Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
4
votes
1
answer
711
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Faltings' category of almost modules
Hi,
Let $V$ be an integral domain with an ideal $m\subset V$ and put $K = S^{-1}V$ where
$S = {1}\cup m$ (a multiplicatively closed subset). Is it true that the category of almost
$(V,m)$-modules is ...
- 2,842
2
votes
0
answers
86
views
Commutative noetherian domains with large fixed rings
Let $R$ be a commutative domain and let $\theta$ be a ring automorphism of $R$. The fixed ring of $\theta$ is defined by $R^{\theta}:=\{r \in R: \ \theta(r)=r \}.$ An ideal $I$ of $R$ is called ...
- 21
1
vote
1
answer
143
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Bound on the weight of the minimum weight generator of [n,k] cyclic codes?
I'm looking at creating sparse generator matrices for cyclic codes of a given length and dimension. A generator matrix of an [n,k] cyclic code can be expressed as
$G = \begin{bmatrix}g_0 & g_1 &...
- 111
0
votes
2
answers
2k
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non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
7
votes
1
answer
1k
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Are squarefree monomial ideals on a regular system of parameters in a regular local ring radical?
Suppose $(R,m)$ is a regular, local ring. Let $x_1,x_2,...,x_n$ be a regular system of parameters. Let $I$ be an ideal generated by squarefree monomials in the $x_i$'s. Is $I$ a radical ideal? The ...
- 225
4
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0
answers
112
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Centers of Noetherian Algebras and K-theory
I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...
- 161
1
vote
0
answers
199
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Criterion for global dimension of subring
All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...
- 5,405
4
votes
1
answer
240
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Duality for rank one modules over a number ring
Let $K$ be a number field, and $R$ an order of $K$. Consider the category $\mathcal{M}$ of all finitely generated $R$-submodules of $K$. If $X$ is an object of $\mathcal{M}$ such that $R=\textrm{End}...
- 2,406
0
votes
1
answer
149
views
$I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$
In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals),
an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$.
[See Comm. Rings by Kaplansky, ...
- 41
5
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0
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194
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How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]
Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...
- 5,637
0
votes
1
answer
164
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Extending derivations to the superposition closure
Let $X$ be a set and $\mathcal{F}\subseteq {\mathbb{R}^X}$ an arbitrary family of functions.
The superposition closure of $\mathcal{F}$ is defined as
$$
\overline{\mathcal{F}}=\{ H\circ(f_1\times\...
- 2,527
1
vote
2
answers
285
views
Decomposition of a quotient module
Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let
$f:R^2 \rightarrow R^2$ denote the map given by the matrix
$$M=\begin{pmatrix}
v & y \\
x & z
\end{...
- 1,207
8
votes
1
answer
742
views
Reasonable implementation of finding Gröbner bases over non-field coefficient rings
Gröbner bases are usually considered in the ring of polynomials over a field. However, there are useful definitions and algorithms for Gröbner bases over other coefficient rings; see, for instance, ...
- 7,338
2
votes
1
answer
224
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Equations for blow-ups non-regular centers
Let $X = \textrm{Spec} A$ be a reasonable scheme and $I\subset A$ an ideal generated by a regular sequence. Then we have a full set of generators/relations for the blow-up of $X$ along $V(I)$.
Are ...
- 3,555
4
votes
1
answer
1k
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Non-existence ofintegral basis of integral closure in a finite extension of Frac(A), A Dedekind.
Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ ...
- 373
0
votes
1
answer
372
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the algebraic closure of strict henselian DVR
Let $A$ be a strict henselian DVR, and $\hat A$ is completion of $A$,
is $K(A)^{alg} \longrightarrow {K(\hat A)}^{alg}$ a isomorphism?
where $K(A)$ and $K(\hat A)$ are quotient fields.
- 1,921
1
vote
1
answer
322
views
Uniqueness of a closed subscheme in a disjoint union
Let $Z =C \cup F$ be a disjoint scheme union of closed subschemes of Pn.
Let $p_C$ be the Hilbert polynomial;
assume
(1) $F$ finite, reduced,
(2) all irreducible components of $C$ are positive ...
