Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
4
votes
1
answer
134
views
Terminology/literature for $\forall I\leq A,\; IB\cap A=I$
I am interested in extensions $A\leq B$ of commutative rings with the property that for all ideals $I\leq A$ we have $IB\cap A=I$. Is there a standard name for this property, or a standard reference ...
- 30.5k
2
votes
0
answers
97
views
References discussing the category of ordered commutative rings
Is there a reference anywhere discussing the category of ordered commutative rings?
I'm thinking of ordered commutative rings and ring homomorphisms preserving the order, but I would also be ...
- 10.1k
7
votes
0
answers
275
views
Split epimorphism of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...
- 696
2
votes
2
answers
417
views
Transition maps in trivial direct limit
If $\{X_i, i\in I\}$ is a directed system of abelian groups such that we have
$$\varinjlim_{i\in I}X_i = 0$$
is it true that for every $i$ and large enough $j\ge i$ the transition map $f_{i,j} : X_i\...
- 1,424
1
vote
0
answers
66
views
Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring
Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
- 111
29
votes
2
answers
7k
views
Elementary proof of Nakayama's lemma?
Nakayama's lemma is as follows:
Let $A$ be a ring, and $\frak{a}$ an ideal such that $\frak{a}$ is contained in every maximal ideal. Let $M$ be a finitely generated $A$-module. Then if $\frak{a}$$M=M$...
- 10.5k
11
votes
2
answers
287
views
Fundamental group under Gelfand duality
Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^*$-algebras is an equivalence of categories. In other words, all topological ...
- 21.5k
18
votes
7
answers
2k
views
Superfluous definitions
It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying ...
- 4,713
1
vote
0
answers
110
views
What is some algebraic intuition behind the fact that the (real part) of the logarithm of Bernoulli umbra plus $1$, is $-\gamma$?
Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus.
Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\...
- 12.9k
9
votes
1
answer
661
views
What are abelian categories enriched over themselves?
As far as I understand, an arbitrary abelian category is not enriched over itself, for example, $\mathrm{ChainComplex}(\mathrm{Ab})$ is, right? On the other hand, the categories $\mathrm{Mod}(R)$ (in ...
- 6,962
15
votes
1
answer
664
views
R-module hom a direct summand of Z-module hom?
$\DeclareMathOperator\Hom{Hom}$Fix a commutative ring $R$. For $R$-modules $M$ and $N$, there is an inclusion of abelian groups $\Hom_R(M,N) \to \Hom_{\mathbb{Z}}(M,N).$ Are there conditions on $R$ ...
- 12.9k
2
votes
0
answers
97
views
Binary unions are effective in abelian categories
Let $\mathsf{A}$ be an abelian category and $S,T\hookrightarrow M$ be two subobjects. We can naturally form the commutative square
and it's surely cartesian. (Since intersections are given by ...
- 721
1
vote
0
answers
130
views
Stein Manifold and sheaf cohomology with support in a point
Let $X$ be a Stein manifold with analytic structure sheaf $\mathscr{O}_{X}$. Let $M$ be a coherent $\mathscr{O}_{X}$-module, $x \in X$, and $U$ a Stein open containing $x$. Write $\mathfrak{m}_{x}$ ...
- 63.9k
1
vote
0
answers
173
views
Calculating multiplication in a finite dimensional algebra over $\mathbb{Q}$
Suppose $ L $ be an extension over $ \mathbb{Q} $ of degree $ n $. Let $\{e_{1},e_{2},\dots,e_{n}\} $ be a basis of this extension. Now I know the product $ e_{i}^{2} $ and $ e_{i}e_{j} $ . So we can ...
- 923
2
votes
0
answers
104
views
Does same group of units imply surjective contraction map on spectra
Let $B\subset A$ be an inclusion of Krull dimension 1 rings, and assume $A$ a Dedekind domain. I think that $\mathfrak{p}\in Spec B$ is in the image of the contraction map $\mathfrak{P} \mapsto \...
- 675
2
votes
1
answer
314
views
Algebraically characterizing morphisms of commutative rings that are a homeomorphism on the prime spectra
Let us say that a morphism $\varphi\colon A\to B$ of rings (commutative, with unit) is a homeomorphism on the (prime) spectra iff the corresponding map $\operatorname{Spec}B\to\operatorname{Spec}A$ (...
- 20.5k
2
votes
0
answers
312
views
Degree $8$ cyclic extension over $\mathbb{Q}$
Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...
- 923
0
votes
1
answer
130
views
Levi-Civita field in unusual basis
Can all elements of the Levi-Civita field be represented as power series of a single element
$$p=\varepsilon^{-1}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}...
