Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
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Axiomatic definition of integers
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ...
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Deformations of free modules
Where can I found a description of the deformation theory for modules?Is it possible to deform a free module in such way that each fibre of the deformation is still free?
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derivative in the ring k[e]/e², chain rule
Let $k$ be a ring and $\overline{k} = k[\epsilon]/\epsilon^2$. For every $f \in k[t]$ there is a unique $f' \in k[t]$ such that $f(t+\epsilon)=f(t)+\epsilon f'(t)$ holds in $\overline{k}[t]$. It ...
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Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories?
Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can fail to be abelian categories.
Poking around on the internet has ...
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Reflexive modules over a 2-dimensional regular local ring
This question comes from reading Washington's proof of Iwasawa's theorem, and wanting to learn the commutative algebra version of the classification of finitely-generated $\Lambda$-modules. I went to ...
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Serre type vanishing theorem of coherent sheaves on quasi-projective variety?
For a projective variety $X$, Serre's vanishing theorem says that $H^i(X, \mathcal{F}(n))=0$ for any coherent sheaf, $i\geq 1$ and sufficiently large $n$. I am wondering, is there a similar type of ...
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Why are finitely generated modules over principal artin local rings direct sums of cyclic modules?
I am looking for a proof of the following fact:
If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic $R$-modules.
(Apparently such rings $...
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Is formal smoothness a local property?
Is the following statement true?
Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally smooth ...
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What is Serre's condition (S_n) for sheaves?
The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
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What is the German translation of "catenary ring"?
I am looking for the correct technical term in German for the notion of catenary ring in commutative algebra.
Does anyone know?
For those who don't know what a catenary ring is but would like to: ...
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Origin of the term "localization" for the localization of a ring
I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source ...
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When is every "solid" perfect complex faithful?
Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy.
...
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Why is the prime spectrum not useful in non-archimedean analytic geometry?
This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the notes of Conrad.
Reading Conrad's notes (and e.g. those of ...
user5117
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Generalized Chinese Remainder Theorem
Let $U,V$ be submodules of a $R$-module $M$. Then the diagonal induces an isomorphism
$M/(U \cap V) \to M/U \times_{M/(U+V)} M/V.$
This is a (useful!) generalization of the Chinese Remainder Theorem ...
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Additive commutators and trace over a PID
I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{...
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Krull dimension of a completion
How does one study Krull dimension of some I-adic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull ...
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Amazing examples in complex Algebraic Geometry
Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least ...
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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 ...
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Obstructions for reduced embedded deformation of Artinian rings
Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$...
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Two questions about Cohen-Macaulay rings
The following questions seem basic, but I can't find them in the literature. I'm also unable to think of a counterexample.
Let $A$ be a local Cohen-Macaulay ring of dimension $d$.
Let $I$ be an ...
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Transformations of integer polynomials under combinations of their roots
I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials f &...
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Krull rings and determinantal invariants
During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
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Non-existence ofintegral basis of integral closure in a finite extension of Frac(A), A Dedekind.
Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ ...
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An example of a series that is not differentially algebraic?
Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an ...
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What's an example of a transcendental power series?
Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$?
I am looking for elementary example (so there should be a proof of transcendence that does ...
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A local ring not a quotient of a regular local ring
In his book Commutative Ring Theory, Matsumura proves that if a local ring is equidimensional, and a quotient of a regular local ring, then its completion is equidimensional.
What is an example of a ...
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When is a blow-up non-singular?
Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the
blow-up $\operatorname{Bl}_{Z}(X)$ non-singular?
The blow-up of a non-singular variety along a non-...
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The structure of the module of Kähler differentials of R[[x]] over R
It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers.
We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results ...
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Maximal ideal of codimension >1
To assuage my conscience over an unsourced statement in a paper I'm writing:
I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or ...
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Is the multiplication between even numbers an associative algebra? [closed]
We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist?
It has been proposed as a counterexample the set of even numbers. ...
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When are intersections of finitely generated field extensions finitely generated?
Let $k$ be a field, and let $E$ and $F$ be fields extending $k$, both contained in some single extension of $k$. If $E$ and $F$ are finitely generated (as fields) over $k$, must $E\cap F$ also be ...
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Do all Dedekind domains have the "Riemann-Roch property"?
Let $R$ be a Dedekind domain with fraction field $K$.
Say that a Dedekind domain $R$ has the Riemann-Roch property if: for every nonzero prime ideal $\mathfrak{p}$ of $R$, there exists an element $f ...
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Noetherian rings of infinite Krull dimension?
Since Noetherian rings satisfy the ascending chain condition, every such ring must contain infinitely many chains of prime ideals s.t. the heights of these chains are unbounded.
The only example I ...
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Lifting results from smooth maps to essentially smooth maps.
Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.
(Note: $R\to S$ is essentially finitely presented provided that $...
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Homological dimension of a graded ring which is like polynomial ring
Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$...
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When is the set of zero divisors equal to the union of the minimal primes in a reduced ring?
It is straightforward to show, for example, that the set of zero divisors of a (commutative unital) reduced Noetherian ring is precisely the union of its minimal primes. When else can we say that the ...
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Ring-theoretic characterization of open affines?
Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
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Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).
Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$.
If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...
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Matrix factorization categories for ADE singularities
What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...
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What is the abstract relationship between an indecomposable representation and a sum of irreducible representations with the same character?
$\mathbb{Z}$ is the simplest example of a group with indecomposable representations which are not irreducible. Over $\mathbb{C}$, the isomorphism classes of $n$-dimensional representations of $\...
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Software for computing multi-graded Hilbert series
The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights $(1,1,-1,-1)$
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$$
\frac{1 - abcd}{(1-...
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Why is Proj of any graded ring isomorphic to Proj of a graded ring generated in degree one?
I have seen it stated that Proj of any graded ring $A$, finitely generated as an $A_0$-algebra, is isomorphic to Proj of a graded ring $B$ such that $B_0 = A_0$ and $B$ is generated as a $B_0$-algebra ...
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A reference: the splitting principle for exterior powers of coherent sheaves?
It's well known that if E is a vector bundle with Chern roots $a_1,\ldots, a_r$,
then the Chern roots of the $p$th exterior power of E consist of all sums of $k$ distinct $a_i$'s. I would like to say ...
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is there a notion of weakly noetherian?
A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left ...
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Infinite collection of elements of a number field with very similar annihilating polynomials
Hello all, let $n$ be an integer $\geq 2$ and let $\alpha$ be an algebraic number
of degree $n$. Let $R$ be the ring of algebraic integers in ${\mathbb Q}(\alpha)$, and
let $B$ be the subset of $R$ ...
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Proving that two local PIDs, one inside the other, with the same field of fractions are equal.
Let $R\subset S$ be two local PIDs that have the same fields of fractions. How to prove that they are equal?
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What is the insight of Quillen's proof that all projective modules over a polynomial ring are free?
One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in ...
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Does every projective A/I-module come from A?
Let $A$ be a Noetherian commutative ring and $I$ an ideal in $A$. It is pretty much trivial to see that every free $A/I$-Module is obtained from a free $A$-module by tensoring over $A$ with $A/I$: ...
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For which fields K is every subring of K…?
This question was inspired by
How to prove that the subrings of the rational numbers are noetherian?
which some people found too routine to be of interest. So I have decided to liven things up a bit ...
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How to prove that the subrings of the rational numbers are noetherian?
I have managed to prove that the aforementioned subrings are in bijection with the subsets of the primes, however, I haven't been able to prove that they are all noetherian. I need help.
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