Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
3
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2
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Modules of finite support
I'm reading Dwyer and Fried's paper "Homology of free abelian covers, I". In it, they make the following claim, which I'm having trouble verifying.
Let $F$ be a field and $A = F[x_1^{\pm 1},\ldots,...
- 33
2
votes
0
answers
254
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Newton polyhedron and product of ideals
Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and
$J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ,\...
- 1,713
4
votes
2
answers
1k
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elementary classification of artinian rings
this may be too elementary for mathoverflow, but I'll give it a try.
rings are commutative here. it is well-known that every $0$-dimensional noetherian ring is artinian. the standard proof uses a ...
- 63.1k
2
votes
1
answer
191
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what are the possible approximations for ideals
(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.)
Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...
- 2,232
3
votes
1
answer
492
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Noetherian descent extension for a given ring
For a homomorphism of rings $R \to S$, the following are equivalent:
a) $(-) \otimes_R S : \mathrm{Mod}(R) \to \mathrm{Mod}(S)$ reflects isomorphisms
b) $R \to S$ satisfies effective descent with ...
- 63.1k
7
votes
1
answer
304
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Explicit formula for associator of commutative power series
Perhaps this question is too elementary, but if it's written down anywhere, I'd love to know about it. Suppose I have a power series $f\in R[[x,y]]$ for some commutative, unital ring. I've recently ...
- 10.5k
2
votes
2
answers
106
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Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein?
$(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite injective dimension($id$). It is known that if $R$ is Gorenstein, then $M$ has finite flat dimension ($fd$). I wonder ...
- 1,355
5
votes
2
answers
367
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Invariant means on commutative magmas
It is a very standard fact that commutative semigroups admit an invariant mean and the proof basically relies on Markov-Kakutani fixed point theorem. Now, it seems to me that the proof of this theorem ...
- 6,034
1
vote
1
answer
2k
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The annihilator of the quotient module
Suppose that $(A,m)$ is a Noetherian local ring, $M$ is an $A$-finite module. Assume that $x_1, ..., x_n$ are elements in $m$. Is the following equality true:
$$
\mbox{ann}(M/(x_1, ..., x_n)M) = (x_1,...
- 149
1
vote
0
answers
49
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Atomicity of the semidomain (under Dirichlet convolution) of arith. fncs to a subrig of the ring of integers of a totally real NF
Let $S$ be a non-trivial subsemiring of the ring of integers of a totally real number field, and denote by $D_S$ the integral semidomain of all arithmetic functions $\mathbf N^+ \to S$ with the ...
- 10.5k
1
vote
0
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487
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Two questions regarding $f$-adic completions of (non noetherian) rings
Lately I've looked at $f$-adic completions of commutative rings. I had posted two questions regarding the topic on math.SE which didn't receive any attention and I think they might be fit for ...
- 189
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votes
1
answer
1k
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Cofibrant Replacement of chain complexes
Hi,
I encountered an issue today that I can't resolve to myself:
Consider the projective model structure on chain complexes over a ring R (Ch(R)), bounded below if you like.
Projectives in Ch(R) ...
- 109
1
vote
1
answer
359
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Cohomology after completion
I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if it'...
- 1,049
11
votes
1
answer
693
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The word problem in the ring of polynomials
This question must be well known but I cannot find it in the literature.
Question: What is the computational complexity of the word problem in a subring of the ring of polynomials in $n\ge 1$ ...
user6976
2
votes
1
answer
173
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formally etale deformations of algebras
Let $A$ be a local artinian ring with residue field $k$, $S$ a $k$-algebra. Suppose there is a formally etale deformation $B$ of $S$ over $A$, i.e. a flat $A$-algebra $B$ such that $S\cong B\otimes_Ak$...
- 620
1
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1
answer
433
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irreducible elements in a ideal of $R[x_1,x_2]$
Let $\mathbf R$ denote the real numbers, let's take a finite number of points in $\mathbf R^2$ and let's take the ideal $I$ of all the polynomials that vanish on this points. Using the Hilbert basis ...
- 21
6
votes
1
answer
616
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Projective modules over free groups
Consider the ring of Laurent polynomials $R := \mathbb{Z}[s,s^{-1}]$ with integer coefficients. Are all projective $R$-modules free? (Let's say left modules by convention.)