- 28.2k
8
votes
3
answers
1k
views
How to prove that two univariate polynomials are always algebraically dependent?
How to prove that two univariate polynomials(over any field) are always algebraically dependent? Also, how to prove the generalization of this question i.e if number of polynomials are more than ...
- 251
3
votes
0
answers
150
views
Finite commutative group schemes whose exponent coincides with its rank
In group theory, a finite commutative group $G$ contains an element whose order is the exponent of $G$. Thus, If the exponent of $G$ is the same as the order of $G$, it must be that $G$ is cyclic. ...
- 329
1
vote
1
answer
128
views
Koszul complex of equations defining a stabilizer
Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0.
Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we ...
- 39.3k
4
votes
0
answers
258
views
Cotangent complex of a formal thickening
Let $R$ be an (animated) commutative ring, with cotangent complex $L_R$ and let $\mathcal{C}(R) = \mathcal{D}(R)_{\Sigma^{-1}L_R/}$ be the category of nice square zero extensions of $R$. A typical ...
- 858
1
vote
1
answer
205
views
Question about linear functionals over C-infinity modules
A remark in Nelson's "Tensor Analysis" implies there are no non-trivial linear functionals on the module of continuous vector fields on a manifold, when considered as a module over the ring ...
- 5,407
1
vote
2
answers
158
views
An example of a commutative ring with a non-zero nil ideal that is idempotent
Recall that an ideal of a commutative ring is said to be a nil ideal if each of its elements is nilpotent. I am looking for a non-zero nil ideal of a commutative ring that is idempotent.
- 9,650
1
vote
0
answers
39
views
When nilradical belongs to a Gabriel filter
Recall that a Gabriel filter of ideals $\mathscr{I}_\sigma$ of a commutative ring $R$ is a non–empty filter of ideals satisfying that every ideal
$I$ of $R$, for which there exists an ideal $J\in\...
- 147
0
votes
1
answer
154
views
Nullstellensatz and nilpotence of a module
Let $\nu : G \rightarrow H$ be a surjective group homomomorphism with kernel $N$, $H$ abelian, and $G$ finitely generated.
The rational abelianization of $N$, $H_1(N)$ is a $\mathbb{C}[H]$-module, ...
- 148k
3
votes
1
answer
452
views
Commutative algebra for the Conway games
I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are ...
- 6,453
0
votes
0
answers
218
views
Cohen-Macaulay modules and connections to Mirror Symmetry
Let $ R $ be a local Noetherian Gorenstein domain. Suppose a module $ M $ fits into an exact sequence $$ 0 \rightarrow K \rightarrow R^n \rightarrow M \rightarrow 0 $$ Then we write $ K = tM $. A ...
- 936
1
vote
0
answers
61
views
When some idempotent ideals belong to the Gabriel filter of ideals for a hereditary torsion theory
Let $\mathscr{I}_\sigma$ be the Gabriel filter of ideals for a hereditary torsion theory $\sigma$ over a commutative ring $R$. I am looking for equivalent conditions on either $\sigma$ or $R$ under ...
- 147
2
votes
1
answer
253
views
Strict henselianization and branches of explicit curve at singularity
Let $A$ be a local ring, which we can assume is reduced. Let $k$ be the residue field of $A$.
In the Stacks project (https://stacks.math.columbia.edu/tag/06DT), I have learned some notion of the ...
- 921
6
votes
1
answer
245
views
Reference request for results that involve the transcendence degree
Currently I'm reading "On the Decidability of the Real Exponential Field" by Macintyre and Wilkie and the Proof of Theorem 1.1 (page 462-464) uses two algebraic results that involve the ...
- 123
0
votes
2
answers
299
views
A question about localization of commutative rings
Given a commutative ring $ R $ and a multiplicatively closed subset $ S $ of $ R $, there are two ways to consturct $ S^{-1}R $:
define an equivalence relation $ \sim $ on $ R\times S $ and then take ...
- 1,424
2
votes
0
answers
235
views
Formally étale maps of animated $k$-algebras
In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
- 301
3
votes
0
answers
292
views
modules over principal ideal rings
Let $R$ be a commutative principal ideal ring (not necessarily Artinian) and let $M$ be a finitely generated $R$-module. Is $M$ a direct sum of cyclic $R$-modules? (i.e. a generalization of the theory ...
- 429
10
votes
3
answers
3k
views
Sum of radical ideals
Let $A$ be a commutative ring and endow the closed subsets of $\operatorname{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This ...
