All Questions
6,057 questions
16
votes
1
answer
1k
views
Are there non-reflexive modules isomorphic to their bi-dual?
Let $M$ be an $R$-module. We say that $M$ is reflexive if the natural map $M\rightarrow M^{**}$ is an isomorphism.
I'd like to know if there exists a module isomorphic to its bi-dual but not ...
- 279
2
votes
1
answer
575
views
Can you suggest a good name for a local homomorphism φ:(R,m)->(S,n) of local rings with the property that φ(m)S is n-primary?
Can you suggest a good name for a local homomorphism $(R,\mathfrak{m})\stackrel{\varphi}{\rightarrow}(S,\mathfrak{n})$ of local rings with the property that $\varphi(m)S$ is $\mathfrak{n}$-primary?
...
- 3,101
7
votes
1
answer
2k
views
The space of valuations of a function field
Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations.
First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of ...
- 1,804
5
votes
2
answers
1k
views
Where does the definition of "Tower of Algebras" come from?
A tower of algebras is a sequence of algebras
$$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$
with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...
- 527
0
votes
1
answer
208
views
How to consider a module over the ring Q[t,t^(-1)] to be a module over the polynomial ring Q[t]? [closed]
Can we view a module over the ring $\mathbb{Q}[t,t^{-1}]$ to be a module over the polynomial ring $\mathbb{Q}[t]$?
where $\mathbb{Q}$ denote any rational number coefficients.
- 161
4
votes
1
answer
382
views
Is the subspace of DVR's of the Zariski-Riemann space still quasi-compact?
If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that ...
- 1,347
0
votes
1
answer
583
views
Question about modules, quotient rings, and polynomial rings? [closed]
Consider an integer polynomial ring, $A = \mathbb{Z}[t]$, and a ring of fractions, $B = \mathbb{Z}[t, t^{-1}]$; obviously, $A$ is a subring of $B$.
Now we consider two modules over $A$ and $B$, $M$ ...
- 161
4
votes
2
answers
634
views
Why bi-module, bi-bundle, etc?
This is perhaps an ill-proposed question. Any way, thank you guys.
We have a lot of bi-stuffs, such as bi-module, bi-bundle, etc. They are basically two commuting actions from two sides, left and ...
- 1,271
2
votes
1
answer
964
views
When are two projective modules of equal rank isomorphic?
Let $R$ be a commutative ring and let $M,N$ be two finitely generated projective $R$-modules which have equal rank (not necessarily constant). What kind of general results are there concerning the ...
- 777
7
votes
5
answers
2k
views
Commutative algebra final project
I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following ...
21
votes
1
answer
2k
views
When does the relative differential $df=0$ imply that $f$ comes from the base?
Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be
defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of
$B \otimes_A B \to B$, as in [Hartshorne II.8].
If $df=0$,...
- 27.9k
5
votes
0
answers
288
views
sheaves with free abelian stalks over compact space
The question concerns a sheaf $S$ of abelian groups over a compact space $X$. Suppose each stalk $S_x$ is finite rank free. Is the group of global sections free?
- 345
4
votes
2
answers
818
views
Double orthogonal complement of a finite module
Crossposted from math.stackexchange since I'm not getting any answer.
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...
- 141
4
votes
1
answer
1k
views
Is the transcendence degree of a domain over a subfield the same as that of the fraction field of that domain?
Consider the inclusion $k\subset A$ of the field $k$ in the domain $A$ and the fraction field $K=Frac(A)$ of $A$.
Obviously if a family $(a_i)_{i\in I}$ of elements $a_i \in A$ is algebraically ...
- 47.5k
3
votes
1
answer
316
views
Polynomials mapping the twisted cubic power series ring to itself
If $f(X) \in \mathbb{F}_2((T))[X]$ and $f(\mathbb{F}_2[[T^2,T^3]]) \subseteq \mathbb{F}_2[[T^2,T^3]]$, then does it follow that $f'(0) \in \frac{1}{T}\mathbb{F}_2[[T]]$?
This might seem like an ...
- 4,995
1
vote
0
answers
531
views
Krull's intersection theorem for commutative local not necessarily noetherian rings
Is there a characterisation of those commutative local not necessarily noetherian rings that satisfy Krull's intersection theorem ? How can the intersection theorem be phrased in terms the module ...
2
votes
1
answer
463
views
Characterization of prime ideals in regular local rings
Let $R$ be a regular local ring of dimension $d$ and let $x_1,x_2,...,x_d$ be a regular system of parameters. Now, for any $y\in R$, the colon ideal $(x_1,x_2,...,x_h):y$ where $h\leq d$ is a prime ...