- 145
0
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0
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355
views
Can we find a Groebner Basis?
I would like to ask the following. Given only the leading terms of an ideal $I$, namely the set $LT(I)$, is it possible to find a Groebner Basis of $I$? If not always, then when is it possible? We ...
- 1
3
votes
0
answers
450
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Ext groups of affine scheme
Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...
- 491
2
votes
1
answer
877
views
maximal Cohen-Macaulay module [closed]
This is from CM Rings (Bruns & Herzog) p 64 2.1.20. Show that a one dimensional Noetherian local ring has a maximal Cohen Maucaulay module.
- 287
5
votes
1
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632
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Showing an Ext^2 element is zero
If we have an extension of bundles $0 \to E \to F \to G \to 0$ on $X$, then to show that this is the zero element in $Ext^1_X(G,E)$, we need to show that this sequence splits. To produce a splitting ...
- 145
5
votes
1
answer
2k
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Length of a module over different rings
Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.
Under what ...
- 1,391
6
votes
1
answer
437
views
"Archimedeanising" an ordered field
If $K$ is an ordered field, let $B$ be the subring comprising the $x \in K$ such that $|x| \le n$ for some $n \in N$, and let $I$ be the ideal of $B$ comprising the infinitesimal elements (i.e. the $x ...
- 879
3
votes
2
answers
788
views
Rees algebra for non-radical ideals
Today in my introductory algebraic geometry class we defined the so-called Rees algebra associated with an ideal $I$ of a ring $R$ (with strong conditions on $R$, if you like: I don't mind restricting ...
- 3,605
2
votes
1
answer
169
views
Open idempotents in modules over a local ring
Let $R$ be a local ring. By an open idempotent I mean an $R$-module $F$ equipped with a homomorphism $e : F \to R$ such that $e \otimes F = F \otimes e$ is an isomorphism $F \otimes F \cong F$ (this ...
- 63.1k
0
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0
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82
views
grade of ideals in non-noetherian rings
Let $R$ be a commutative ring with unity, and $M$ an $R$-module. Assume that $I$ and $J$ are finitely generated ideals and $K$ another ideal of $R$. Let $\textbf{x}$ be a sequence of generators of $I$...
- 1,355
2
votes
0
answers
466
views
Is every Artinian local ring a quotient of a nilpotent algebra over a field?
This question arose from a conversation with a friend where we tried to classify all one-point-schemes. Apologies if it's a totally stupid question.
If $R$ is a Noetherian commutative ring with ...
1
vote
0
answers
82
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If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?
Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is commutative,...
- 1,479
2
votes
1
answer
194
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Name and references for a "twisted" addition in a ring
This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...
- 3,513
4
votes
1
answer
427
views
Is this height-transcendence-degree inequality true without AC ?
Let $R$ be a $k$-algebra ($k$ a field) and a domain of finite Krull dimension. In
$\quad$ Krull dimension less or equal than transcendence degree?
it is shown that
$$\text{Krull-dim}(R) \le \text{...
- 16.2k
5
votes
0
answers
260
views
Degree bounds for Grobner Basis
Let $I= \langle f_1, \ldots f_n \rangle \subset K[x_1,\ldots, x_n]$ be a homogeneous ideal and $\operatorname{deg}(f_i) \leq d$ then it has Grobner Basis where degree of each generator is less than or ...
- 1,277
7
votes
1
answer
735
views
Can we make Buchberger's algorithm faster for a given ideal if we are allowed to vary the monomial order?
Suppose we have a finite set of generators for an ideal $I \subset R := \Bbbk[x_1,\dotsc, x_n]$, where $\Bbbk$ is a field. If we choose a monomial ordering, then Buchberger's algorithm allows us to ...
- 7,338
2
votes
1
answer
796
views
How does torsion behave under the direct image functor?
Assume we have a finite morphism $f: X\rightarrow Y$ of smooth projective varieties of degree $d$ over $k=\mathbb{C}$. Then $f_{*}$ induces an equivalence between the categoy of coherent $O_X$-modules ...
