Questions tagged [commutative-algebra]

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

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Koszul algebras

Why we are interested to find whether an algebra is Koszul or not? What is the significance of being Koszul? Only this that we will know certain homological invariants of the algebra like projective ...
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17 views

Noetherian commutative ring with finite but not discrete spectrum

I know this is probably not that hard but I don't know how to properly approach this. So I am asked to give an example of a ring fulfilling the properties in the title of the question. Now I know ...
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Ext and Tor duality

In the appendix of this paper of Felix, Halperin and Thomas, Proposition $A.6$ is the following: Let $R$ be a differential graded algebra, and let $M$ be an $R$-module. Then $$\text{Tor}^R(\...
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if $\operatorname{Dim}(A)=0$, then every prime ideal is maximal [duplicate]

Given a ring $A$, By the chain of prime ideals of $A$, we mean a sequence of prime ideals of $A$ such that $P_0 \subsetneq P_1 \dots \subsetneq P_{n-1}\subsetneq P_n$. Further this chain has a length $...
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23 views

Motivation for separated and proper schemes

Hartshorne mention at the beginning of section 4 in chapter 2 that the definition of separated is similliar to hausdorff. We all can see that. That is also what I found in google. Again - we all can ...
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39 views

The completion of localization

This is the example 5.6.3 in chapter I of Hartshorne's Algebraic Geometry. For the reducible variety $$ Y = \Big\{ (x,y) \in {\mathbb{A}}^2 ~:~ xy = 0 \Big\} $$ it follows that the local ring ${\...
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1answer
32 views

Depth of $R/I$ as an $R$-module versus as a ring

Let $(R, \mathfrak m,k)$ be a Noetherian local ring. Let $I\subseteq \mathfrak m$ be an ideal of $R$. Then $(R/I, \mathfrak m/I, k)$ is a Noetherian local ring but also $R/I$ is a finitely generated $...
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1answer
36 views

Weil divisors associated to Cartier divisors

Let $X=\{x_3^2=x_1^2+x_2^2\}\in \mathbb{P}^3$, let $L_1=Z(x_2,x_1+x_3)$, and $L_2=Z(x_2,x_1-x_3)$. I don't quite understand how to get that $\operatorname{div}(x_2)$ is associated to $[L_1]+[L_2]$ ...
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1answer
47 views

Cohomological criterion for non-triviality of negative part of graded module

Let $R$ be a graded ring and $M$ a graded module. Then for sufficently large $n$, we have $$H^0(\operatorname{Proj}(R), \widetilde{M}(n))\cong M_n.$$ Hence if I want to show that $M_{>0}$ is non-...
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1answer
19 views

Vector spaces over an integral domain and the canonical isomorphism between the tensor products

Let $A$ be an integral domain and write $S=A-\{0\}$. Then the total ring of fractions $S^{-1}A$ of $A$ is an abelian field. Note that $\varepsilon:A\rightarrow S^{-1}A,\,a\mapsto a/1$, is an injective ...
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Struggling while learning commutative algebra

I took a course in abstract algebra (till galois theory), topology (with some very basic algebraic topology), smooth manifolds, complex analysis and never did I struggle even epsilon close to how I am ...
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33 views

A stalk of the direct image of a finite morphism

Let $f : X \to Y$ be a finite morphism of schemes and $y \in Y$. Then $$(f_* \mathscr{O}_X)_y \cong \bigoplus_{f(x) = y} \mathscr{O}_{X, x}.$$ How can I show it? Or equivalently, Let $\...
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1answer
15 views

A module annihilated by a maximal ideal is semisimple

I want to show that if $M$ is a module over a commutative ring $R$ that is annihilated by a maximal ideal $I$ of $R$, then $M$ is a semisimple $R$-module. What I have in mind is the following: if $M$ ...
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24 views

“Understanding” the techniques involved in this proof that $F[x, y]/\langle y^2 - x^3 \rangle$ is a domain and is not integrally closed

Let $A = F[x,y] / \langle y^2 - x^3 \rangle$. In a past exam question (from last year), students were asked to show that $A$ is an integral domain, and also that it is not integrally closed in its ...
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Computing whether a set of polynomials cuts out a homogeneous variety