More generally, let $G$ be ...
- 1,243
2
votes
1
answer
996
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Count the number of homogeneous polynomials
Is there a general way of counting the number of homogeneous polynomials of certain degree in a complex projective space or a weighted complex projective space, mod the ideal generated by some ...
- 248
4
votes
2
answers
207
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Splitting of primes in extension of integral domains
Let $A$ and $B$ be integrally closed, commutative Noetherian integral domains, and let $f: A \to B$ be a finite étale injective homomorphism. Let $d$ be the degree of $f$ (i.e. the rank of $B$ ...
- 519
2
votes
2
answers
666
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Z_p flatness and irreducible components.
I just used the following.
Lemma. Let $A$ be a $\mathbb{Z}_p$-flat ring, of finite type over $\mathbb{Z}_p$, and suppose that $A \otimes \mathbb{F}_p$ is a domain. Then $A$ is a domain.
Proof: ...
- 1,209
0
votes
0
answers
350
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Reference: A nowhere vanishing section of a vector bundle is locally split
Well-known fact:
If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
- 7,338
1
vote
1
answer
200
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local cohomology of Buchsbaum ring
Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?
- 127
3
votes
1
answer
226
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Buchberger algorithm question
I'm interested in Buchberger's criterion for determining if G={g_1,....,g_n} is a Grobner basis for the ideal it generates. In the procedure, I consider the S-polynomial S(g_i,g_j) and check if it has ...
- 41
5
votes
1
answer
562
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Castelnuovo-Mumford regularity and degree of generator.
Let $R$ be a polynomial ring over a field $k$,: $k[x_{1},..x_{n}]$, $\mathfrak{m}=(x_0,...,x_{n})$ and $M$ be a finitely generated $R$ module.
In a paper of Kodiyahlam, he define the Castelnuovo-...
- 325
1
vote
0
answers
74
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Macaulayfications and standard blow-ups
Let $X$ be an affine variety such that $X-p$ is Cohen-Macaulay,
and let $\pi \colon \widetilde{X} \rightarrow X$ be a standard blow-up of $X$
with respect to $\mathcal{O}_X$, centered at $p$.
It is ...
- 1,595
0
votes
2
answers
1k
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Generic rank of a coherent sheaf on a projective variety vs. generic rank on the cone
Let $S:=k[X_1,\ldots,X_n]$ be a polynomial ring over a field $k$, with its natural grading, and let $\mathfrak{p}$ be a homogeneous prime ideal of $S$. Also, let $M:=\bigoplus_{i} M_i$ be a finitely ...
- 3,101
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1
answer
108
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Generalization of a Result about degree bounds of invariant rings
A theorem of Knop states that if $G$ is semisimple and connected acting on a vector space $V$ over a field $K$ of characteristic 0, then the degree of the Hilbert series of $K[V]^G$ is less than or ...
- 928
2
votes
1
answer
403
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The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
- 1,788
1
vote
1
answer
235
views
Can the property of essential finite type checked at a point?
Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian.
Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has ...
- 21
5
votes
1
answer
959
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Multiplicative Structures On Free Resolutions
Hello,
this question is related to Differential graded structures on free resolution?.
Given a regular local ring $S$ and $f\in{\mathfrak m}_S\setminus\{0\}$, I am interested in studying $R$-modules ...
- 2,756
1
vote
0
answers
97
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A property of minimal prime ideals in rings with finite chromatic number
Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
- 271
5
votes
2
answers
689
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Irreducible components of quotients of Cohen-Macaulay rings of the "correct" dimension
Suppose $R$ is a Cohen-Macaulay ring. It is well known that if $I$ is an ideal of $R$ generated by $n$ elements, and $I$ has codimension $n$, then $R/I$ is also Cohen-Macaulay.
Now suppose that $I$ ...
- 3,093
2
votes
1
answer
495
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On the divided power ring over the integers
Consider the divided-power ring $A := \mathbb Z \langle x_1, \ldots, x_n \rangle$ consisting of $\mathbb Z$-linear combinations of divided-power monomials of the form $x_1^{(a_1)} \cdots x_n^{(a_n)}$; ...
