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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Serre subcategories of the category of chain complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$. Let $R$ be a commutative $k$-algebra. We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
Walterfield's user avatar
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Krull dimension of ring of invariants

Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
kindasorta's user avatar
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Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings

Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
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Topological modules over a locally compact ring

Let $R$ be a locally compact, separably metrizable ring (commutative with an identity) and let $M$ be a closed submodule of $R \oplus R$. Is the projection of $M$ onto the first coordinate closed?
Nik Weaver's user avatar
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Equivalences of categories of complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$. Let $R, S$ be two commutative $k$-algebras. Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
Walterfield's user avatar
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number of representations by sums of three squares (with coefficients)

There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for $$ \#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\} ?$...
Dr. Pi's user avatar
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Smoothness of locus of triples $(B_1,B_2,i)$ in Nakajima's notes

In section 1.4 of Nakajima's notes on Lectures on Hilbert Schemes, it is mentioned that $(\mathbb A^2)^{[n]}$ is identified with the space of triples $\{(B_1,B_2,i)\}/GL_n$. Here $B_1,B_2$ are $n\...
Rex's user avatar
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An alternative proof that Buchsbaum rings are generalized Cohen-Macaulay

Let $(R,\mathfrak{m})$ be a Noetherian local ring. $R$ is said to be Buchsbaum if, for each ideal $\mathfrak{q}$ generated by a full system of parameters, the number $\lambda_R(R/\mathfrak{q})-e_{\...
walkar's user avatar
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A direct proof that every projectivity between parallel lines is affine

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
Taras Banakh's user avatar
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Completion of $\mathbb F_q(T)$

It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...
joaopa's user avatar
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Relation between row space and column space resp. null space and left null space over general rings

Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
Thomas Preu's user avatar
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Geometrizing saturation construction

Edit: My original question quickly got one close request, so I edited it to add some context and motivation. Consider homogeneous polynomials $J_1,\dots,J_r\in k[\bar x]$. I want to construct a ...
user347489's user avatar
11 votes
2 answers
744 views

Are topological PID's Noetherian?

Romain Giquaud has given a counterexample to the general form of the question. The bounty is for a solution for locally compact, metrizable rings. (I suspect the answer may be positive with this ...
Nik Weaver's user avatar
4 votes
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328 views

GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials

This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$. Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
Martin Brandenburg's user avatar
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362 views

Subalgebras of quadratic algebras that are not quadratic

Suppose $A=k\oplus A_1 \oplus A_2\oplus \cdots$ is a quadratic algebra over a field $k$. Let $B$ be the subalgebra generated by a subspace $V\subseteq A_1$. What are the examples of such subalgebras $...
Lorenzo Del Vecchiopontopolos's user avatar
2 votes
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63 views

Number of solutions of overdetermined quadratic polynomial equations

Given $m$ linearly independent quadratic polynomials over the complex field in $n$ variables with $m>n$ and such that the number of zeros, say $N$, is finite, is there a known or conjectured strict ...
Alm's user avatar
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Extracting implications in polynomial constraint system from Groebner basis

Given a Groebner basis for a system of polynomial constraints over $\mathbb{Q}$, are there any known methods for extracting the low degree factorable polynomials in the ideal generated by that basis? ...
PPenguin's user avatar
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Embedding noetherian domains in a PID with finite index

The starting point of this post is the following question: Embedding number fields in fields with class number 1 It is shown that in the answers that , given an number field $K$, we cannot necessarily ...
GreginGre's user avatar
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Is the projective dimension of finite torsion-free modules over regular ring of dimension $n$ smaller that $n$?

Let $R$ be a Noetherian regular integral domain of Krull dimension $n$. Let $M$ be a finite torsion-free $R$-module. Is this true that $M$ has projective dimension $<n$ ? This would be a ...
Stabilo's user avatar
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Representation of a number as a product of $\sqrt{n^2 + 1} + n$

Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and $$ \prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
Pavel Gubkin's user avatar
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1 answer
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When is the Tor-dimension of $R/(r)$ strictly smaller than that of $R$?