3
votes
1
answer
636
views
Left Adjoint to the Forgetful Functor on $\lambda$-rings?
The forgetful functor from the category of $\lambda$-rings to that of rings has a right adjoint in the form of the universal $\lambda$ functor $\Lambda$, which is isomorphic to the big Witt vectors ...
- 822
0
votes
2
answers
511
views
When are seminormal rings Cohen-Macaulay?
I know that not every local seminormal ring is Cohen-Macaulay. But are 1-dimensional local seminormal rings Cohen-Macaulay?
- 179
19
votes
2
answers
5k
views
Are morphisms from affine schemes to arbitrary schemes affine morphisms?
To put this question in precise language, let $X$ be an affine scheme, and $Y$ be an arbitrary scheme, and $f : X \rightarrow Y$ a morphism from $X$ to $Y$. Does it follow that $f$ is an affine ...
- 387
2
votes
2
answers
2k
views
Torsion-free modules over a general ring
I want to know how to prove that a torsion-free module over a general ring is flat. In Lectures on Rings and Modules, T.Y. Lam proves this in the case where your ring is an integral domain. Can you ...
- 17
6
votes
2
answers
567
views
Given 2 towers of fields, when are these fields isomorphic?
Let $F_0 \subset F_1 \subset F_2 \subset \cdots$ and $K_0 \subset K_1 \subset K_2 \subset \cdots$ be two towers of fields. Also, let $F = \cup_{i=0}^\infty F_i$ and $K = \cup_{i=0}^\infty K_i$.
Now ...
- 1,197
7
votes
2
answers
1k
views
Invariants and base change
Suppose $R$ is a Noetherian commutative ring, and $M$ a finite free $R$-module, with an action of a finitely generated discrete group $G$ by $R$-linear maps.
Is there any homological condition on ...
- 71
1
vote
0
answers
345
views
Is the ideal of denominators preserved under flat pullback?
Let $\phi \colon A \to B$ be a flat homomorphism of rings (commutative, with unit). Let $R$ be the total ring of fractions of $A$ (obtained by inverting all nonzerodivisors), and let $S$ be the total ...
- 7,338
11
votes
2
answers
4k
views
Definition of a Grothendieck ring
I've been looking at some definitions of Grothendieck rings. However I've not found a good definition that I've understood. Any recommendations?
I'm referring to the definition in tensor categories, ...
- 379
0
votes
1
answer
223
views
Equivalent functors
Let $R$ be a commutative Noetherian ring, $M$ is a finitely generated $R$-module. If $F: Mod \to Mod$ is a left exact functor and $R^iF(E)=0$ where $E$ is injective module. Assume that $F(-) \cong Hom(...
- 259
2
votes
2
answers
605
views
(non-trivial) isotrivial family of elliptic curves over C^{\times}
So How does one prove (rigorously) that
$$
Frac(\mathbb{C}[x,y,t]/(y^2-x^3-t)) \not\simeq Frac(\mathbb{C}[t][x,y]/(y^2-x^3-1))?
$$
So here $Frac$ denotes the fraction field of an integral domain.
...
- 7,609
4
votes
1
answer
138
views
Pulling back roots from the Completion
Consider the following diagram of regular local rings
$\begin{matrix}
\hat{A} & \xrightarrow{\quad\hat\varphi\quad} & \hat{B} \\
\ \uparrow\scriptstyle\alpha & \circlearrowleft & \ \...
- 4,902
2
votes
2
answers
974
views
Smoothness of hypersurfaces in Grassmannians
I have a general question, and then the specific version of that question I need for research. All vector spaces over $\mathbb{C}$.
Grassmanians of planes
The $(2,n)$-Grassmannian, denoted $Gr(2,n)$...
- 13k
3
votes
4
answers
874
views
Extension of Tate's result regarding Tor
In a 1957 paper (Link), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R then there is a graded commutative DGA $X$ over $R$ with $H_i X=0$ except $H_0 X= R/I$ (I guess R should ...
- 3,726
3
votes
3
answers
519
views
Cohen-Macaulay property for reducible schemes
I have the following question about certain schemes being Cohen-Macaulay.
Let $X$ be the union of all coordinate $k$-planes in
${\mathbb A}^N$. Is it CM?
Let $R$ be a collection of $k$-element ...