- 3,637
1
vote
0
answers
185
views
Free resolutions of affine (non-projective!) varieties
Say, you have an ideal $I$ of a polynomial ring $R = K\lbrack X_1,\ldots,X_n \rbrack$ over an algebraically closed field $K$ (you can assume $K = \mathbb{C}$). What does a minimal free resolution of $...
- 11
1
vote
0
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125
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A question from Hilbert's Nullstellensatz [closed]
From the Hilbert's Nullstellensatz, we know that for any algebraic closed field $K$ and any prime ideal $p$ of $K[X_1,X_2,\cdots,X_n]$, the intersection of all the maximal ideals containing $p$ is $p$....
- 1,997
1
vote
1
answer
361
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Algorithm for Polynomial Reduction in a Quotient Ring
Any reference or suggestion for the following problem would be greatly appreciated.
I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want ...
- 1,256
1
vote
0
answers
29
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Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (...
- 10.5k
11
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1
answer
946
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Is ΩΣ in {simplicial commutative monoids} group completion?
Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory ...
- 25.2k
3
votes
2
answers
667
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Normality and rational singularities via Hilbert series
Let $A$ be a finitely generated ${\mathbb Z}_{\geq 0}$-graded algebra over a field without zero divisors;
assume that all graded components are finite-dimensional and that $Spec(A)$ is smooth
outside ...
- 7,037
2
votes
0
answers
106
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A problem related to parametrizing $\operatorname{rank}\le r$ matrices and Segre embedding
Given a field $k$. We denote $A_{mn}=k[\{X_{ij}\}_{1\le i\le m,1\le j\le n}]$ a polynomial ring of $mn$ variables. Given $m,n,r>0$, we have a natural homomorphism $\phi\colon A_{mn}\to A_{mr}\...
user20948
2
votes
1
answer
170
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Matching power series to infinity
As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ...
- 18.7k
3
votes
1
answer
258
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How to prove this algebra is flat?
Hi,
Let $S = R[T_1,\dots,T_n]/(f_1,\dots,f_r)$ where $\det(\partial f_i/\partial T_j)_{i,j=1,\dots,r}\in S^\times$. Then $S$ is flat over $R$. How to prove it?
I am not looking for an answer like: "$...
- 2,842
2
votes
1
answer
320
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Order of ring automorphisms of localizations of polynomial rings over finite fields
Suppose that $F$ is a finite field and $S\subset F[t]$ is a (finite) set of primes. Is is true that any ring automorphism of $R:=F[t][S^{-1}]$ has finite order?
A ring automorphism of $R$ is ...
- 11.1k
1
vote
0
answers
110
views
Grobner basis for a general algebra
Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...
- 2,384
2
votes
1
answer
404
views
Module structure of the abelianization of the commutator subgroup
Let $G$ be a (non-abelian) group, and let $G_2$ denote its commutator subgroup. Then the abelianization $G_2^{ab} = H_1(G_2,\mathbb{Z})$ is a module over the group ring $\mathbb{Z}[G^{ab}]$. The ...
- 879
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0
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151
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book for help on problems with noetherian rings
Can you please introduce to me a book which would help me to prove the two following problems?
In a noetherian ring, every integrally closed ideal is unmixed.
Let $R$ be a noetherian ring, $P$ a ...
- 565
3
votes
0
answers
46
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What is an example of an integral domain with a module that is 1-separable but not separable?
Let R be an integral domain. All modules under discussion are torsion free unital left R-modules.
An R-module is completely decomposable if it is the direct sum of rank 1 submodules.
An R-...
- 29
3
votes
1
answer
422
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Tor dimension in polynomial rings over Artin rings
I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
- 165
2
votes
1
answer
284
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about the local ring of $\mathbb{Z}_p[T]/(pT^2+T+1)$ at the prime p
is the localisation of the ring $$A:=\mathbb{Z}_p[T]/(pT^2+T+1)$$ at the prime ideal (p) isomorphic to $\mathbb{Z}_p$?
If not, how to understand this ring very explicitly?
- 997
0
votes
0
answers
292
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regular locus of an affine domain
Let $A$ be an affine domain over a field $k$ (need not be algebraically closed). Let $\mathfrak{p}$ be a prime ideal of $A$, such that $A_{\mathfrak{p}}$ is a regular local ring. Does there always ...