Let $R$ be a ring (commutative with unit) which I assume Noetherian and regular. In particular, the homological dimension of $R$ is the same as its Krull dimension. I am looking for results in ...
Stabilo's user avatar
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polynomials with no repeated factors

Assume that $F(x_1,\ldots, x_n)$ is a polynomial with integer coefficients that is "square-free" over $\mathbb Q$, i.e. it does not have repeated polynomial factors whose coefficients are in ...
Dr. Pi's user avatar
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Gorenstein property from initial ideal

My question is: If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...
Chess's user avatar
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If some powers of polynomials are linearly independent, does it imply higher powers are also independent?

Let $P_1,\dotsc,P_k$ be polynomials. Assume they are pairwise non-proportional (i.e., any two of them are linearly independent). Suppose $N$ is a power such that $P_1^N,\dotsc,P_k^N$ are linearly ...
Zach Teitler's user avatar
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2 votes
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Gorenstein projective module over commutative local algebras

Let $A$ be a local commutative finite dimensional algebra over a field $K$. An $A$-module $M$ is called Gorenstein projective if $M$ is reflexive, $Ext_A^i(M,A)=0=Ext_A^i(M^{*},A)$ for all $i>0$ ...
Mare's user avatar
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2 votes
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$K_0((k[x]/(x^2))[y])$

Let $K_0(R):= K_0(P(R))$ where $P(R)$ is the category of finitely generated projective $R$-modules, where $R$ is a commutative ring with unity. Now if $R = k[x]/(x^2)$, $R$ is a local ring thus all ...
user443060's user avatar
4 votes
0 answers
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Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$

EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect. Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
Libli's user avatar
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-1 votes
1 answer
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Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?

The idea for the following question came from Joachim König's last comment appearing here, namely, the example with $u=x+y^3,v=x^3+y$. Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...
user237522's user avatar
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1 vote
0 answers
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Invariant polynomials under a non-standard group action

There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...
Jan-Willem van Ittersum's user avatar
1 vote
1 answer
129 views

If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k[t]$?

Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$. Let $f,g,h \in k[x,y]$, $g \neq h$, satisfy the following two conditions: (1) $(f,g)$ is a maximal ideal of ...
user237522's user avatar
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4 votes
1 answer
172 views

When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$

Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
user443060's user avatar
1 vote
1 answer
329 views

Is a proper map of varieties which is a bijection on points an isomorphism?

Suppose that I have a proper morphism $f: X \to Y$ of varieties (i.e. reduced separated schemes of finite type). I am given that (a) on a dense open $U \subseteq Y$, $f$ is an isomorphism (i.e. $X\...
Inna's user avatar
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1 answer
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$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ iff there exist $a_i,b_j \in k$ such that $\langle F_i-a_i \rangle_{i=1}^{n} = \langle G_j-b_j \rangle_{j=1}^{m}$

Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$ and let $F_1,\ldots,F_n,G_1,\ldots,G_m \in \mathbb{C}[x,y]$, $n,m \in \mathbb{N}-\{0\}$. Claim: $\mathbb{C}(...
user237522's user avatar
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5 votes
0 answers
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Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)

Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
Gro-Tsen's user avatar
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7 votes
2 answers
240 views

Double dual of free $\mathbb{Z}_{(p)}$-modules

For an abelian group $A$, put $DA=\text{Hom}(A,\mathbb{Z})$ and $D_{(p)}A=\text{Hom}(A,\mathbb{Z}_{(p)})$. It is a theorem of Specker that when $A$ is free abelian of countable rank, the natural map $...
Neil Strickland's user avatar
1 vote
1 answer
144 views

Ideals: If $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle$, then $\langle f_1-\lambda,f_2-\mu \rangle = \langle g_1-\delta,g_2-\epsilon \rangle$?