- 12.9k
2
votes
0
answers
115
views
When $k(f(x), g(x)) = k(x)$? In other words, when is a given polynomial parametrization of an affine planar (rational) curve proper?
If $f, g \in k[x]$, where $k$ is a field, then $k(f, g) = k(h)$ for some rational function $h \in k(x)$ (this is a special case of Lüroth's theorem).
Question 1: Under what conditions does the above ...
- 5,339
0
votes
1
answer
128
views
About cluster variables obtained by (sequentially) mutating at exchangeable variables from an initial cluster
Let $\Sigma=(X,ex,B)$ be a seed, $\mathcal{A}(\Sigma)$ a corresponding geometric cluster algebra and $\mathcal{X}_{\Sigma}$ the set of all cluster variables of $\mathcal{A}(\Sigma)$. We call a ...
- 3,587
4
votes
0
answers
356
views
Jet Nestruev's proof that the exterior derivative $d$ on a real line is not a Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}$
The relation between the exterior derivative $d:C^\infty(\mathbb{R})\to\Omega^1\mathbb(\mathbb{R})$ and the Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}:C^\infty(\mathbb{R})\to\Omega_{C^\...
- 2,385
1
vote
2
answers
311
views
A variation on Abhyankar–Moh–Suzuki theorem
The well-known theorem of Abhyankar–Moh–Suzuki says the following:
Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero.
If $k[f,g]=k[t]$, then $\deg(f) \mid \deg(g)$ or $\deg(g) \mid \...
- 5,339
5
votes
0
answers
276
views
Analysis proof of dual number spectral theorem
Let the dual numbers be $\mathbb R[\varepsilon]/(\varepsilon^2)$. Write a general dual number as $a + b \varepsilon$ (where $\varepsilon^2 = 0$). Given a symmetric matrix $M$ over the dual numbers (i....
- 4,943
1
vote
1
answer
203
views
Special cases of the embedding problem
Embedding problem. Let $I$ be the ideal of polynomial algebra $A=K^{[n]}$, such that $A/I$ is also a polynomial algebra with smaller number $k$ of variables. Is it true that $I$ is generated by $n-k$ ...
CommunityBot
- 1
10
votes
1
answer
327
views
Proving that polynomials belonging to a certain family are reducible
In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone.
Assume that $\mathbb F_p$ is ...
- 22.5k
0
votes
0
answers
112
views
Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
- 923
6
votes
1
answer
442
views
If power of an ideal is locally free then it is locally free
Let $X$ be a noetherian scheme and $\mathcal{I} \subset \mathcal{O}_X$ a coherent sheaf of ideals. Suppose that $\mathcal{I}^d$ is locally-free for some power $d$. Then the blowing up $\mathrm{Bl}_{\...
- 148k
3
votes
0
answers
234
views
Non-flat base change
Let $(R,\mathfrak{m},k,E)$ be a Noetherian local ring with $E$ an injective hull of $k$, let $M$ be a finitely generated $R$-module and $M^{\vee}=\mathrm{Hom}_{R}(M,E)$. Is there a non-flat $R$-...
- 63.9k
2
votes
0
answers
132
views
Pushforward of a Cohen-Macaulay module
Let $f \colon R \to S$ be a homomorphism between (local Noetherian) rings which turns $S$ into a finitely generated $R-$module and let $M$ be a finitely generated over $S$.
Is $M$ is Cohen-Macaulay ...
9
votes
2
answers
417
views
Over which (graded) rings are all modules decomposable into indecomposables?
A module is decomposable if it is the direct sum of two modules. The process of splitting summands off of a decomposable module does not need to terminate, so infinitely generated modules do not ...
- 35.2k
2
votes
0
answers
112
views
Abelian approximation of fields
Given a field extension $K/k$ of finite degree and a norm $d$ on $\overline{k}$, what is the smallest real number $\alpha_{d}^K$ such that for every element $z$ of $K$ there is an element $z^a$ of $k^...
- 21
2
votes
0
answers
160
views
Descent of codimension of a closed subscheme
Let $X$ be an affine integral normal scheme, $Z\subset X$ a constructible closed subscheme of codimension $\geq 2$. We can write $X$ as a limit of schemes $(X_{\lambda})_{\lambda\in \Lambda}$ of ...
- 119
4
votes
0
answers
85
views
Do generators of the radical of an ideal generated by polynomials with rational coefficients have rational coefficients?
Let $I\in\mathbb{C}[x_1,\dots,x_n]$ be a an ideal generated by polynomials with rational coefficients. Is $\sqrt{I}$ also generated by polynomials with rational coefficients?
- 41