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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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 $B-B$ ...
truebaran's user avatar
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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 ...
მამუკა ჯიბლაძე's user avatar
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 $l(I+(x)/(x))?$
tessellation's user avatar
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 \...
Kumar's user avatar
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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
Kumar's user avatar
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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 ...
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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 (...
Matemáticos Chibchas's user avatar
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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 = ...
Pierre's user avatar
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2 votes
1 answer
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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 $X_3$ ...
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4 votes
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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 $A$, ...
Pierre-Yves Gaillard's user avatar
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 ...
Fred Rohrer's user avatar
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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 ...
Turbo's user avatar
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8 votes
1 answer
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Which ideals have standard Hilbert series?

Let $m$ and $d$ be two positive integers. Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and ...
Antoine's user avatar
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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: $...
Mare's user avatar
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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 ...
mayer_vietoris's user avatar
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 ...
user35486's user avatar
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1 answer
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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?
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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 ...
Peter Heinig's user avatar
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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 $L/K$ ...
Sean Eberhard's user avatar
3 votes
1 answer
2k views

Tensor product of field extensions

Let $K$ be a field of characteristic 0 and $L$ a finite extension of $K$. Denote by $m$ the natural multiplication map from $L \otimes_K L$ to $L$. Denote by $I$ the kernel of the morphism $m$. Is $I$ ...
user43198's user avatar
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2 votes
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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?
user2831784's user avatar
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 ...
user avatar
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 ...
Vincenzo Zaccaro's user avatar
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 $(\...
stupid_question_bot's user avatar
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 &...
Mohammad Safdari's user avatar
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\...
Fred Rohrer's user avatar
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1 vote
1 answer
774 views

On the set of non-zero elements in an integral domain whose generating principal ideal is of a special kind

Let $R$ be an integral domain. Consider the set $$S := \big\{a \in R\smallsetminus \{0\} : Ra+Rx \text{ is a principal ideal } \forall x \in R \big \}.$$ Is $S$ a saturated multiplicative closed ...
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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....
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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 ...
jack's user avatar
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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 ...
John D Evans's user avatar
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 ...
user avatar
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. ...
Salvo Tringali's user avatar
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 ...
Felix Y.'s user avatar
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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 $...
Fallen Apart's user avatar
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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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1 vote
0 answers
272 views

Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?

Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
Sergei Ivanov's user avatar
1 vote
2 answers
327 views

Finding all submodules

Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra ...
Mare's user avatar
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1 vote
0 answers
514 views

An infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$

We shall define a infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$ as follows$\colon$ ${\Bbb F}_p[[X_1,\ldots]]\colon= \underset{n \geq 1}{\varprojlim}\, {\Bbb F}_p[[X_1,\...
Pierre's user avatar
  • 563
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.)
ThiKu's user avatar
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4 votes
1 answer
323 views

An analogue of rational functions for Hahn series

For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which ...
Mike Shulman's user avatar
10 votes
0 answers
836 views

Scholze's infinite to finite type ring theory reductions?

In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking. The most virtuosic pages in Scholze's ...
Stan's user avatar
  • 119
10 votes
2 answers
1k views

Is it possible to describe the image of the $p$-adic logarithm on $1+\mathfrak{m}$, where $\mathfrak{m}$ is the maximal ideal of a $p$-adic field?

Let $R$ be the ring of integers of some finite extension of $\mathbb{Q}_p$. In particular, I'm interested in the case $R = \mathbb{Z}_p[\zeta_{p^k}]$ (the totally ramified extension of $\mathbb{Z}_p$ ...
stupid_question_bot's user avatar
7 votes
1 answer
274 views

Uniquely ordered commutative rings

I am wondering whether there are reasonable necessary and/or sufficient conditions to dedice whether a commutative ring can be uniquely ordered (like for instance $\mathbb{Z}$) or not. In the field ...
Alice's user avatar
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1 vote
0 answers
36 views

Quadratic suborders of an imprimitive quartic order

Let $Q$ be an irreducible quartic order; that is, $Q$ is a subring of the ring of integers $\mathcal{O}_K$ in a quartic extension $K$ over $\mathbb{Q}$ such that the fraction field of $Q$ is equal to $...
Stanley Yao Xiao's user avatar
4 votes
1 answer
308 views

Understanding full set of sections as in Katz-Mazur

I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence: Being a "full set of sections" of $Z/S$ is something which is ...
aytio's user avatar
  • 371
1 vote
1 answer
242 views

minimal number of generators of a k-algebra over commutative rings

I have that $R$ is the $k$-algebra ($k$ is a field) finitely generated by $S=\{f_1, ..., f_m \}\subset k[x_1, \cdots, x_n]$ and this set of polynomials is minimal with respect to inclusion (i.e., e ...
Lucas Reis's user avatar
28 votes
4 answers
4k views

What are traces?

Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
Anonymous Coward's user avatar
1 vote
1 answer
233 views

Localization of a maximal Cohen-Macaulay module

Let $(R,m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $M$ an module with depth$M=d.$ Is there any example of $M$ such that $(1)$ $M_p$ is not free for some $p\in Ass(R)$ and $(2)$...
tessellation's user avatar
9 votes
2 answers
210 views

Which elements of $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]^\times$ are in $\langle 1+x_1,1+x_2\rangle$?

It's a classical fact that the commutative power series ring $\mathbb{Z}_p[[x_1,x_2]]$ is isomorphic to the completed group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times\mathbb{Z}_p]]$, the isomorphism ...
stupid_question_bot's user avatar
-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 ...
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