Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,316
questions
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Can states on commutative Banach algebras be understood as probability measures?
Suppose $\mathcal{A}$ is a commutative Banach algebra (over $\mathbb{R}$) or commutative Banach *-algebra (over $\mathbb{C}$). Is there always a measurable space $(\Omega,\mathcal{F})$ such that there ...
4
votes
1
answer
994
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An fppf cover with trivial Picard group
What is an example of a scheme $X$ for which there does not exist any morphism $f : Y \to X$ which is faithfully flat and locally of finite presentation and where $\operatorname{Pic} Y = 0$?
Remark: ...
1
vote
0
answers
60
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$Ext^i(D(R),R)$ for a certain commutative algebra
Let $k$ be a field which is not algebraic over a finite field and $a \in k$ an element of infinite multiplicative order.
Let $R=k[V,X,Y,Z]/I$ with $I=<V^2,Z^2,XY,VX+aXZ,VY+YZ,VX+Y^2,VY-X^2>.$
...
7
votes
0
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335
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Short exact sequence in nonabelian group cohomology and finitness condition
Let $1\to A\to B\to C\to 1$ be an exact sequence of (nonabelian) $G$-groups. Then there is a well-known exact sequence of pointed sets
$
1\to A^G\to B^G\to C^G\to H^1(G,A)\to H^1(G,B)\to H^1(G,C)
$
...
1
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0
answers
124
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Algebra Invariants of Schubert Calculus
For the Grassmannian Gr[N,k] of $k$-planes in $\mathbb{C}^N$, the cohomology ring $H^*(Gr[N,k])$ is a much studied object in an area called Schubert calculus. As a complex algebra, $H^*(Gr[N,k])$ is ...
4
votes
0
answers
142
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Deminormal and Gorenstein
Let X be an irreducible deminormal variety such that the normalisation is Gorenstein. Does it follow that X is also Gorenstein?
for deminormal definition, see https://arxiv.org/pdf/1506.02002.pdf
0
votes
1
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350
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Strange subscheme in ${\mathrm{Spec}} R \times {\Bbb A}^1_{\Bbb C}$
Let ${\Bbb C}[X_1,\ldots,X_n]$ be a $n$-variable polynomial ring over a complex number field ${\Bbb C}$. For its maximal ideal $(X_1,\ldots,X_n)$, we define the geometric regular local ring as
$R \...
3
votes
1
answer
178
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Definability of nilradical in the model theory of rings
I am looking for a reference dealing with the first-order definability of the nilradical of a commutative ring. The only thing I have found so far is an exercise in Wilfrid Hodges' book Model Theory (...
2
votes
0
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252
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Proof of this ‘lemme connu’
In the proof of Corollary 10.12 of Exposé I of SGA 1 something like the following is asserted as a ‘known lemma’:
Let $k$ be an infinite field and $B$ a finite $k$-algebra. If $B$ is not a product ...
3
votes
1
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525
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On integral domains over which special kind of modules are projective
For an integral domain $R$ let $\mathrm{Frac}(R)$ denote its field of fractions. Then $R$ is embedded in $\mathrm{Frac}(R)$ and we can consider $\mathrm{Frac}(R)$ as an $R$-module.
Can we ...
1
vote
1
answer
118
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Basic elements and localizations
Let $(R, \mathfrak{m})$ be a local domain and $x$ is a basic element of $\mathfrak{m}$, that is $x \in \mathfrak{m} \setminus \mathfrak{m}^2$. Let $P$ be a prime ideal containing $x$. Is it true that $...
6
votes
1
answer
423
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Morita equivalence and isomorphisms in cohomology theories
Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that
$$P \otimes_{B} Q \cong A, Q \otimes_A P \cong B$$
(as $A-A$ and $B-B$ ...
14
votes
0
answers
1k
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A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
12
votes
1
answer
720
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Determinant is to Pfaffian as resultant is to what?
This is an irresponsible question: I do not have done any thinking on it, or even literature search.
I just became curious whether there is some modification of the notion of a common root of two ...
2
votes
0
answers
74
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Analytic spread of an ideal after reduction
Let $(R,m)$ be a local ring and $I$ an ideal in $R.$ Let $l(I):=\dim \bigoplus_{n\geq 0}(I^n/mI^n)$ and $x\in R\setminus I.$
My question is
what is the relation between $l(I)$ anf $l(I+(x)/(x))?$
2
votes
1
answer
626
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Gluing Schemes along subschemes
Let X be the union of two planes in $\mathbb{A}^4$ touching at origin. Blow up X at the origin. Call it $\overline{X}$. It has two disjoint copies of $\mathbb{A}^2$ blown up at the origins. Their ...
