i,j}\hookrightarrow X\times\text{Proj}(k[t_1,\ldots,t_k])$ if the ...
1
vote
0answers
70 views

A subspace of a set of bounded and continuous functions is closed in this set and the ideal of this set

Question: Let $(X, d)$ be a compact metric space. For a given $x_0 \in X$, define $C_{x_0}(X,\mathbb{R})$ by $$C_{x_0}(X,\mathbb{R}) = \{f \in C(X, d):f(x_0) = 0\}$$ Note that $C(X,\mathbb{R})$ ...
1
vote
1answer
102 views

A set of bounded and continuous functions under supnorm is a commutative algebra with a multiplication identity

Question: Let $(X, d)$ be a metric space. Denote all bounded and continuous functions from $X \rightarrow \mathbb{R}$ as $BC(X,\mathbb{R})$, the topology of which is supnorm. Show that $BC(X,\mathbb{...
0
votes
1answer
64 views

Localizations of $k[y,z]/(1-y^2+z^2)$ UFDs

Let $k$ be a non algebraically closed field with $i \not \in k$; equivalently the polynomial $T^2+1 \in k[T]$ is irreducible over $k$. How to prove or disprove that for the ring $R:=k[y,z]/(1-y^2+z^...
0
votes
1answer
34 views

List of equations in MAGMA

I have some integer $n$, some ambient affine space $\mathbb{A}^n$, and a list $L$ of equations $f_{ij}$ cutting out a variety $X$ in the ambient space. I have problems defining the list $L$ correctly. ...
1
vote
0answers
55 views

Tensor product restriction of scalars and isomorphism

Let $R$ be a commutative ring and $S\subset R$ a subring. Let $M$ and $N$ be two $R$-modules. 1) Via restriction of scalars, we can also view $M$ and $N$ as $S$-modules. Show that we have a ...
1
vote
1answer
48 views

Checking irreducibility of polynomials in two variables

There are a few exercises in Hartshorne about checking singularity of an affine curve. For example, $Y$ defined by $x^2 = x^4 + y^4$ over a field $k$ (with ${\mathrm{char}}k \neq 2$). This is easy. ...
0
votes
0answers
21 views

Coprime polynomials of consecutive degrees

Let $f,g \in k[t]$ be two polynomials, $k$ a field of characteristic zero. Assume that $f$ and $g$ satisfy the following two conditions: (i) $f$ and $g$ are coprime, namely, they have no common root (...
0
votes
0answers
66 views

Stalks of Plane Conic $C \cong \mathbb{P}^1$ are UFD

Assume $k$ is a alg closed field. Then it is easy to check that the morphism $\phi: \mathbb{P}^1 \to \mathbb{P}^2, (x_0:x_1) \mapsto (x_0^2: x_0x_1:x_1^2)$ induces an isomorphism between $\mathbb{P}^1$...
0
votes
1answer
28 views

going between property

Let $R'\in R$ be an integral extension of rings with $R'$ a $K$-algebra finitely generated . Consider a chain of different prime ideals in $R'$ , $P_{1}\subsetneq P_{2}\subsetneq P_{3}$ Such that ...
-1
votes
1answer
38 views

Can commutative local rings have any non-zero zero divisors? [closed]

Can commutative local rings have any non-zero zero divisors? Is this possible?
2
votes
0answers
60 views

Computing projective dimension over hereditary rings (modules).

I want to prove this example from Rotman's which I have not found proved in literature yet and Im curious about. Let $_{R}M$ be a left module over a left herditary ring $R$, then $p.d(_{R}M) \leq 1$,i....
0
votes
2answers
53 views

$\mathbb{R}[X]$ is an integral extension of $\mathbb{R}[X^2-1]$

I am trying to prove that every polynomial of $\mathbb{R}[X]$ satisfies a monic polynomial equation with coeffients in $\mathbb{R}[X^2-1]$ that is every polynomial $b(x)= x^m+b_{m-1}x^m-1+...+b_{0}$ ...
2
votes
0answers
47 views

Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? [duplicate]

Let $U$ and $V$ be modules over a commutative ring $K$. Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? I'm a differential geometer so I'm usually dealing with finite-...
0
votes
1answer
25 views

Proving the analogous of $a\subseteq r(a)$ for $r_M(N)$

I am trying to solve the problem number 20 of chapter 4 from the book of Atiyah and Macdonald's Introduction to Commutative Algebra. I solved everythng but this: to prove the analogous of $a\subseteq ...
1
vote
1answer
91 views

On regular sequence in generating set in a homogeneous ideal in polynomial ring of maximum height

