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Questions tagged [ac.commutative-algebra]

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

2
votes
0answers
21 views

PID expressed as finite union of subrings

There is a classical theorem that no field can be expressed as finite union of proper subfields. In contrast, there is an example of an integral domain that can be expressed as finite union of proper ...
0
votes
0answers
25 views

Is a family of Cohen-Macaulay modules again Cohen-Macaulay (non-noetherian case)

Let $A$ be a local non-noetherian $\mathbb{C}$-algebra, $B$ a finitely generated, regular $\mathbb{C}$-algebra and $M$ a finite $B \otimes_{\mathbb{C}} A$-module, flat over $A$. Suppose that $M \...
0
votes
0answers
59 views

Do Plucker relation follow from a subsystem of equations?

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$, \begin{align*} P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} ...
2
votes
0answers
51 views

Krull dimension of completions in non-noetherian setting (especially completed perfections)

What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology? An example of the sort of "nice" topological ring I'm looking for is a ...
0
votes
0answers
60 views

Completion of localization of completion

Let $(A,m)$ be a noetherian local ring, and let $p \subseteq A$ be a prime ideal. From this data, we can construct two rings: 1. We may localize $A$ at $p$, and then complete, obtaining the $pA_p$-...
4
votes
1answer
66 views

Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?

Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $I\subseteq m$ a proper homogeneous ideal in $S$. Is this true that we always have: $$[Im:(x)][Im:(y,z)]\subseteq Im \ ?$$ In a paper we ...
2
votes
1answer
198 views

Is the support of a flat module generically flat?

Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y ...
2
votes
1answer
101 views

Are unique prime ideal factorization domains locally noetherian?

I asked this question on Mathematics Stack Exchange but got no answer. Here is the question: Let $A$ be a domain (that is, a commutative ring with one in which the condition $ab=0$ implies $a=0$ or $...
2
votes
0answers
81 views

A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
3
votes
1answer
138 views

Is $A[b/a]$ a Krull domain?

Is $A[X]/(aX+b)$ a Krull domain when $A$ is and when $a\in A-\{0\}, b\in A-Aa$ are such that $Aa$ and $Aa+Ab$ are prime ideals of $A$? This is stated as Proposition 8 in Pierre Samuel, "Sur les ...
8
votes
1answer
240 views

Description of p-adics tensor the reals

What is $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{R}$ equivalent to? where $\mathbb{Z}_p$ are the p-adic integers. I am specially interested in the case $p=2$. Do know that $\mathbb{Z}_p\otimes_{\...
15
votes
3answers
386 views

Graded analogues of theorems in commutative algebra

Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word commutative ...
10
votes
1answer
528 views

Is the formal power series ring integrally closed?

Let $k$ be a field and $s$ and $t$ be variables. Is the ring $k[s][[t]]$ integrally closed in $k[s,s^{-1}][[t]]$?
2
votes
0answers
178 views

Are there enough curves (to connect 'points' of f.g. algebras)?

(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the ...
6
votes
0answers
311 views

$\mathbf{Q}_p$ versus $\mathbf{C}$

Let $\sigma_p : \mathbf{Q}_p\to\mathbf{C}$ and $\sigma_{\ell} : \mathbf{Q}_{\ell}\to\mathbf{C}$ be two field homomorphisms, with $\ell\neq p$. Can one describe the compositum field $K$ of $\mathbf{Q}...
1
vote
0answers
36 views

Primary decomposition with parameters

$\newcommand\QQ{\mathbb{Q}}$ Considering the polynomial $$f = x^2 - a y$$ one notes, that it is irreducible in $\QQ[x,y]$ for all $a \neq 0 \in \QQ$ and factors for $a = 0$. More generally, let $A=...
1
vote
0answers
73 views

Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
0
votes
0answers
143 views

Behavior of Ext under base change

Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules. How to show the existence of the following exact sequence $\cdots\longrightarrow ...
5
votes
1answer
228 views

A question on dominant morphism of affine schemes

Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A_{ \mathfrak p}$ is regular local ...
-2
votes
0answers
56 views

Conditions on a local ring for f.g modules to have full support

Let $A$ be a noetherian local ring of Krull dimension $d$, and let $M$ be a finitely generated $A$-module. Assume $M$ also has Krull dimension $d$. What are some conditions on the ring $A$ that will ...
2
votes
0answers
120 views

Graded Betti numbers $\beta_{n,j}$ for points in $\mathbb{P}^n$

Let $S = \mathbb{C}[z_0, \dots, z_n]$, and let $X$ be a set of points in $\mathbb{P}^n$. I'm looking for references concerning results for the graded Betti numbers $\beta_{n,j}(S/I(X))$, i.e., the ...
5
votes
2answers
616 views

Algebra for algebraic topology

My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back ...
1
vote
0answers
65 views

Notions of connected components in a finite family fibration

Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...
0
votes
1answer
156 views

Counting the number of poles for rational functions in a coordinate ring of a curve

I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...
1
vote
0answers
54 views

Relation between lifts of simple roots and lifts of idempotents (Henselian property)

Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
2
votes
0answers
90 views

Making explicit the local structure theorem of étale maps in a very simple case

Making explicit the local structure theorem of étale maps in a very simple case. First I recall the following items from Stacks. \smallskip \textbf{Lemma 10.141.2.} Any étale ring map is standard ...
1
vote
0answers
63 views

