All Questions
6,057 questions
3
votes
1
answer
321
views
spurious torsion under compositions of linear maps
Say we have a PID $R$, integers $1 \leq a \leq b$, and $R$-homomorphisms $R^a \stackrel f\to R^b \stackrel g\to R^a$ with $g \circ f$ of full rank.
For $h = f, g, g \circ f$, let $c(h)$ be the ...
- 735
2
votes
1
answer
323
views
Presentation of finite modules with null annihilator
Let $R$ be a noetherian local ring and let $M$ be a finite $R$-module. Assume that the annihilator of $M$ is zero. Consider a minimal presentation of M as follows: $R^n\stackrel{\varphi}{\...
- 3,101
1
vote
1
answer
2k
views
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
2
answers
872
views
Rational power series
If we let $R=\mathbb{Z}[x]$ and $D=\mathbb{Z}[[x]]$. We say that $z\in D$ is rational if there is $g\in R$, $g\ne 0$ such that $zg\in R$. Let $S$ be the set of all rational elements in $D$. Then $S$ ...
- 233
3
votes
0
answers
461
views
Krull dimension of non-integral extensions
Some hours ago, a question was posted, asking (citation by heart, not literally)
Let $R$ be a commutative ring (with unit). What can be said about the Krull dimension of an algebraic, non-integral ...
- 16.2k
1
vote
2
answers
355
views
Is there a relationship between the right global dimensions of R and R[1/v]?
A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...
- 30.3k
8
votes
2
answers
1k
views
Algebra Counterexample Request: Linear Quotients
A result of Herzog, Hibi, and Zheng in "Monomial ideals whose powers have a linear resolution" states that:
Theorem: Let $I\subseteq\Bbbk[x_1,\ldots,x_n]$ be a monomial ideal generated in degree 2. ...
- 1,324
2
votes
1
answer
325
views
Gersten for homotopy invariant K-theory of non-singular varieties.
Here is the question:
if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective?
If $X$ is ...
- 1,347
11
votes
1
answer
949
views
Magma "actions" (or alternatively, "What is the Yoneda lemma for magmas?")
Arguably the most import thing about groups, semigroups and more generally categories, is that they can act on sets (or even collections of sets in the case of a category). This is the basis for all ...
- 2,392
52
votes
3
answers
5k
views
What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?
On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...
- 30.3k
6
votes
1
answer
517
views
Growth zeta-functions of regular languages
Dear All,
my following question may be known and ought to be known, so in case it is folklore please could you give me the references.
To start, it is obvious that growth of rational languages are ...
- 1,437
12
votes
3
answers
3k
views
Can we say anything about the Krull dimension of a localization?
I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...
- 30.3k
12
votes
2
answers
1k
views
Cohen-Macaulay domain with non-Cohen-Macaulay normalization?
Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one ...
- 10.7k
29
votes
2
answers
5k
views
Regular, Gorenstein and Cohen-Macaulay
All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;
It is well-known that every regular ring is Gorenstein and every Gorenstein ring is ...
- 1,661
2
votes
1
answer
505
views
graded noetherian module
Let M be a R graded module with $M= \oplus M_i$. If M is noetherian then $M_i=0 $ for i << 0. My question is this, isn't $M_i = 0$ for all i >> 0 as well? If $(M_{n_i})_{i} \neq 0, n_i > 0$ ...
- 131
2
votes
3
answers
2k
views
Algebraic extensions of p-adic closed fields
I have been working with p-adically closed fields and there are two results that are used time and times again in what I am reading, but I cannot find any references where they are proved...
The ...
7
votes
1
answer
2k
views
structure theorem for modules
Can structure theorem for modules be extended to modules over UFDS , to modules over Neotherian rings ? if yes then can one get the statement and reference?
Since operations on matrices with ...
- 201
2
votes
2
answers
1k
views
Irreducible component of a Cohen-Macaulay variety
Is it true that an irreducible component of a Cohen-Macaulay variety is also Cohen-Macaulay? If not, then in what cases does this fact hold?
- 23
1
vote
0
answers
273
views
Depth of intersection
Let $I$ be an ideal in $S=K[x_1,\dots,x_n]$. Can we compute $\operatorname{depth}(I\cap K[x_1,\dots,x_r])$ with $r \leq n$? Is there any relation between depth $I$ and $\operatorname{depth}(I\cap K[...
- 287
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
515
views
Cohen Macaulay, free and finitely generated module
Here is an unsolved problem for me in Kaplansky's "Commutative rings" p. 103, no. 18.
Let $R$ be a Cohen-Macaulay ring. Let $T$ be a ring containing $R$ and suppose that as an $R$-module it is free ...
- 1,661
16
votes
1
answer
5k
views
Minimal primes and zero-divisors
The non-zero elements of the minimal prime ideals of a noetherian commutative ring are zero-divisors.
The proof of this fact I know of uses primary decomposition. Are there alternative (e.g. more ...
- 16.2k
24
votes
3
answers
3k
views
Automorphisms of a weighted projective space
What is the automorphisms group of the weighted projective space $\mathbb{P}(a_{0},...,a_{n})$ ?
Consider the simplest case of a weighted projective plane, take for instance $\mathbb{P}(2,3,4)$; any ...
- 8,998
6
votes
3
answers
787
views
Trace of the identity map in a projective module
Let $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $m_1,\ldots,m_n$ be a set of generators of $M$. The Dual ...
- 777
3
votes
0
answers
289
views
Terminal quasi-affine varieties?
