Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
2
votes
0
answers
629
views
Induced map on algebraic de Rham cohomology
Let $X/k$ and $Y/k$ be two smooth affine varieties over a field $k$ with $\mathrm{char}(k) = 0$ and $\varphi: X \rightarrow Y$ be a morphism. I would like to know under what conditions, the induced ...
- 1,109
1
vote
1
answer
114
views
Decomposition of skew-symmetric maps
Let $A$ be a ring and let $F$ be a finitely generated, free $A$-module. Let $\alpha: F \to \textrm{Hom}_A (F, A)$ be a skew-symmetric homomorphism, i.e. $\alpha(x)(y)=-\alpha(y)(x)$ for all $x,y \in F$...
- 3,031
0
votes
0
answers
138
views
Profinite Local Ring inside Polynomial Ring
This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
- 61
5
votes
0
answers
101
views
Is there a positive integer k such that any endomorphism of any free module over any commutative ring is a linear combination of k idempotents?
Consider the following Condition (C) on a positive integer $k$:
(C) If $R$ is a commutative ring, if $F$ is a free $R$-module, and if $f$ is an endomorphism of $F$, then $f$ is an $R$-linear ...
- 4,416
5
votes
1
answer
499
views
software for computations on flag varieties in arbitrary characteristic
Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?
The only one I know of is Macaulay2, via the Schubert2 package, but it works with what ...
- 5,846
3
votes
0
answers
2k
views
Cohomology and tensor product
Let $G$ be a profinite group, $A$ a free $\mathbb{Z}_p$-module of finite rank with a continuous action of $G$ and $B$ any $\mathbb{Z}_p$-module (I am not supposing it to be free), with the trivial ...
- 657
3
votes
1
answer
374
views
Composition and intersection of residue fields
Let $A$ be a normal ring with quotient field $K$. Let $L/K$ be a finite separable extension.
Let $E_1/K$ and $E_2/K$ be extensions of $K$ contained in
$L$. Let $B_1$ (resp. $B_2$) be the normalization ...
- 1,673
4
votes
1
answer
375
views
Regular sequence of elements of degree 1 for a homogeneous Cohen-Macaulay ring
Assume that a positively graded ring R is generated in degree 1. Is it true that, if R is Cohen-Macaulay, then there exists a regular sequence x of elements of degree 1 so that R/x is zero dimensional?...
- 157
2
votes
0
answers
1k
views
Why is scalar extension important?
What I want to know is maybe not as dumb as the bare question.
Suppose B is a commutative unital ring and C is a category of B-modules. Suppose that f : A --> B is a homomorphism, and F is ...
- 403
1
vote
0
answers
148
views
Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$
Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$.
Definition(Super-Gorenstein ideal): $...
- 87
1
vote
0
answers
206
views
Example for 1-dim, Noeth., local domain which is unibranched but not analytically irreducible
Does somebody know an example for an 1-dim., Noeth., local domain $D$ which is unibranched (that is, its integral closure $D'$ is local) but not analytically irreducible (that is, its $\mathfrak{m}$-...
- 11
1
vote
0
answers
2k
views
Formal power series ring & completion
I encountered the following passage in Matsumura's Commutative Ring Theory :
A a Noetherian ring, $B=A[[x]]$ a formal power series ring. $M\subset B$ a maximal ideal, $\mathfrak{m}=M\cap A$. Then $(...
- 2,857
3
votes
1
answer
475
views
Can the factorisation of (p) in a number field K be described by the minimal polynomial of a primitive element?
Let $K$ be a number field, with ring of integers $O_K$, and let $\alpha\in O_K$ be a primitive element for the extension $K/Q$, with minimal monic polynomial $f(x)\in Z[x]$. If $p$ is a prime number, ...
- 2,406
2
votes
1
answer
286
views
Modules with small support have big depth - reference wanted
Hello,
I would appreciate an exact reference / proof of the following fact, which I am almost able to prove, but not really:
Let $A$ be a regular Noetherian comm. ring, of finite Krull dimension. ...
- 5,562
2
votes
1
answer
336
views
Extension of radical ideal after adjunction of roots
I want to apologize in advance if this is blatantly trivial, but I already posted on math.stackexchange.com and got no answer at all.