- 7,037
0
votes
0
answers
179
views
semigroup actions of groups on regular rooted trees
If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
- 125
4
votes
3
answers
835
views
Field with cyclic product group
If a field has a cyclic multiplicative group, is it necessarily finite?
- 1,441
1
vote
0
answers
383
views
Size of an abelian permutation group with generators of order 2 [closed]
Let $g_1, \ldots, g_k$ be distinct permutations on a set $\Omega$. Suppose that $G = \langle g_1, \ldots, g_k \rangle$ is an abelian permutation group with only elements of order at most 2. Is it ...
- 11
6
votes
1
answer
623
views
When is the cofibrant replacement of a product the product of the cofibrant replacements?
I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE \...
- 30.3k
5
votes
3
answers
1k
views
adjoint of multiplication operator in a commutative algebra
Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\...
- 6,614
12
votes
1
answer
949
views
Discrete version of Nullstellensatz?
Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very ...
- 902
2
votes
0
answers
363
views
A simple problem on commutative algebra related to G.I.T
Let $G$ be a geometrically reductive algebraic group over an algebraically closed field $k$. Let $X$ be an affine variety over $k$ on which $G$ acts regularly. Then $G$ acts on the coordinate ring $A$ ...
- 1,804
6
votes
2
answers
5k
views
Periodic matrices
A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix.
Can we characterize the periodic matrices in $\mathcal{M}_n(\mathbb{Z})$?
If we replace $\mathbb{...
- 2,829
6
votes
3
answers
644
views
Line bundles on fibrations
Let $f:Y \to X$ be a flat morphism with positive dimensional fibers. Is it always true that line bundles that are trivial along each fiber are of type $f^*L$ for $L$ a line bundle on $X$?
- 95
6
votes
3
answers
949
views
Torsion-free tensor powers
Let $R$ be an integral domain. If $M$ is an $R$-module such that every tensor power of $M$ over $R$ is $R$-torsion-free, then is $M$ necessarily flat as an $R$-module? If not, then does this ...
- 4,995
2
votes
2
answers
310
views
Can normalisations of curves over a perfect field change residue fields?
Does anyone know an example of a curve $X$ over a perfect field $k$ such that if $\tilde{X}$ is its noramlisation, there exists a point $x \in X$ and a point $y \in \tilde{X}$ over $x$ such that $k(y) ...
- 1,347
9
votes
0
answers
349
views
Computing Ext for toric divisors
Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute
$$
\...
- 20.5k
10
votes
1
answer
607
views
Are cluster variables prime elements?
Cluster algebras introduction
A cluster algebra is a subalgebra $A$ of $k[x_1^{\pm1},...,x_n^{\pm1}]$ generated by a set of cluster variables, which are elements which can be generated from the set $\{...
- 13k
3
votes
1
answer
191
views
Local coordinate system under finite integral extension
Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras (of the same dimension) which makes $B$ integral over $A$.
Let $\mathfrak{m}=(x_1,\ldots,...
- 4,902
4
votes
0
answers
303
views
Connecting group ring, abelianization
For reasons arising in algebraic topology, I'm wanting to better understand the relations between two functors from groups to abelian groups, $\mathbb{Z}[\cdot]$ and $\operatorname{ab}$; group ring ...
- 1,987
10
votes
1
answer
866
views
Current status of a conjecture of Bloch
In the seminal paper $K_2$ and algebraic cycles, Bloch make the following conjecture :
Suppose $A$ is a local Noetherian integral domain with quotient field $F$
$K_2(A)$ → $K_2(F)$ is injective
...
- 951
3
votes
0
answers
495
views
An elementary proof of a criterion for $k$ sufficiently large that $M_k = \Gamma(\mathbb{P}^n, \widetilde{M}(k))$
It is well known that coherent sheaves on $\mathbb{P}^n$ are equivalent, as a category, to finitely generated graded modules over the polynomial ring, provided that in the latter category, morphisms ...
- 7,338
6
votes
2
answers
563
views
If the discriminant of a binary quadratic form has high valuation, is the form "almost a square".
For a binary quadratic form $ax^2+bxy+cy^2$ over a field (characteristic not 2), the discriminant $b^2-4ac$ is 0 if and only if that form is the square of a linear form. I am curious about an ...
- 2,955
12
votes
4
answers
688
views
Conjugacy for p-adic matrices of finite order II
Question: Say $p$ is an odd prime, and take two matrices $A,B\in GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that if their reductions mod $p$ are conjugate in $GL_n({\mathbb F}_p)$ then they ...
- 5,363