Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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5 votes
1 answer
461 views

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 views

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 views

$Ext^i(D(R),R)$ 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 answers
335 views

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 vote
0 answers
124 views

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 views

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 answer
350 views

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 views

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 answers
252 views

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 answer
525 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 answers
1k views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 vote
0 answers
57 views

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 vote
0 answers
126 views

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 views

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 views

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 views

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 votes
0 answers
165 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 ...

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