Let $A$ be a Noetherian domain containing an algebraically ...
- 4,902
3
votes
1
answer
546
views
Center of the category of $R$-algebras
Let $R$ be a ring (assumed to be commutative and with unit). What is the center of the category of $R$-algebras, i.e. $\text{Z}(R\text{-Alg})$? This is a commutative monoid. See this recent question ...
- 63.1k
0
votes
2
answers
563
views
Primary decomposition of zero-dimensional modules
(I removed my motivation because it may be misleading :) )
Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) ...
- 5,243
1
vote
1
answer
375
views
Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?
Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-...
- 5,349
2
votes
1
answer
196
views
Relations between a set of inner products of vectors
Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A_1\rangle,\dots,|A_n\rangle$, $\langle A_i|A_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products $\...
- 23
3
votes
0
answers
258
views
On comparison of various linear topologies on a noetherian local ring
In what follows we will always use this notation:
$R$ will be a commutative noetherian ring with unity, $X=\mathrm{Spec}\:R$, $f\colon X\rightarrow X$ a self-morphism of schemes, $\varphi\colon R\...
- 3,101
0
votes
1
answer
119
views
Colon operation after adjoint variables
Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Let $I$ an ideal of $R$. We have
$$0:_MI = \cap_x(0:_Mx),$$
where $x$ runs a set of generators of $I$.
Now set $S = ...
- 2,773
1
vote
1
answer
234
views
Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$
What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
- 189
2
votes
2
answers
669
views
Maximal Cohen Macaulay modules over regular factor rings.
Hi,
my question is simple. Let (R,m) be a commutative regular local noetherian ring. Is it true that for every prime p \in Spec(R), the factor ring R/p has maximal cohen-macaulay R/p-module?
Best ...
- 203
5
votes
0
answers
296
views
Minimal Koszul-Tate resolutions
In what generality of commutative associative algebras does there exist a minimal Koszul-Tate resolution? Or what is the most general condition known?
- 3,880
2
votes
1
answer
81
views
Degenerations and spanning monomials
Let $R = \mathbb{C}[x_1,…,x_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear ...
- 2,537
2
votes
1
answer
99
views
Homocyclic primary module over PID
I posed the question here, but get no answers yet.
Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime ...
- 767
4
votes
2
answers
670
views
term for a "faithful" module
Is there a term for an $A$-module $M$ such that $M \otimes_A -$ takes nonzero modules to nonzero modules?
Motivation: It is a standard theorem that if $B$ is faithfully flat over $A$, then $\hbox{...
- 7,338
0
votes
1
answer
303
views
Completion of a completion
Let $A$ be a commutative ring (not necessarily noetherian).
Let $I\subseteq J\subseteq A\,$ be two finitely generated ideals.
Let us denote the completion functor by $\Lambda_K (M) = \varprojlim_n M/...
- 1
3
votes
0
answers
916
views
Unibranch rings
Let us call a Noetherian local ring $A$ unibranch if it is a domain and the normalization map is finite and induces a bijection on spectra.
My question is as follows: is this property preserved when ...
- 1,209
1
vote
1
answer
221
views
4
votes
1
answer
244
views
Unique matrix satisfying a system of equations
Assume I have a $n\times n$ positive semidefinite matrix $G$ of rank $p$ satisfying a set of $np - p(p-1)/2$ equations $v^T_jGv_j = 1$, $j = 1 \ldots np - p(p-1)/2$ for some given vectors $v_j$. It is ...
- 149
2
votes
3
answers
1k
views
Commutative Noetherian Domains of Krull Dimension One
k is an alegraically closed field and A is a commutative k-algebra. We also know that A is a Noetherian domain and its Krull dimension is one. Are there any necessary and sufficient conditions on A ...
- 21
1
vote
0
answers
155
views
Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber
Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
- 89
5
votes
1
answer
541
views
Tensor product of regular ring (with some conditions)
Basically, my question is whether this answer is correct. Here is the point. Let $R$ be a ring, and let $A$ and $B$ be $R$-algebras. Suppose that $A$ is regular and $B \otimes_R B$ is regular too. ...
