Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
8
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Doing explicit computations with coordinate rings
Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of ...
- 3,918
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votes
1
answer
2k
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The space of valuations of a function field
Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations.
First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of ...
- 1,804
6
votes
2
answers
541
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Positive matrices matrices over commutative rings
Assume that $R$ is a commutative ring with a ring compatible ordering and let $A$ and $B$ be symmetric $n\times n$ matrices with entries in $R$ such that $\sum x_iA_{ij}x_j\geq 0$ and $\sum x_iB_{ij}...
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2
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511
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When are seminormal rings Cohen-Macaulay?
I know that not every local seminormal ring is Cohen-Macaulay. But are 1-dimensional local seminormal rings Cohen-Macaulay?
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3
votes
1
answer
307
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operations on ideals in a subring of number field
For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$,
does this equality hold in general?
$(I+J) \cap K = (I \cap K) + (J \cap K)$
I have no counterexample yet but I couldn't prove ...
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2
votes
1
answer
248
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A class of rings related to rings with IBN property [closed]
A ring $R$ (with 1) has IBN property if free $R$-modules have unique rank (e.g., commutative rings). In the same fashion, lets call $R$ a good ring if in every free $R$-module any independent set can ...
user39121
6
votes
0
answers
3k
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Tensor product of two algebras [closed]
Let $A$ and $B$ be algebras over a field $K$. The ideals of the tensor product $A\bigotimes_K B$ are of the form $I\bigotimes_K J$ where $I$ is an ideal of $A$ and $J$ is an ideal of $B$?
- 545
2
votes
1
answer
404
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Module structure of the abelianization of the commutator subgroup
Let $G$ be a (non-abelian) group, and let $G_2$ denote its commutator subgroup. Then the abelianization $G_2^{ab} = H_1(G_2,\mathbb{Z})$ is a module over the group ring $\mathbb{Z}[G^{ab}]$. The ...
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votes
1
answer
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checking if F[x]/I is isomorphic to F[x]/J
Let $F$ be a field. Let $p$ and $q$ be monic members $F[x]$. Let $I = \{p\cdot r : r\in F[x]\}$ and $J = \{q\cdot r : r\in F[x]\}$. I know that if $F[x]/I$ is isomorphic to $F[x]/J$ then ($\...
user5810
4
votes
0
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225
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Intersection numbers on blow ups of toric varieties
Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
- 357
1
vote
1
answer
292
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Lifting a direct summand of a free module
[EDIT]: After getting a nice counter example provided by Steven Landsburg I realize that I forgot to impose an important condition...namely $R$ is supposed to be complete w.r.t. the $I$-adic topology. ...
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6
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1
answer
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Seeking examples or proof: injectivity of Cartan homomorphism for commutative rings?
This question is motivated by some issue raised by David Speyer in this question.
Let $R$ be a ring. Let $K_0(R)$ and $G_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules ...
- 30.5k
4
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0
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Existence of algebraic closure and Axiom of choice [duplicate]
Possible Duplicates:
Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?
algebraic closure of commuting pairs of matrices
we need zorn's ...
- 1,788
0
votes
3
answers
565
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Homology of koszul complex is finitely generated?
$A$ a local ring and $a_{1}$, ..., $a_{n}$ elements in its maximal ideal, $M$ a finitely generated $A$-module. In this case apparently the homologies from the Koszul complex are finitely generated as $...
- 2,857
2
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1
answer
652
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Searching for polynomials with squarefree discriminant
In connection with the Galois theoretic results surrounding the irreducibility of $f(x)= x^{N}-x-1$ over $\mathbb{Q}$, I've been trying to prove for a while that the discriminant of $f$ is actually ...
- 625
5
votes
1
answer
458
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Ideals of $C(X)$ with only finitely many number of zerosets
We denote the rings of all real valued continuous functions on compeletely regular Hausdorff space $X$ by $C(X)$.Let $I$ be a ring ideal of $C(X)$. define $$Z[I]:=\lbrace Z(f):\;f\in I\rbrace$$ where ...
- 1,788
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votes
1
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826
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Rings with finitely generated nilradical
Let $\mathfrak{a}$ be a monomial ideal in a polynomial algebra over some commutative ring $R$. If $R$ is reduced, then the radical $\sqrt{\mathfrak{a}}$ of $\mathfrak{a}$ is again a monomial ideal, ...