The following question appears in MSE without answers. Let $f_1,f_2,g_1,g_2 \in \mathbb{C}[x,y]-\mathbb{C}$. Assume that $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$, ...
user237522's user avatar
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1 vote
0 answers
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Why N-1 and N-2 rings are called like that?

In the Stacks Project, Tag 032F, we find: Definition. Let $R$ be a domain with field of fractions $K$. We say $R$ is N-1 if the integral closure of $R$ in $K$ is a finite $R$-module. We say $R$ is N-...
Elías Guisado Villalgordo's user avatar
4 votes
0 answers
229 views

A question about Euclidean domains

An integral domain $R$ is a Euclidean domain if there is a degree function $$\deg : R-\{0\} \to \mathbb{Z}_{\ge 0} $$ such that For every $a,b\in R$ with $b\ne 0$ there are $q,r\in R$ such that $$ a=...
Mohammad Safdari's user avatar
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1 answer
132 views

$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$?

The following question is a direct continuation of this question: Let $u,v \in \mathbb{C}[x,y]$. Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$ (excluding the cases where $...
user237522's user avatar
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1 vote
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For a finite locally free $A\to B$ when does the different equal the Noether different?

(Cross-posted from MSE.) All rings commutative with $1$. Let $A\to B$ be an $A$-algebra which is finite projective, meaning $B$ is finitely generated projective as an $A$-module, so there is the trace ...
P-addict's user avatar
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4 votes
1 answer
234 views

If $\mathbb{C}(u(x,y),v(x,y),f(x))=\mathbb{C}(x,y)$, for every $f(x) \in \mathbb{C}[x]-\mathbb{C}$, then already $\mathbb{C}(u,v)=\mathbb{C}(x,y)$?

The following question is a direct continuation of this elaborate question; it is mentioned there at the end: Let $u,v \in \mathbb{C}(x,y)$ or $u,v \in \mathbb{C}[x,y]$, if it is easier to answer in ...
user237522's user avatar
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6 votes
1 answer
182 views

Ideal-like filter on a ring not generated by ring ideals

Suppose one has a filter (a collection of subsets closed under increasing the size of the set and under finite intersection) $F$ on a ring $R$. Say that $F$ is (ring) ideal-like if for every set $U \...
Charles Wang's user avatar
1 vote
0 answers
93 views

Examples of compressed Gorenstein ring

Let $(R,\mathfrak{m},k)$ be a Gorenstein local Artinian ring of socle degree $s$ and embedding dimension $e>1$. We set $$ \varepsilon_i=\min\left\{ \binom{e-1+s-i}{e-1}, \binom{e-1+i}{e-1}\right\} \...
SKS's user avatar
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0 answers
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On certain definition of arithmetical ring

The definition of an arithmetical ring states that A ring $R$ is arithmetical if the ideal lattice is distributive or equivalently $R$ is locally a valuation ring. I was reading a paper where ...
Amit Phogat's user avatar
9 votes
0 answers
312 views

History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring

This is a repost. So far, I've received no answers on HSM Stack Exchange; maybe I do in MO. In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (for a ...
Elías Guisado Villalgordo's user avatar
2 votes
1 answer
139 views

Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$

Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
uno's user avatar
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2 votes
0 answers
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Help with Macaulay2 computation of invariant ring

Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y)=...
It'sMe's user avatar
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8 votes
1 answer
335 views

Homological conjectures for finite dimensional commutative algebras

$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
Mare's user avatar
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4 votes
2 answers
349 views

$p$-divisibility of Picard groups

Let $p$ be a prime number and let $k$ be a field with $char(k)\neq p$ such that all finite extensions have degree coprime to $p$. (For example, we can take $k=\mathbb{R}$ and $p\neq 2$ or let $k$ the ...
Boaz Moerman's user avatar
3 votes
1 answer
194 views

On Flat and Projective Modules over integral domain

Is this true that finitely generated flat module over an integral domain is projective. If Yes, please provide a proof.
Amit Phogat's user avatar