Module of Kahler differentials for manifolds [duplicate]

Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...
6
votes
1answer
180 views

Existence of isomorphism mod every power of the maximal ideal

This problem is a continuation of Hensel lemma and rational points in complete noetherian local ring. Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal. Assume $...
-1
votes
0answers
118 views

Is the integral closure of a polynomial ring a UFD? [migrated]

Let $\mathbb{C}(x)$ be the field of rational functions over the complex numbers and $F$ a finite extension of $\mathbb{C}(x)$. Suppose $B$ is the integral closure of $\mathbb{C}[x]$ (the ring of ...
4
votes
0answers
112 views

Explicit description of injective hulls

Let $k$ be a field, let $R:=k[x_1,\ldots,x_n]$, and consider the $R$-module $M:=R/{(x_1,\ldots,x_n)}\cong k$. Then the injective hull $I_M$ of $M$ admits the following explicit description: $$ I_M = k[...
5
votes
0answers
269 views

Completion of a local ring of an arithmetic surface

Vaguely, an arithmetic surface is an algebraic curve with coefficients in some "ring of integers". More formally it is an integral normal scheme $X$ of dimension 2 of finite type over a Dedekind ...
3
votes
1answer
90 views

Summing complete intersections

Suppose we have polynomials $f_1,\dots,f_r\in k[X_1,\dots,X_N]$ defining a complete intersection in $\mathbb{A}^N$. I suspect that it is then true that $f_1(X)+f_1(Y),\dots,f_r(X)+f_r(Y)\in k[X_1,\...
1
vote
0answers
137 views

Rank 2 vector bundle with trivial first chern class is self-dual

I saw a statement used in the paper that E of rank 2 with $c_1(E) = 0$ is self-dual. I was wondering, how does one prove this statement? If it makes a difference, let the underlying variety be ...
1
vote
0answers
53 views

What's the probability a random module element has prime annihilator?

I'm going to pose two versions of my question---an ill-defined version and a well-defined version. Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...
11
votes
3answers
543 views

Are archimedean subextensions of ordered fields dense?

Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is $F$-archimedean if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$. Is it true that if $E$ is $F$-...
1
vote
0answers
116 views

Behavior of regularity under base change

Let $(R,\mathfrak{m})$ be a noetherian local domain. Let $(A,\mathfrak{n})$ be a regular local ring contains $R$ such that $\mathfrak{n}\cap R=\mathfrak{m}$ and $A$ is essential finitely generated as ...
4
votes
2answers
216 views

Ideals invariant under ring automorphisms

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties: $I$ is generated by two homogeneous elements; $I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
3
votes
1answer
135 views

On a special type of subring of $\mathbb C[x_0,…,x_{q-1}]$

Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let $$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
14
votes
2answers
503 views

Is a matrix similar to its transpose over $\mathbb{Z}_p$?

Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$? On the one hand, I know the analogous fact is false ...
6
votes
1answer
191 views

Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module?

Let $R$ be a one-dimensional, reduced and noetherian $k$-algebra (we may also assume that $R$ is a finite $k[x]$-algebra). Let $M$ be a finitely generated, torsion-free module over $R$, i.e. no ...
11
votes
1answer
169 views

Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overline{k}/k)$ stabilize a compatible system of roots of $t$?

There may be some technical issues with the question, but hopefully what I mean is clear... Let $k$ be a number field (or maybe any finitely generated field over $\mathbb{Q}$ of characteristic 0) ...
13
votes
0answers
268 views

Hensel lemma and rational points in complete noetherian local ring

Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal. If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
4
votes
1answer
179 views

Henselianizations over countable index sets

Let $A$ be a ring, $I\subset A$ a finitely generated ideal. The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...
11
votes
2answers
834 views

Is a scheme Noetherian if its topological space and its stalks are?

Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
0
votes
1answer
89 views

Relation between Hilbert function and complete intersection ideals

Consider $T=k[x_1,\ldots,x_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x_1,x^{a_2}_2,\ldots,x^{a_n}_n)$$ with $2\leq a_2 \leq\ldots\leq a_n$. I want to prove that $$\sum_{i=...
0
votes
0answers
47 views

Restriction of fractional ideal sheaf to irreducible component is torsion-free

I translate the question into commutative algebra: Let $R$ be a one-dimensional, reduced ring (which is also finite free over some PID since the considered curve corresponding lies finitely over the ...
3
votes
1answer
152 views

Is a factorial scheme with Noetherian stalks locally Noetherian?

Motivation: The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct ...
6
votes
0answers
152 views

Maximal subalgebras in polynomial ring $\mathbb{R}[x]$ over the field $\mathbb{R}$ of real numbers

Question. What are the maximal subalgebras of polynomial ring $\mathbb{R}[x]$ over the field of real numbers?
3
votes
1answer
55 views

example of a non-finitely generated co-Hopfian module over a commutative QF ring

Can anyone provide an example of such a module? or show that no such module exists? For semisimple rings, we have co-Hopfian if and only if finitely generated. Perhaps the fact that QF rings (...
0
votes
0answers
66 views

Localization of Hom module in an advanced setting (0-dualizing module)

Let $A,\,B$ be noetherian rings such that and let $M$ be an $A$-module. Let $g:A \to B$ be a ring homomorphism which makes $B$ into a finite free $A$-algebra. Now we can regard the $A$-module $\...