Questions tagged [commutative-algebra]
Questions about commutative rings, their ideals, and their modules.
12,417
questions
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23 views
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 ...
0
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0answers
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 ...
3
votes
0answers
33 views
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
$$\text{Tor}^R(\...
0
votes
0answers
21 views
if $\operatorname{Dim}(A)=0$, then every prime ideal is maximal [duplicate]
Given a ring $A$, By the chain of prime ideals of $A$, we mean a sequence of prime ideals of $A$ such that $P_0 \subsetneq P_1 \dots \subsetneq P_{n-1}\subsetneq P_n$. Further this chain has a length $...
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0answers
23 views
Motivation for separated and proper schemes
Hartshorne mention at the beginning of section 4 in chapter 2 that the definition of separated is similliar to hausdorff. We all can see that. That is also what I found in google. Again - we all can ...
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0answers
39 views
The completion of localization
This is the example 5.6.3 in chapter I of Hartshorne's Algebraic Geometry. For the reducible variety
$$
Y = \Big\{ (x,y) \in {\mathbb{A}}^2 ~:~ xy = 0 \Big\}
$$
it follows that the local ring ${\...
1
vote
1answer
32 views
Depth of $R/I$ as an $R$-module versus as a ring
Let $(R, \mathfrak m,k)$ be a Noetherian local ring. Let $I\subseteq \mathfrak m$ be an ideal of $R$. Then $(R/I, \mathfrak m/I, k)$ is a Noetherian local ring but also $R/I$ is a finitely generated $...
0
votes
1answer
36 views
Weil divisors associated to Cartier divisors
Let $X=\{x_3^2=x_1^2+x_2^2\}\in \mathbb{P}^3$, let $L_1=Z(x_2,x_1+x_3)$, and $L_2=Z(x_2,x_1-x_3)$.
I don't quite understand how to get that $\operatorname{div}(x_2)$ is associated to $[L_1]+[L_2]$ ...
2
votes
1answer
47 views
Cohomological criterion for non-triviality of negative part of graded module
Let $R$ be a graded ring and $M$ a graded module. Then for sufficently large $n$, we have
$$H^0(\operatorname{Proj}(R), \widetilde{M}(n))\cong M_n.$$
Hence if I want to show that $M_{>0}$ is non-...
1
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1answer
19 views
Vector spaces over an integral domain and the canonical isomorphism between the tensor products
Let $A$ be an integral domain and write $S=A-\{0\}$. Then the total ring of fractions $S^{-1}A$ of $A$ is an abelian field. Note that $\varepsilon:A\rightarrow S^{-1}A,\,a\mapsto a/1$, is an injective ...
0
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0answers
44 views
Struggling while learning commutative algebra
I took a course in abstract algebra (till galois theory), topology (with some very basic algebraic topology), smooth manifolds, complex analysis and never did I struggle even epsilon close to how I am ...
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0answers
33 views
A stalk of the direct image of a finite morphism
Let $f : X \to Y$ be a finite morphism of schemes and $y \in Y$.
Then
$$(f_* \mathscr{O}_X)_y \cong \bigoplus_{f(x) = y} \mathscr{O}_{X, x}.$$
How can I show it?
Or equivalently,
Let $\...
1
vote
1answer
15 views
A module annihilated by a maximal ideal is semisimple
I want to show that if $M$ is a module over a commutative ring $R$ that is annihilated by a maximal ideal $I$ of $R$, then $M$ is a semisimple $R$-module. What I have in mind is the following: if $M$ ...
0
votes
0answers
24 views
“Understanding” the techniques involved in this proof that $F[x, y]/\langle y^2 - x^3 \rangle$ is a domain and is not integrally closed
Let $A = F[x,y] / \langle y^2 - x^3 \rangle$. In a past exam question (from last year), students were asked to show that $A$ is an integral domain, and also that it is not integrally closed in its ...
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0answers
29 views
Computing whether a set of polynomials cuts out a homogeneous variety
I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a homogeneous variety. My first idea is to compute the radical of the ideal $I$ that they ...
1
vote
1answer
52 views
extending ring homomorphism into fields
Let $A$ be a subring of $B$ such that $B$ is integral over $A$.
Show that every ring homomorphism $f:A\rightarrow K$ with $K$ an
algebraically closed field can be extended to a ring ...
1
vote
1answer
21 views
an irreducible affine curve is normal if and only if it is nonsingular
An is normal if and only if it is nonsingular.
This statement comes from Kemper, A Course in Commutative Algebra. He says to use Proposition 8.10 and Theorem 14.1.
Theorem 14.1. A Noetherian local ...
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0answers
24 views
Intersection of $I$-adic neiborhoods equal to intersection of kernel of localization map
Let $R$ be a noetherian ring and $I$ is an ideal. Let $M$ be a finite $R$-module. I want to show
$$\bigcap_{n=0}^\infty I^nM=\bigcap_{I\subseteq \mathfrak{m}}\ker(M\to M_\mathfrak{m})$$
where $\...
1
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0answers
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)}\...
1
vote
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 ...
2
votes
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 ...
1
vote
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 $...
4
votes
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.
0
votes
3answers
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 ...
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
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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0answers
52 views
+50
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
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0answers
23 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
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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. ...
0
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0answers
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$ ...
2
votes
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, ...
1
vote
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 ...
1
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0answers
27 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 ...
1
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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: ...
0
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0answers
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\}...
1
vote
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 ...
1
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0answers
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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0answers
54 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
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 ...
1
vote
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 ...
0
votes
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 \{\...
1
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0answers
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.
2
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0answers
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 ...
0
votes
0answers
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$.
1
vote
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
2
votes
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$-...
1
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0answers
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 ...
1
vote
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 $C(X,\mathbb{R})$ ...