- 3,704
2
votes
1
answer
308
views
Question on bigraded modules
Let $R$ be a polynomial ring $k[x_1,...,x_n]$, let $f_1,...f_s$ be some non-zero polynomial sin $R$ of degree $p_1,...,p_s$ respectively.Define $S$ by $k[X_1,...,X_n, T_1,...T_s]$ with bigrading ...
- 325
0
votes
0
answers
244
views
Properties of Gorenstein ideal
Fix an integer $k>4$. For any integer $r>0$, denote by $S_{r}:=\mathbb{C}[X_0,X_1,X_2,X_3]_{r}$ the vector space of degree $r$ polynomials in $X_i$ with coefficients in $\mathbb{C}$. Let $W$ be ...
- 1,040
3
votes
1
answer
216
views
Simple reference for valuative criterion of integrality?
I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles ...
- 27.9k
4
votes
2
answers
759
views
What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?
As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...
- 25.8k
2
votes
0
answers
81
views
variants of ramification groups - need terminology and sources
I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question.
Let $L/K$ be a Galois extension, and $w$ be a valuation of ...
- 687
2
votes
0
answers
89
views
Orders of certain quotients of power series rings
Let $\Lambda_d := \mathbb{Z}_p[[T_1, \ldots, T_d]]$ denote the ring of formal power series in $d$ variables over the ring of $p$-adic integers. Suppose that $g \in \Lambda_d$ is an irreducible element,...
- 21
1
vote
1
answer
482
views
Injective hulls of residue fields of a local ring and its ring invariants by finite group action
Let $R$ be a local ring, $m$ its maximal ideal and $k:= R/m$ its residue field.
Suppose that a finite cyclic group $G= \mathbb{Z}/ m \mathbb{Z}$ has a linear nontrivial action on $R$.
Let $R^G$ be a ...
- 909
1
vote
0
answers
143
views
Automorphism on F_2[[X,S]]
Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that
$\sigma \colon S \mapsto S + S^2 + S^3$
$\sigma \colon X \mapsto X + S$.
It is easy to see that the ideal $(S)$ is stable ...
- 19
4
votes
1
answer
552
views
Factorization of schemes
Let $k$ be a ring (perhaps a field). Let $M$ be the "set" of isomorphism classes of $k$-algebras and regard it as a commutative monoid with multiplication $\otimes_k $ and unit element $k$. There is ...
- 63.1k
6
votes
1
answer
272
views
Is a certain symmetric power reprsentation of GL(m) cyclically generated
Let $V_m$ be the $m$-dimensional complex vector space with basis $\{e_1, \dots, e_m\}$ and let $i\leq m$. Consider the element ${v}_0^i \in S^i(S^m(V_m))$, where ${v}_0$ is the element $e_1\dots e_m \...
2
votes
2
answers
492
views
Model Theoretic Localization
This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, \times)...
- 761
1
vote
3
answers
467
views
$\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$
Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a_1,\dots,a_n$. Then there is an isomorphism $\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$ ...
- 671
1
vote
1
answer
963
views
Question on an exercise in Hartshorne: Equivalence of categories
This is a slight reformulation of exercise II.5.9.(c) in Hartshorne's "Algebraic Geometry" which I don't understand.
Let $K$ be a field and $S=K[X_0,\ldots,X_n]$ a graded ring. Set $X=Proj(S)$ and ...
- 2,782
0
votes
1
answer
111
views
Explicit representation of $R[\frac{x}{y}]$ where $x, y\in R$ for non-Euclidean PIDs $R$?
It's a fact proven by Pendleton, Gilmer, and Ohm (as an obvious corollary of their work, anyways) that PIDs are QR-domains, meaning every overring (ring between the domain and the quotient field) is a ...
- 147
2
votes
0
answers
164
views
Flags of varieties
I was wondering if there is a generalization of flags in the following way: Suppose you have a series of inclusions of affine varieties $V_1\hookrightarrow V_2\hookrightarrow\cdots\hookrightarrow V_n$ ...
- 928
1
vote
0
answers
255
views
Fitting ideal/ determinantal variety
Let $R$ be an integral domain, "nice" (regular for instance). Consider a homomorphism
$$
f: R^m \to R^m
$$
of two rank $m$ free $R$ modules. Assume that $\ker f =0$ and that the cokernel is $M$. Now ...
- 131