All Questions
25 questions
9
votes
1
answer
325
views
Nonzero module with vanishing derived fibers
What's an example of a nonzero $R$-module with vanishing derived fibers at all points of $\mathrm{Spec}(R)$?
This was asked in When does a quasicoherent sheaf vanish?
but the answer there only says ...
- 2,346
0
votes
2
answers
387
views
Torsion of modules
Given a left module $M$ over a domain $R$, I am interested in irreducible elements $r\in R$ such that $r\cdot m=0$ for some $m\in M-\{0\}.$ I think "torsors" would be perfect name for such $...
- 2,390
4
votes
1
answer
357
views
Surjectivity of natural map of rings
$\DeclareMathOperator\Hom{Hom}$Let $A$ be an integral domain and $P$ be a prime ideal in $A$. We denote $B=A/P$ then is the following natural map
$$\Hom_A(P,A)\otimes_A B\to \Hom_A(P,B)$$ surjective?
...
- 585
2
votes
1
answer
394
views
When the annihilator of each nonzero submodule is prime
Let $M$ be a fixed faithful $R$-module over integral domain $R$. Is there any equivalent condition (on $R$ or on $M $) under which the annihilator of any nonzero submodule of $M$ to be a prime ideal ...
- 21
1
vote
0
answers
99
views
Finding an injective envelope containing another injective envelope
Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
- 147
4
votes
1
answer
669
views
When every proper submodule of a free module is contained in a maximal submodule
Let $(R, m)$ be a commutative local ring (it is not Noetherian in general) and $F$ be a free $R$-module. Under what conditions every proper submodule of $F$ is contained in a maximal submodule.
- 71
1
vote
1
answer
161
views
Geometric meaning of colocalization of modules?
Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...
- 10.5k
7
votes
2
answers
1k
views
Beauville-Laszlo for schemes
For a commutative ring $A$ and $f\in A$ a non-zero divisor, the Beauville-Laszlo theorem gives a gluing statement for vector bundles on $A$ in terms of a vector bundle on $A\big[\frac{1}{f}\big]$, a ...
- 3,472
8
votes
2
answers
542
views
Basis for free modules over an affine domain
Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$.
Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...
- 5,901
8
votes
1
answer
359
views
Global to local principle for f.g. $\mathbb{Z}[x]$ modules
In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...
8
votes
1
answer
299
views
Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module?
Let $R$ be a one-dimensional, reduced and noetherian $k$-algebra (we may also assume that $R$ is a finite $k[x]$-algebra). Let $M$ be a finitely generated, torsion-free module over $R$, i.e. no ...
- 435
1
vote
0
answers
294
views
Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
3
votes
0
answers
432
views
When is every submodule of a module a direct sum of indecomposable submodules?
Is there any reference for modules over a commutative ring with identity such that every submodule of them is a direct sum of indecomposable submodules? Or is there any characterization of such ...
- 41
2
votes
0
answers
122
views
Descent for Dualizable Modules
It's known that a pure morphism of commutative rings $\phi:A\to B$ is of effective descent for the stack of modules. In other words if $\phi$ is pure one can recover $Mod(A)$ as the 2-limit of a ...
- 10.4k
4
votes
0
answers
74
views
self-cogenerator rings
Let $\mathbb{U}$ be a non-empty set (class) of objects of a
category $C$. An object $B$ in $C$ is said to be cogenerated by
$\mathbb{U}$ or $\mathbb{U}$-cogenerated if, for every pair of
distinct ...
- 41
0
votes
1
answer
131
views
Radical of modules [closed]
Let $R$ be a local ring with the unique maximal ideal ${\frak m}_R$ and $M$ be a $R$-module. Define
$I(M) \colon= \cap ~({\mathrm{all~ proper~ maximal ~submodules~ of}}~M)$,
where proper means ...
- 781
9
votes
0
answers
420
views
Geometric interpretation of minimal number of generators of a module
Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, \...
- 243
0
votes
0
answers
165
views
on the ``generic" modules of finite length (skyscrapers)
Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.)
Let $M$ be a finitely generated $R$-module ...
- 2,232
2
votes
1
answer
191
views
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
25
votes
5
answers
3k
views
is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?
This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
- 1,162
3
votes
3
answers
484
views
Support of a module over a polynomial algebra
In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u_1,...,u_l]$
$$D \to E \to F,$$
we have that
Supp $E \subset$ ...
- 502
8
votes
2
answers
397
views
A criterion for freeness over a local ring
Let $A=K[[X_1,\dots,X_n]]$ where $K$ is a field. Let $M$ be a finitely generated torsion-free $A$-module, such that
for all $k$, the $A[1/X_k]$-module $M[1/X_k]$ is free of rank $d$;
for every $i \...
- 6,924
28
votes
3
answers
3k
views
Equivalent definitions of invertible modules
Let $R$ be commutative unital ring, and $M$ an $R$-module. $M$ is called invertible (a.k.a. projective module of rank one), if it is finitely generated, and $M_{\mathfrak{p}} \cong R_{\mathfrak{p}}$ ...
- 523
1
vote
2
answers
702
views
Is a reduced, torsion-free module of finite rank over an Henselian ring free?
Let $R$ be an Henselian discrete valuation ring with field of fractions $K$. Let $M$ be a torsion-free $R$-module of finite rank (i.e. $dim_K(M\otimes_RK)<+\infty$). Let $D$ be the maximal ...
- 73
13
votes
2
answers
3k
views
Wikipedia's definition of 'locally free sheaf'
Let $R$ be a, say, noetherian ring and $M$ an $R$-module. The Wikipedia article on 'locally free sheaf' tells me that the following two statements are equivalent:
The module $M$ is locally free (Edit:...
- 2,782