Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
- 30.5k
2
votes
1
answer
118
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Dual of a semilinear morphism
Let $R$ be a commutative ring and let $M$ and $N$ be $R$-modules. Let $\sigma:R\rightarrow R$ be a ring automorphism.
Let $f: M\rightarrow N$ be a $\sigma$-semilinear map, i.e. a map of abelian ...
- 11.1k
0
votes
1
answer
465
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What is lim⟶ I^n M?
Let $R$ be a commutative ring, $I$ is an ideal of $R$, $M$ is an $R$-module.
$$IM\supset I^2M\supset I^3M\supset\cdots$$
What is $\mathop {\lim }\limits_{\begin{subarray}{c}
\longrightarrow \\
\...
- 259
6
votes
0
answers
217
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which automorphisms of a subring extend to those of a ring
(Probably a silly question, but..)
Consider the ring $R=k[[x_1,\dots,x_n]]/I$, (e.g. char(k)=0), and its subring, $R_1$, generated by some of $x_i$'s. In general, an automorphism of $R_1$ does not ...
- 2,232
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votes
1
answer
400
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ideal transform
Let $I$ be an ideal of a commutative ring $R$. $M$ be an $R$-module. In Local cohomology: an algebraic introduction with geometric applications of Brodmann M. P., Sharp R. Y we have
$$D_I(M)=\mathop {\...
- 259
1
vote
0
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131
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Rings with the property $\dim R-\dim R/p\leq \text{const}$ for all minimal $p$
I'm curious if there exists a class of rings generalizing quasi-unmixed rings. I guess a generalization of quasi-unmixed rings can be done as follows:
For a fixed integer $i$
$$\forall p\in\...
- 189
4
votes
1
answer
382
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"extend a functor"
Hi,
I have probably a basic question. I have a functor $F: Sch \rightarrow Set$, an algebraic stack $M$ with a "universal family" $G\rightarrow M$ and a representability property like this: for every ...
- 41
2
votes
2
answers
326
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Where did the multigraded Segre product appear in the literature?
Let $k$ be a field and $A\subset \mathbb{N}^d$ a vector configuration. Let $R,S$ be commutative $k$-algebras, both graded by the affine semigroup $\mathbb{N}A$. Is the 'multidgraded Segre product' $R ...
- 1,961
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1
answer
232
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What are sources for pathological and non-so-pathological Gabriel filters on commutative rings?
The heavy lifting in the theory of Gabriel filters is for noncommutative rings, and discussions I've been able to find all focus there.
I am trying to develop a theory of Gabriel-filter localization ...
- 403
3
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0
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461
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Krull dimension of non-integral extensions
Some hours ago, a question was posted, asking (citation by heart, not literally)
Let $R$ be a commutative ring (with unit). What can be said about the Krull dimension of an algebraic, non-integral ...
- 16.2k
0
votes
1
answer
251
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What is a certain cartesian product of algebras?
Suppose $F$ is a field and $A$ the $F$-algebra $F[X]\times_{(F\times F)} F$ given by the missing corner of a cartesian square in $F$-algebras
\begin{equation}
F~\xrightarrow{\Delta} ~F\times F~ \...
- 1
1
vote
0
answers
257
views
level of rings and stable range of rings
The level $s(A)$ of a ring $A$ with unity $1$ is the smallest natural number $s$
such that $-1$ is a sum of $s$ squares in $A$. (If $-1$ is not a sum of squares in $A$, we
say that $s(A) =\infty$.). ...
- 11
3
votes
1
answer
320
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Decision problem about the existence of solution for an integer matrix equation
Given $A,B,C \ $ integer matrices of dimensions $l \times m$, $l \times n$ and $l \times m$, we want to decide (algorithmically) about the existence of $X$ (unimodular) and $Y \ $ integer matrices ...
- 61
0
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1
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379
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Is a tensor product of two dvrs semilocal?
Under what conditions is the tensor product of two dvrs semilocal?
The same question about being reduced.
Tensor product is taken over another dvr or over a field to make things simpler.
For ...
- 1
2
votes
0
answers
506
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Zariski's Main theorem [closed]
Sir,
I am studying Zariski's Main theorem. May i know good source of problems related to Zariski's main theorem to understand it better?
Thanks in advance.