3
votes
0
answers
349
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Transverse intersection of two divisors
Let $X$ be a smooth variety and $D_1$ and $D_2$ are two smooth divisors which intersect transversely. Assume also that $D_1\cap D_2$ is irreducible. Is it true that $\mathcal{O}(-D_1)|_{D_2}\cong \...
11
votes
1
answer
843
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Proving that a variety is not (isomorphic to) a toric variety
Is there an algorithmic (or other) way to prove that a (projective)
variety is not isomorphic to a toric variety?
I'd be happy with an algebraic answer (for affine or projective varieties),
using the ...
2
votes
1
answer
257
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Union of Cohen-Macauley components
Let $X=X_1\cup X_2\cup X_3$ be a variety with three irreducible equidimensional components which are normal and Cohen-Macauley. Suppose $X_1$ and $X_3$ intersects trasversally and $X_2$ and $X_3$ ...
0
votes
0
answers
100
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Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
12
votes
2
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1k
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An elementary lemma of commutative algebra
Let $R$ be a commutative ring, $M$, $N$ $R$-modules, and $f: M\rightarrow N$ a homomorphism. It is known that $f$ is injective (surjective) if and only if $f_m$ is injective (surjective) for all ...
2
votes
1
answer
194
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Why is $C^\infty(M)$-module homomorphism $P\mapsto\Gamma(P)$ surjective?
$\DeclareMathOperator{\Id}{Id}
\require{cancel}$
Jet Nestruev's "Smooth Manifolds and Observables" contains following exercise:
Exercise. Show that $P$ is geometric if and only if the two modules $...
5
votes
2
answers
170
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T-nilpotency and quasinilpotency of ideals
Let $R$ be a commutative ring and let $\mathfrak{a}\subseteq R$ be an ideal. The ideal $\mathfrak{a}$ is called T-nilpotent if for every sequence $(r_i)_{i\in\mathbb{N}}$ in $\mathfrak{a}$ there ...
4
votes
0
answers
474
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Endomorphisms of free modules and extension of scalars
I asked this question on Mathematics Stackexchange, but got no answer.
Let $B$ be a commutative ring with $1$, let $A$ be a subring such that any unit of $B$ which belongs to $A$ is a unit of $A$, ...
2
votes
1
answer
326
views
Representation of elements in affinoid algebra
Let $K$ be a complete, algebraically closed non-Archimedean field, and let $p \in K[x]$ be of degree $d > 0$ and norm 1. (Here the norm of a polynomial is the maximum of the norms of its ...
4
votes
1
answer
568
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Dualizing complex definition ubiquity
The following definition is the one, I found here http://stacks.math.columbia.edu/tag/0A7A. But let me recall it (out of consistency's sake):
Definition
For $A$ a Noetherian ring, a dualizing complex ...
1
vote
1
answer
383
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Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?
Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ are irreducible and of equal degree what is the best ...
11
votes
1
answer
846
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Detailed modern references for basic properties of Pfaffians over commutative rings
Pfaffians are important to algebraic combinatorics, at least.
This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing against ...
2
votes
0
answers
135
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Ext over a certain commutative algebra
Let $A$ be the algebra $K[x,y]/(x^2,y^2,xy,yx)$. Then $A$ is a 3-dimensional commutative algebra and $Ext^i(M,A) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$. Namely: $...
1
vote
1
answer
342
views
singular locus of semi-normal variety
Let X be a seminormal variety and S be the singular locus of X. Is Blow$_S X=$ the normalisation of X?
Is the singular locus given by the conductor ideal?
1
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0
answers
57
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Finite presentation for left-exact endofunctors
For a commutative unital ring R and a left-exact endofunctor L on the category of R-modules, what is, or should be, the definition of finite presentation for L ? Possibilities:
L( R ) is ...
1
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0
answers
126
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Nielsen--Schreier for fields
Is it true that a subextension of a purely transcendental extension is itself purely transcendental?
In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ ...
1
vote
1
answer
448
views
Finitely Generated Commutative Z-algebra
Let $R$ be a commutative finitely generated $\mathbb{Z}$-algebra. Then the nilradical is equal to the Jacobson radical.