- 1,391
1
vote
1
answer
196
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Polynomial analogue of "prime independence"
In number theory a well-known fact is that congruence modulo distinct primes are 'independent'. That is, to know that $n \equiv a \pmod{p}$ does not change the probability as to what $n \equiv x \pmod{...
- 26.9k
4
votes
0
answers
312
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Dimension of a commuting nilpotent variety
Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
- 768
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1
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340
views
Length of a module
Let R be a commutative ring, M an R-module of finite length and let N be an Injective R-module with zero socle. Then why $ \text{Hom}_R(M, N) $ is zero?
- 11
2
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75
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Analogue of Bass's Lemma 2.4 on when inverse images of free modules are free
Let $R$ be a Noetherian integral domain. Let $x\in R$ be a prime element. Let $\overline{R}=R/Rx$.
Let $P$ be a finitely-generated projective $R$-module.
Assume that $\frac{P}{xP}$ is a free $\...
- 1,644
0
votes
0
answers
145
views
Lifting points of étale group scheme
Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the ...
- 445
4
votes
1
answer
141
views
Which power of $2$ kills $W(k)$?
Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...
- 16.9k
1
vote
0
answers
699
views
Is a complete intersection satisfying Jacobian matrix smooth criterion a smooth variety?
Every scheme here is over complex number.
Let $X \subset (\mathbb{C}^*)^n$ be a complete intersection with $X$ defined by the ideal $I \subset \mathbb{C}[x_{1}^{\pm},\dots,x_{n}^{\pm}]$ generated ...
- 3,472
4
votes
0
answers
245
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When does a commutative DGA have a finitely generated quasi-free resolution?
Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question is,...
- 15.5k
2
votes
2
answers
203
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Tychonoff spaces and ideals
Let $X$ be a tychonoff space and let $T$ be the set of all $f \in C(X)$ such that for any $g$ the equation $fg = 1$ has at most finitely many solutions. Under what conditions on $X$, the set $T$ is an ...
user40679
1
vote
0
answers
67
views
Modification of nonfree locus
Let $ R $ be a commutative noetherian ring with identity. Let $ M $ be an $ R $-module. By definition the nonfree locus $ NF(M) $ of $ M $ is defined as the set of prime ideals $ {\mathfrak p} $ of $ ...
- 51
2
votes
2
answers
798
views
Determinantal rings are Cohen-Macaulay
Consider a $n\times n$ matrix $M$ with entries in $R=\mathbb{C}[x_1,\dots,x_n]$. Let $I$ be the ideal of $(n-1)\times(n-1)$ minors of $M$. Is $\mathcal{O}_{\mathbb{C}^n}/I$ Cohen-Macaulay?If not, ...
- 671
5
votes
1
answer
327
views
When is the projective line the seminaive projective line?
Excuse the possible naivete of this question. Since reading a nice survey article by Daniel Biss a few years ago, I'm always worried about what $P^1(R)$ is, for a ring $R$.
So that I stop worrying, ...
- 13.3k
2
votes
0
answers
135
views
Lifting of Commuting Maps of Vector Bundles
Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is ...
- 431
7
votes
2
answers
567
views
Rational powers of ideals in Noetherian rings
Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We define $I_a = \{x \in R: x^q\in \overline{I^p}\}$, ...
- 805
1
vote
1
answer
448
views
Bounding Castelnuovo-Mumford regularity in a short exact sequence
Let $R$ be a commutative ring with unity, $I$ be an ideal and $a\in R$ be an element in $R$. We have the following short exact sequence:$$0\rightarrow R/(I:a)\rightarrow R/I\rightarrow R/(I+(a))\...
user25585
1
vote
0
answers
124
views
Condition for a finite group scheme to be étale [closed]
My question comes from the reading of Tate's paper $p$-divisible groups. In the last few pages there is an argument which gives as trivial the following fact. If we take a $p$-divisible group over a ...
- 445
6
votes
1
answer
137
views
Injective dimension over enveloping algebra
Let $k$ be a field, and let $A$ be a commutative noetherian $k$-algebra.
If a finitely generated $A$-module $M$ has finite injective dimension over $A$, does this imply that $M\otimes_k M$ has finite ...
- 61