- 161
3
votes
0
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130
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On avoiding a linear subspace of an algebra
Let $R = \bigoplus_{n \geq 0} R_n$ be a standard graded algebra over an infinite field $K = R_0$. Suppose $depth \ R > 0$. Thus there exist linear forms which are non-zero divisors of $R$.
Let $a \...
7
votes
0
answers
518
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An elementary question in singularities
The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
- 3,758
2
votes
1
answer
651
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Splitting matrix of rank one
Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc
Let A=(a i j ) a matrix with elements in R and dimension n x m.
Suppose
rank A=1 ↔ all 2 x ...
5
votes
0
answers
817
views
morphism which is open but not universally open
In someone's note, I have seen such an example, but I can't show that it is not universally open. Here is the example:
Let $k$ be a field and $A = k[T]_{(T)}$, the discrete valuation ring obtained ...
- 1,109
3
votes
1
answer
495
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universal finite differential module of affinoid algebra
Let $k$ be a value field (archimedean), for example $k = \mathbb{Q}_p$, the p-adic field.
The free Tate algebra is $$ T_n := \left\{ \ \sum a_I X^I, \ a_I \in k, \ a_I \rightarrow 0 \text{ as } |I| \...
- 1,109
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0
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544
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isomorphism between vector spaces and modules - Commutative Algebra
Hi, Let $M_i$ be A modules. Then we know that $Ass (\oplus M_i) = \bigcup Ass(M_i) $. We consider here isomorphisms between modules.
Now consider a stanley ...
- 287
4
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0
answers
188
views
A non-matroidal notion of dependence on a set of ideals
Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ ...
- 1,961
3
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0
answers
141
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Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$. Is then $F\cap k[x_1,\dots,x_n]=k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n]$ for polynomials $f_i$?
Let $F\subseteq k(x_1,\dots,x_n)$ be a subfield with $k\subseteq F$. I know that $F=k(\psi_1,\dots,\psi_r)$ for rational functions $\psi_i\in k(x_1,\dots,x_n)$. I'm interested in the intersection $F\...
- 339
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votes
1
answer
177
views
Laurent series with analytic coefficients
Let $A=H(D(0,1))$ the ring of holomorphic functions on the open unity disc.
I consider the function $f$:
$$f (t)=\sum f_{i}t^{i} \in A[[t]]$$
I suppose that the $t$-adic valuation of it is less or ...
- 3,472
3
votes
1
answer
191
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Local coordinate system under finite integral extension
Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras (of the same dimension) which makes $B$ integral over $A$.
Let $\mathfrak{m}=(x_1,\ldots,...
- 4,902
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0
answers
383
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Size of an abelian permutation group with generators of order 2 [closed]
Let $g_1, \ldots, g_k$ be distinct permutations on a set $\Omega$. Suppose that $G = \langle g_1, \ldots, g_k \rangle$ is an abelian permutation group with only elements of order at most 2. Is it ...
- 11
2
votes
1
answer
255
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What does the d-slice of a weighted polynomial algebra look like?
This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves.
...
- 33.8k
3
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answers
591
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Algebraic description of double vector bundles.
It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated ...
- 51
2
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1
answer
345
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A weaker form of Zariski's connectedness principle
Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ ...
- 10.9k
8
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494
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"Consecutive" irreducible polynomials
If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then
it is easy to see that for any integer $m$, at least one of the polynomials
$P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb Z}...
- 3,595
1
vote
0
answers
233
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projective dimension of finitely generated modules over char 0
In the paper "On modules of finite projective dimension over complete intersection" Dutta proved that a finitely generated module over a local complete intersection ring over char $p>0$ has finite ...
- 2,444
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votes
0
answers
243
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strict henselian and excellent henselian
Hello, everyone. I want to ask a problem about strict henselian ring.
Let $A$ be a strict henselian DVR.
Dose there exist subrings $A_{i}$ of $A$, such that $A=lim_{i} A_{i}$ and where $A_{i}$ are ...
- 1,921
1
vote
1
answer
518
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Cardinality of a linear independent subset of a free module over a commutative ring which is not an integral domain
If R is a commutative ring with unity and not an integral domain and F is a free R-module with rank k,is there a linear independent set with cardinality > k?
I prooved that this is not true if R is an ...