I am not able to make much traction on this, nor can I find this result in any ...
11
votes
1
answer
1k
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Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?
I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \...
2
votes
0
answers
307
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Endomorphism algebra of a coherent sheaf is locally free
What is an example of a Noetherian ring $A$ and a finitely generated $A$-module $M$ such that the endomorphism algebra $\mathrm{Hom}_{A}(M,M)$ is flat as an $A$-module but $M$ is not flat?
2
votes
1
answer
54
views
A question on isomorphism between factor modules over commutative semi-simple ring
Let $R$ be a commutative semi-simple ring with unity (i.e. all modules over $R$ is semi-simple) , let $P \le N \le M$ be a chain of $R$ modules such that $M \cong M/N$ ; then is it true that $M \cong ...
1
vote
0
answers
234
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Quotient of polynomial ring over a Dedekind domain
Let $A$ be a Dedekind domain and let $B:=A[X]/(f(X))$, where $f(X)\in A[X]$ is some monic polynomial such that $B$ is a domain. If I take the canonical map $A\longrightarrow B$, then it nduces a ...
9
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0
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165
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When is the rank 2 free metabelian group of exponent $n$ center free?
Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free?
The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\...
3
votes
1
answer
198
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Torsion submodules of non-noetherian modules
Let $R$ be a commutative ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. The $\mathfrak{a}$-torsion submodule of $M$ is defined as $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\...
4
votes
0
answers
104
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A Euclidean domain in which radix expansion is not possible
It is well known that given two nonconstant polynomials $f,g\in F[x]$ where $F$ is a field, there are unique polynomials $r_0,\dots ,r_n$ such that
$$f=r_n g^n +\dots+r_1 g +r_0,$$
where $\deg r_i &...
15
votes
1
answer
1k
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Applications of cluster algebras
Why are so many algebraists nowadays interested in cluster algebras?
(This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)
1
vote
1
answer
100
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Functions on rings and polynomials with coefficients in a certain kind of localisation
Let $R$ be a commutative ring with unity and let $S$ be a multiplicatively closed subset of $R$ such that $S$ contains no zero divisor . So the canonical map $f : R \to S^{-1}R$ is invective , hence w....
1
vote
0
answers
182
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generators for a graded algebra
I have a graded commutative algebra over a field $K$ and I also know that it is not generated by the degree 1 elements. Somehow I could guess a set of generators, is there a criterion to check that ...
7
votes
0
answers
211
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Invariant theory over rings
Apologies if this is a silly question, but I have had cause to briefly introduce myself to invariant theory. I have noticed that authors primarily work over (algebraically closed) fields. I was ...
5
votes
3
answers
364
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Minimal "subset" of a set of homogeneous polynomials with same solution space
Suppose $A:=\{f_1,\dots,f_m\}\subset \mathbb{C}[x_1,\dots,x_n]$ with $m>n$ is a set of homogeneous polynomials of equal degree $d>0$. Suppose further that the variety they define consists of a ...
-1
votes
1
answer
434
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tangent bundle of $\mathbb{P}^1$
Let V be a two dimensional vector space over k. Consider the set of isomorphism classes of all rank 1 $k[x]/x^2$-submodules of $V\otimes k[x]/x^2$. Does this set has a variety structure? Is it ...
2
votes
0
answers
97
views
If $H$ is an atomic, unit-cancellative monoid such that the set of atoms of $H$ is finite up to associates, then $H$ is BF
In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. ...
3
votes
2
answers
801
views
What is the divisibility theory for Bezout Domains?
There are many facts about integer gcds which can be proved by appealing to unique prime factorization (up to sign). for example $\gcd(a^2,b^2)=\gcd(a,b)^2$. One way to get the machinery (if that is ...
2
votes
0
answers
60
views
Different term's contribution in zonal polynomial
I am very interested in the contribution of different terms in a zonal polynomial. Let's focus on the simplest case. In (20) in "Distributions of matrix variates and latent roots derived from normal ...
5
votes
0
answers
762
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Picard group of non reduced scheme
Let X be a non reduced scheme. $X_{red}$ be the reduced scheme. Is it true that Picard group of $X_{red}$=Picard group of X?
Is this map surjective $Pic (X)\rightarrow Pic(X_{red})$?
Is there a ...