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a homogeneous variety. My first idea is to compute the radical of the ideal $I$ that they ...
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1answer
52 views

extending ring homomorphism into fields

Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Show that every ring homomorphism $f:A\rightarrow K$ with $K$ an algebraically closed field can be extended to a ring ...
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1answer
21 views

an irreducible affine curve is normal if and only if it is nonsingular

An is normal if and only if it is nonsingular. This statement comes from Kemper, A Course in Commutative Algebra. He says to use Proposition 8.10 and Theorem 14.1. Theorem 14.1. A Noetherian local ...
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Intersection of $I$-adic neiborhoods equal to intersection of kernel of localization map

Let $R$ be a noetherian ring and $I$ is an ideal. Let $M$ be a finite $R$-module. I want to show $$\bigcap_{n=0}^\infty I^nM=\bigcap_{I\subseteq \mathfrak{m}}\ker(M\to M_\mathfrak{m})$$ where $\...
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42 views

Bijection between $\text{Spec}(A\otimes_R B)$ and $\text{Spec}(A)\times_{\text{Spec}(R)}\text{Spec}(B)$ for $R$-algebras A and B.

I'm looking for a bijection between $\text{Spec}(A\otimes_R B)$ and $\text{Spec}(A)\times_{\text{Spec}(R)}\text{Spec}(B)$ for $R$-algebras $A$ and $B$. Where $\text{Spec}(A)\times_{\text{Spec}(R)}\...
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1answer
43 views

Why in a von Neumann regular ring do we have that $ax(1+x)=1+x?$

Why is it true that in a commutative von Neumann regular ring, we have that $ax(1+x)=1+x?$ Definition: We say that a unital ring $R$ is von Neumann whenever for every $a \in R,$ there exist an $x \in ...
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1answer
60 views

Is $k[x,xy] \subseteq k[x,y]$ a flat ring extension?

Let $k$ be a field of characteristic zero. Is $k[x,xy] \subseteq k[x,y]$ a flat ring extension? I guess that the answer is no? Though I am not sure how to prove this. Perhaps applying this ...
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1answer
22 views

Examples where $1+I$ is inverted but $I$ is not mapped into the Jacobson radical

Let $f:A\to B$ be a commutative ring morphism. Let $I\vartriangleleft A$ be an ideal. If $f(I)\subset \mathrm J(B)$ is contained in the Jacobson radical then $f(1+i)=1+f(i)\in B^\times$ is a unit so $...
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2answers
65 views

Example of the non-commutative ring with the set of units are commutative

I was looking for an Example of the non-commutative ring with the set of units are commutative. it will be a great help. Thanks in advance.
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57 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 ...
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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 $B$ ...
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1answer
51 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 $\mathbb{C}$, ...
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52 views
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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)...
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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{...
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1answer
29 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. ...
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48 views
+50

Global section of vector bundle of rank $1$ on punctured spectrum

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth $\ge 2$. Consider the punctured spectrum $U=\operatorname{Spec}(R)\setminus \{\mathfrak m\}$ . Let $\mathcal L$ be an invertible sheaf on $U$ ...
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1answer
23 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, ...
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1answer
35 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 ...
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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 ...
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1answer
42 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: ...
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111 views
+50

Global-section functor distributes over tensor product of vector-bundles on punctured spectrum?

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth $\ge 2$. Let $\mathcal F, \mathcal G$ be Algebraic vector bundles on the punctured spectrum $U=\operatorname{Spec}(R)\setminus \{\mathfrak m\}...
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1answer
76 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 ...
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46 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?
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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 ...
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1answer
49 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 ...
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1answer
57 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 ...
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1answer
36 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 \{\...
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40 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.
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68 views
+50

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 ...
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39 views

$A/I$ is flat if and only if $I=0$ [closed]

Let $A$ be a domain and $I$ own ideal. $A/I$ is flat if and only if $I = 0$.
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1answer
63 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 ...
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1answer
43 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 ...
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41 views

Find two irreducible polynomials such that their intersection is disconnected. [closed]

Let $k$ be an algebraically closed field, $R=k[x,y,z].$ Find two irreducible polynomials $f,g\in R$ such that $V(f)\cap V(g)$ is disconnected where $V(f)=\{P\in Spec(R)|f\in P\}$. My personal ...
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1answer
29 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 $A$-...
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52 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 ...
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0answers
66 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 $C(X,\mathbb{R})$ ...

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