Let $J$ be a homogeneous ideal in $S=k[x_1,...,x_d]$, where $k$ is an infinite field, such that $J$ has height $d$ i.e. $\dim (S/J)=0$. Then $\mu(J)\ge d$ and $\operatorname{grade}(J)=\operatorname{ht}...
0
votes
0answers
84 views

Exercise 4.5.H in Vakil

Vakil's 4.5.H reads as follows Suppose $I$ is any homogeneous ideal of $S_•$ contained in $S_+$, and $f$ is a homogeneous element of positive degree. Show that $f$ vanishes on $V(I)$ (i.e., $V(I) ⊂ ...
2
votes
0answers
36 views

Flatness of the $p$-th power roots ideals in a perfect ring

Let $A$ be a perfect ring of characteristic $p>0$. If $x\in A$ is a nonzero divisor then $$ (x^{1/p^{\infty}})=\bigcup_{e=1}^{\infty}(x^{1/p^{e}})A $$ is an $A$-flat ideal. Reading Hochster's ...
0
votes
1answer
28 views

empty set in zariski topology

I stumbled across the following assertion: Let $A$ be a commutative ring and Spec($A$) given with the Zariski topology. In this topology all closed sets are of the form: $V(\mathfrak{a}):=$ {$\...
1
vote
1answer
48 views

Ideals whose union is an ideal [duplicate]

Does someone have an example of three ideals such that no one is contained in another, yet their union is an ideal?
0
votes
0answers
27 views

well-definedness of closed set in Zariski topology

Let $A$ be a commutative ring and $\mathfrak{p} \subset$ Spec($A$). Then for an ideal $\mathfrak{a} \subset A$, we have a closed set in the Zariski topology defined by $V(\mathfrak{a})$:={$\...
0
votes
1answer
59 views

$A$ is a ring and $I$ is an ideal of $A$ with $I \neq A$. Then, are (1) and (2) equivalent?

$A$: commutative ring with unit. (1) $I$ is a primary ideal of $A$. (2) If $a,b \in A$, and $ab \in I$, then there exists a positive integer $n$ such that $a^n \in I$ or $b^n \in I$. I notice ...
4
votes
1answer
78 views

Kernel of $\mathbb{Z}_p \to \mathbb{Z}/p^{n}\mathbb{Z}$ equals to $p^n \mathbb{Z}_p$.

Let $\mathbb{Z}_p$ denotes the ring of $p$-adic integers, i.e., $\mathbb{Z}_p:= \varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$. Then consider the projection map $\pi_{n}: \mathbb{Z}_p \to \mathbb{Z}/p^{n}\...
0
votes
1answer
49 views

Direct image sheaf under an open embedding

Let $i: U=\mathbb A^2-0\to\mathbb A^2=X$ be the inclusion (over a base field $k$). It's well-known that $\mathbb A^2-0$ is not an affine variety and the restriction map $\mathcal O_X(X)\to\mathcal O_X(...
0
votes
0answers
24 views

Regularity preserves under passing to some neighbourhood

Given a ring $A$ and an $A$-module $M$, an element in $A$ is said to be $M$-regular if it's not a zero divisor of $M$. A sequence $y_1,\dots,y_n$ of elements of $A$ is said to be $M$-regular if the ...
1
vote
1answer
36 views

$f,g \in k[t]$ with $k(f,g)=k(t)$, $\deg(f)=2$ and $\deg(g)=3$

Let $f=f(t),g=g(t) \in k[t]$, $k$ is a field of characteristic zero. Assume that the following two conditions are satisfied: (i) $\deg(f)=2$ and $\deg(g)=3$. (ii) $k(f,g)=k(t)$. Question: Is it ...
1
vote
0answers
46 views

When is $K[X_1,X_2,…,X_n] \to K[Y_1,Y_2,…,Y_m]$ a flat morphism

Let $K$ be a field and $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ a polynomial ring morphism. Assume $n, m \ge 2$. By definition $\varphi$ endows $K[Y_1,Y_2,...,Y_m]$ with a $K[X_1,X_2,...,...
1
vote
1answer
35 views

Miles Reid Commutative Algebra exercise 3.3 (Noetherian rings)

I would like how could one prove the following Let $R$ be a ring, $\mathfrak{a}_1,\dots,\mathfrak{a}_r$ ideals of $R$ such that each $R/\mathfrak{a}_i$ is a Noetherian ring. Then $\bigoplus_{i=1}^rR/...
0
votes
2answers
44 views

Use of “$A$ is a domain” in the proof that $Q$ is an injective $A$-module iff it is divisible

Let $A$ be a PID. Then, an $A$-module $Q$ is injective iff $Q=rQ$ for every $r\neq 0$ in $A$. My question is, where is the property "A is a domain" used in the proof of the above? Can someone please ...

1
3 4
5
6 7
252