- 6,700
2
votes
0
answers
171
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Irreducibility of a general fibre
Let $A\subseteq B$ be an inclusion of affine domains over an algebraically closed field $k$ of characteristic $0$. Can someone give me a reference for the following fact?
If $A$ is algebraically ...
1
vote
1
answer
222
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composition of Puiseux series?
Formal series over a field (or ring) k can be composed in the following sense: given $y \in k[[x]]$, $y$ lying in the maximal ideal, there exists a unique map of topological rings $k[[x]] \to k[[x]]$ ...
- 4,070
5
votes
2
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Structure theorem of f.g. modules over a (non) PID
I am looking for an example of a commutative ring with $1$, in which every ideal is generated by a single element, for which the conclusion of the structure theorem for finitely generated modules is ...
- 11.1k
1
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2
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574
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commutative rigs and the Grothendieck Group
If I start with a commutative rig, and apply the Grothendieck Group construction to it, twice, once to the additive structure and once to the multiplicative structure, is the result well-known? Does ...
- 11.8k
5
votes
1
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module of differentials of formal power series ring and of its field of quotiens
For any $A$-algebra $B$ ( commutative ring with 1 ), we have the existence of $\Omega_{B/A}$, the module of relative differentials of $B$ over $A$, which can be defined by an universal property. In ...
- 1,109
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votes
2
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206
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h^0 of a sheaf supported at a point
$C$ : a projective curve over an algebraically closed field, $I$ : the ideal sheaf of $\mathcal{O}_C$ defining a point $p\in C$.
Why is the following hold?
$h^0(C,(\mathcal{O}_C/I^{r})\otimes\mathcal{...
- 1
1
vote
0
answers
132
views
Degrees of polynomials defining a Jacobian of maximal rank on a variety
Let $f_1,\ldots,f_{n-k} \in \mathbb{R}[x_1,\ldots,x_n]$ be polynomials of degree at most $d$ defining an algebraic set $A \subseteq \mathbb{C}^n$ which contains an irreducible component $V \subseteq A$...
- 121
1
vote
0
answers
260
views
Ideals in a reduced ring
In a finite reduced commutative ring, is every minimal prime ideal is generated by an idempotent?
The number of idempotent elements and the number of minimal prime ideals are same.
- 11
4
votes
0
answers
94
views
A dimension condition on the cohomology of a homogeneous space
The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...
- 2,995
1
vote
0
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106
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Upper bound for the minimum number of generators of the canonical module
Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$.
The question is that is there any upper bound for the ...
4
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2
answers
308
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When is a torsionfree subgroup contained in a torsionfree direct summand?
Let $F$ be a torsionfree subgroup of a commutative group $G$. Are there nontrivial conditions known under which there exists a torsionfree direct summand of $G$ containing $F$?
I would already be ...
- 6,700
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0
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520
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Getting a bound via polynomial equations
When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$,
\begin{cases}
&\sum\limits_{j=0}^{m-1}x_jx_{2k-j}=...
- 767
2
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1
answer
874
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Automorphism theorem
Help me please to find reference for the proof of the following theorem:
Theorem. Let $\theta$ be a Leibniz cocycle on the Leibniz algebra L with values in $V,$ and assume $\theta^{\bot}\cap C(L)=(0)....
- 21
2
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1
answer
203
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Vertices of a polytope as algebra generators
I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...
- 1,488
3
votes
2
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415
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Is $E_R(k) = E_{\hat{R}}(k)$?
Let $(R, m, k)$ be a Noetherian local ring. Let $E_R(k)$ be the injective hull of $k$ as an $R$-module. It is well known that $E_R(k)$ is Artinian and is an $(R, \hat{R})$-bimodule (see, Brodmann-...
- 2,773
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2
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673
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are deformations of torsion modules always torsion?
Let's say I have a field $\mathbb{K}$ and a flat family of $\mathbb{K}[t]$-modules $M$ over the formal disk $Spec \mathbb{K}[[h]]$.
Now, assume that $M/hM$ is torsion as a $\mathbb{K}[t]$-module (...