Let $U$ be a normal, irreducible, quasi-affine variety over the algebraically
closed field $k$ and consider the ring $A = \mathscr{O}(U)$ of regular
functions on $U$. Write $Max(A)$ for the ...
- 31
4
votes
0
answers
811
views
$Ext$ functor, filtered complexes and spectral sequences
Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
5
votes
0
answers
331
views
Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
3
votes
1
answer
1k
views
Does totally ramified extension really exist?
The answer is certainly "Yes", but this is the problem I met in Algebraic Number Theory by Neukirch. I guess that I must be doing something wrong, since otherwise I will get the statement "There are ...
- 95
15
votes
3
answers
4k
views
Elementary Luroth theorem proof?
Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
- 1,422
5
votes
2
answers
367
views
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
5
votes
0
answers
190
views
"Unknot" algebraic set defined by two mutually dependent set of variables
Let $n$ be an integer $\geq 4$, and let $V \subseteq {\mathbb C}^{2n-1}$ be the set of all
$(a_1,a_2, \ldots ,a_n,b_1,b_2, \ldots ,b_{n-1}) \in {\mathbb C}^{2n-1}$ such that the derivative of the ...
- 3,595
6
votes
1
answer
530
views
Does a variety contain a cartesian product of two curves?
We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. ...
- 7,836
0
votes
0
answers
237
views
resolution of singular points on plane curves and base change
Let $k$ be a field and $C/k$ be an affine plane curve over $k$, namely $C = \mathrm{Spec}(A)$ for some $A = k[x,y]/(f(x,y))$, here $f(x,y) \in k[x,y]$ is an irreducible polynomial. Let $B$ be the ...
- 1,109
4
votes
1
answer
2k
views
Primary decomposition and finitely generated abelian groups
In a question asked by Ben Webster, Harry Gindi commented that it is possible to prove the classification theorem from finitely generated abelian groups by appealing primary decomposition.
I have ...
- 3,830
15
votes
1
answer
4k
views
is residue field ever flat over its local ring?
Let R be a local ring with maximal ideal m and residue field k. Is k ever flat over R? What conditions are needed on R?
Sorry, it's not a very profound question. It came up in a derived functor ...
- 499
2
votes
3
answers
1k
views
Algebraic structures of greater cardinality than the continuum?
Are there interesting algebraic structures whose cardinality is greater than the continuum? Obviously, you could just build a product group of $\beth_2$ many groups of whole numbers to get to such a ...
- 187
0
votes
2
answers
617
views
An element in the product of schemes
Let $ f : X \to S, g : Y \to S$ be the scheme morphisms, and $ X \times_S Y$ be the product of shemes. Let $ z \in X \times_S Y$ , and $ x=p(z) , y=q(z) ,$ where $ p: X \times_S Y \to X, q: X \times_S ...
- 95
1
vote
1
answer
1k
views
resolution of singular points on curve
After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of ...
- 1,109
6
votes
2
answers
541
views
Positive matrices matrices over commutative rings
Assume that $R$ is a commutative ring with a ring compatible ordering and let $A$ and $B$ be symmetric $n\times n$ matrices with entries in $R$ such that $\sum x_iA_{ij}x_j\geq 0$ and $\sum x_iB_{ij}...
- 777
5
votes
2
answers
754
views
Do all finitely generated nilpotent semigroups have polynomial growth?
The notion of nilpotency passes nicely from groups to semigroups. Define $q_1(x,y)=xy$ and $$q_{i+1}(x,y,z_1,\cdots,z_i)=q_i(x,y,z_1,\cdots,z_{i-1})z_iq_i(y,x,z_1,\cdots,z_{i-1})$$ inductively for all ...
- 1,437
7
votes
1
answer
433
views
Powers of maps on finite sets
Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. It is a monoid under the operation of composition. Let $m$ be a positive integer. How many maps in $...
- 385
9
votes
3
answers
1k
views
Is $k[x_1, \ldots, x_n]$ always an integral extension of $k[f_1, \ldots, f_n]$ for a regular sequence $(f_1, \ldots, f_n)$?
The elements of a regular sequence in $k[x_1, \ldots, x_n]$ are algebraically independent over $k$ (see for example Matsumura ex. 16.6), and so for a length n regular sequence $(f_i)$ of homogeneous ...
- 93
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
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
1
answer
325
views
Is a particular element of a particular ring a nonzerodivisor?
Let $A$ be the ring $\Bbbk[\alpha_0, \alpha_1, \alpha_2, x_0, x_1, x_2]$ (where $\Bbbk$ is an infinite field, algebraically closed if it matters). Let $g \in \Bbbk[\alpha_0, \alpha_1, \alpha_2]$ be a ...
- 7,338
4
votes
3
answers
1k
views
Polya's theory of counting and commutative algebra
Do you know if there exist algebraic studies of the ring of the power series which emerge when using the theory of Polya for enumeration of sets with certain symmetries? For instance if some ideals ...
- 902
7
votes
4
answers
2k
views
product of rings
I feel a need to apologies for this question, since it seems to be to basic to be asked.
in this question I am primarily concerned with commutative rings and therefore all rings here are assumed to ...
- 2,027
7
votes
1
answer
912
views
Optimal reference for tensor product of symmetric bilinear forms?
This is just a reference request on a relatively elementary level (for which I apologize in advance), but every time I bump into this question I suspect I'm missing the "correct" conceptual setting. ...
- 52.9k
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
447
views
Commutator tensors and submodules
Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$.
For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical ...
- 33.8k