- 345
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0
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68
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Let $(R, m)$ be noetherian local, $\dim(R)=1$. Show $CH^1(R)=\mathbb{Z}/(\gcd([k_i, R/m]))$, where $k_i$ are residue fields of normalization
Let $R$ be a $1$-dimensional noetherian local domain. Then we have that $CH^1(R)=\mathbb{Z}/(\gcd([k_i, k]))$ where the $k_i$ are residue fields of the normalization and $k$ is the residue field of $R$...
- 841
3
votes
0
answers
313
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Tor over non-noetherian local ring
Let $(A,m,k)$ be a local ring and let $M$ be a finite torsion $A$-module.
Is ${\rm Tor}^A_1(M,k)$ finite over $k$?
I am aware that the conclusion holds for any finite $A$-module $M$ when $A$ in ...
- 53
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0
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417
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Absolute Irreducibility in Characteristic 2
Let $\mathbb F$ be a field and $\mathbb F[x_1,\dotsc,x_n]$ the ring of multivariate polynomials in $n$ variables over $\mathbb F$. A polynomial $P\in\mathbb F[x_1,\dotsc,x_n]$ is said absolutely ...
- 456
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1
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183
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relation between Min(R) and Min(R^)
Let $\hat{R}$ is $m$-adic completion of a local ring $(R,m)$.What is the relation between $Min R$ and $Min \hat{R}$. we know that $\hat{R}$ is faithfully flat $R$-module.
$Min R$=set of all minimal ...
- 418
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0
answers
189
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Maps of free modules over a ring [closed]
(This is exercise 10 of chapter 2 of Atiyah and Macdonald.)
The exercise starts by asking me to prove that if $A^n\cong A^m$ then $n=m$ for any nonzero ring A. I managed to do that (by tensoring with ...
- 639
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0
answers
102
views
semicontinuity of the conductor defined by Temkin
We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...
- 3,472
1
vote
2
answers
364
views
Rig of fractions, including zero denominators
For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect ...
- 1,872
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0
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797
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How can I prove R[x] is integrally closed iff R is integrally closed ? (R: integral domain) [closed]
$R$ is a integral domain. How can I prove $R[x]$ is integrally closed iff $R$ is integrally closed?
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79
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Saturation of a subalgebra over the Tate-algebra inside the power series ring
Let $A$ be a discrete valuation ring and $\pi$ a uniformizer.
Over $A$ we consider the Tate-algebra
$$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert a_n\...
2
votes
0
answers
363
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A simple problem on commutative algebra related to G.I.T
Let $G$ be a geometrically reductive algebraic group over an algebraically closed field $k$. Let $X$ be an affine variety over $k$ on which $G$ acts regularly. Then $G$ acts on the coordinate ring $A$ ...
- 1,804
3
votes
1
answer
590
views
Adjunction for underlying reduced subschemes
Let $k$ be a perfect field (so reduced = geometrically reduced) and $f:X\rightarrow \mathrm{Spec}(k)$ a Cohen-Macaulay morphism. Denote by $i:X_{red}\rightarrow X$ the underlying reduced subscheme ...
- 1,609
4
votes
1
answer
210
views
Explicitly generating 1 in an ideal without prime support
The Question
Let $R$ be a unital commutative ring, and let $a,b_1,b_2\in R$. The following is a basic commutative algebra exercise.
Lemma. If $Ra+Rb_1=R$ and $Ra+Rb_2=R$, then $Ra+Rb_1b_2=R$.
Proof. ...
- 13k
0
votes
0
answers
123
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Irreducibility of superelliptic curves
Let $k$ be an algebraically closed field of characteristic zero, let $a,d$ be integers, and let $f\in k[x]$ be a separable polynomial of degree $d$.
Question: a) Is the affine plane curve $y^a=f(x)$ ...
- 23
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0
answers
73
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K-Theory and completion [duplicate]
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the $\...
- 165
1
vote
0
answers
136
views
de Rham complex of closed immersion between smooth schemes
Hi,
Let $R$ be a $\mathbb Q$-algebra and let $P$ and $Q$ be (EDIT: smooth) $R$-algebras such that there is a surjective
map of $R$-algebras $Q\to P$. The following proof cannot possibly be correct, ...
- 2,842
3
votes
0
answers
474
views
Jacobson-Bourbaki correspondence
The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications of ...
3
votes
0
answers
473
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Infinite Galois correspondence "according to Artin"
Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and ...