- 44.7k
4
votes
1
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Uncountable Reduced ring $R$ with $R[x]$ has only a countable number of maximal left ideals
The question is following:
Is there an uncountable reduced ring (i.e., a ring with no non-zero nilpotent element) $R$ (with identity) such that
$R[x]$ has only a countable number of maximal left ...
user39982
1
vote
1
answer
118
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question about a particular Polynomial ring [closed]
Let K be a field, let $T = K[X_1, X_2,...]$ be a polynomial ring, let $R=K[X_1^{2}, X_1X_2,..,X_i X_j,..]$, and let $L = Frac(R)$ = field of fractions of R. How can we prove that $R =T \cap L$ ?
- 13
2
votes
1
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138
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Relation between intersection multiplicities
Consider $(f_1,\dots,f_n), (g_1,\dots,g_n)\in \mathbb{C}[z_1,\dots,z_n]\ $ such that:
i) $\{f_1=\dots=f_n=0\}= \{g_1=\dots=g_n=0\}=\{0\}\in \mathbb{C}^n\ $ and
ii) $f_1g_1+\dots+f_ng_n\equiv0$.
...
- 93
2
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1
answer
284
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Dimension of An Ultraproduct Field as a Vector Space over Another Ultraproduct Field
Suppose $F \subseteq K$ are fields with $G$ an ultrafilter on an infinite set $X$. If $F^{\ast}$ and $K^{\ast}$ represent the ultraproducts respectively of $F$ and $K$, it is easy to see that $[K : F]...
- 43
2
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1
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683
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Structure theorem for etale maps
I have been trying to understand the structure theorem for etale maps for rings. Let $A\to B$ be a local homomorphism of Noetherian local rings which is etale. Then the structure theorem says that $B=...
- 1,563
1
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2
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301
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Galois extension of a semi-local ring
Hi, i would like to know if weather or not a Galois extension of a commutative semi-local ring is also a semilocal ring.
- 11
4
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2
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What are non-trivial examples of non-singular blow-ups of a non-singular variety?
This question arose from the responses to this question. The references to the comments of Karl Schwede and VA are to comments made there.
The blow-up of the variety $X=\mathbb{A}^2$ along the ...
- 3,284
6
votes
2
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850
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Decomposition of finite algebras over finite fields
Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...
- 32.6k
1
vote
3
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466
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Can you compute the quotient set below?
Let $K$ be a field of characteristic $2$ ($2$ is very important in the statement -- otherwise I can do it myself :) ). On the set $K \times K$ we define the following equivalent relation: $(a, b) \...
-2
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1
answer
187
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behavior of multiplicity in exact sequences
Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions:
Question1. Many concepts in commutative algebra have ...
- 1,355
2
votes
2
answers
406
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Extending a polynomial function from an open subset
I am a bit embarrassed to ask this question, but still: assume that I have
a finite morphism $\pi:X\to Y$ of affine algebraic varieties over a field (probably
finiteness is too strong an assumption, ...
- 7,037
-1
votes
1
answer
1k
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Correspondence between submodules and quotient modules
What is the (natural) bijection between the isomorphic class of sub modules and isomorphic class of quotient modules of a finitely generated torsion module over a PID.
Is there any inclusion relation ...
- 1,269
0
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0
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549
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Fitting ideal sheaves and determinant bundles
I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...
- 5,857
5
votes
4
answers
388
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Familiar equations in more general settings
What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:
...
- 3,612
0
votes
1
answer
630
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Useless question on rank
What is the rank of $A^{n}$ if A is the zero ring? It's clearly not $n$ as many careless authors claim, since it's not even invariant. I don't think it's 0 either because it does have a linearly ...
- 2,857
5
votes
0
answers
401
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Epimorphisms between external tensor products
Let $k$ be a field, $R,S$ commutative $k$-algebras. If $\mathsf{Mod}_{fp}$ denotes the category of finitely presented modules, the external tensor product $\mathsf{Mod}_{fp}(R) \otimes_k \mathsf{Mod}_{...
- 63.1k
1
vote
2
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703
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Is a reduced, torsion-free module of finite rank over an Henselian ring free?
Let $R$ be an Henselian discrete valuation ring with field of fractions $K$. Let $M$ be a torsion-free $R$-module of finite rank (i.e. $dim_K(M\otimes_RK)<+\infty$). Let $D$ be the maximal ...
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