Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,310
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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 ...
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108
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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 $K$-...
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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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1
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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?
5
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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 $R$-...
0
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1
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169
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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\}
?$...
2
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120
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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\...
2
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111
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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_{\...
2
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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\...
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1
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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 ...
5
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1
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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 (...
2
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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 ...
11
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2
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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 ...
4
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1
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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 > ...
3
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1
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362
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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 $...
2
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63
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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 ...
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0
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56
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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?
...
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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 ...
1
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1
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104
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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 ...
4
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1
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195
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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 + \...
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1
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104
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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 ...
2
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110
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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 ...
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63
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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 ...
6
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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 ...
2
votes
1
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134
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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 $i>0$ ...
2
votes
1
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153
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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 ...
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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,...
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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 ...
1
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0
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112
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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 ...
1
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1
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129
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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 ...
4
votes
1
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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 $K_0(R)$ ...
1
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1
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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. $X\...
0
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1
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$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}(...
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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 ...
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2
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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 $...
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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 $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$,
...
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0
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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-...
4
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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=...
0
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1
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$\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 $...
1
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0
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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 ...
4
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1
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234
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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 ...
6
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1
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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 \...
1
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0
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93
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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\} \...
1
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0
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145
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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 ...
9
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0
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312
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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 ...
2
votes
1
answer
139
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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 ...
2
votes
0
answers
155
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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)=...
8
votes
1
answer
335
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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 $...
4
votes
2
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349
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$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 ...
3
votes
1
answer
194
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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.