All Questions
Tagged with limits-colimits homological-algebra
45
questions
1
vote
2
answers
60
views
Is it true that $A[2]\cong \varinjlim_i (A_i[2])$ if $A\cong \varinjlim_{i\in I} A_i$?
Let $I$ be a directed set. Let consider the direct limit in the category of abelian groups. Suppose $A\cong \varinjlim_{i\in I} A_i$. Then, is it true that $A[2]\cong \varinjlim_i (A_i[2])$ ? Here, $[...
- 6,351
1
vote
0
answers
67
views
Inverse limit of a quotient space (simple question)
Setup:
I have a tower of abelian groups $\hspace{1em} \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0$.
There are similar towers for $B_i, C_i$, and $D_i$.
There ...
- 187
0
votes
1
answer
35
views
Direct and inverse limit after deleting some groups
Suppose we have a preordered set $(I,\le)$ and a sequence of abelian groups $\{G_i\}$ with a homomorphism $\alpha_{i,j}:G_i\rightarrow G_j$ if $i\le j$ in $I$.
Let $G$ be the direct limit of this ...
- 3,047
3
votes
0
answers
50
views
Writing an enriched graph as a colimit
I am working with enriched directed graphs (aka, directed graphs/quivers such that the edges are objects in a category V). I can write every graph as a filtered colimit of finite graphs, and I can ...
- 31
3
votes
0
answers
198
views
$\mathrm{Ext}$ and direct limit
Let $R$ be a commutative Noetherian ring. Then, for $R$-modules $\{X_i\}_i$ and $Y$, do we always have $$\mathrm{Ext}^n_R(\varinjlim X_i,Y) \cong \varprojlim \mathrm{Ext}^n_R( X_i,Y)$$ for all $n\geq ...
- 433
3
votes
1
answer
151
views
Is every short exact sequence the direct limit of short exact sequences between finitely presented modules?
Let $R$ be a unitary, associative ring. It is well-known that every $R$-module is the direct limit (filtered colimit) of finitely presented $R$-modules. Is it also true that every short exact sequence ...
- 635
2
votes
1
answer
114
views
Intuition behind cokernel pair
I'm trying to find some insights about cokernel pairs. I understand the definition and I know how to calculate them in concrete categories, but while the kernel pairs seem to have a very sensible ...
- 789
4
votes
1
answer
69
views
Projective Limits of Compact Groups: Exact or Not?
I am reading the following lemma from Washington's book "Introduction to Cyclotomic Fields":
On the other hand, there is a counterexample, given by this answer. The comments below this ...
- 77
1
vote
0
answers
37
views
Is a scalar extension of an $(t)$-adically complete module complete?
Let $M$ be a $(t)$-adically complete $\mathbb C[[t]]$-module, say $M$ is a topologically free $\mathbb C[[t]]$-module. Let $\mathbb C((t))$ be the field of formal Laurent series. Is then the scalar ...
- 924
1
vote
1
answer
114
views
Is the ring of formal power series in $n$ variables the colimit of the powers of its maximal ideal
Let $\mathbb C[[t_1, ..., t_n]]$ be the noetherian local ring of formal power series in $n$ variables with a unique maximal ideal $\mathfrak{m}=(t_, ..., t_n)$. Then, there is a descending chain of ...
- 924
0
votes
0
answers
128
views
Meaning and examples of Grothendieck condition AB4*
What are some examples of AB4* categories? In particular, for a thesis I am writing I need to know if cochain-complexes form an AB4* category
- 566
1
vote
0
answers
45
views
When $\mathrm{Hom}$ functor commutes with colimits in a category of modules? [duplicate]
I was looking through this question, and there's a thing I don't understand why it holds. I mean the next statement in the answer by @Peter McNamara:
Since $R$ is Noetherian, $I$ is finitely ...
- 432
1
vote
0
answers
96
views
Compute $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$
I know that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})=\prod_p({\widehat{\mathbb{Z}_p}/\mathbb{Z}})$. But the following steps took me a long time to figure out what was wrong with ...
- 68
4
votes
2
answers
347
views
Is the limit of a family of sheaves a sheaf?
So, I can prove that the kernel of a morphism of sheaves or a product of sheaves is a sheaf, but I do not know how to prove in general that $lim F_{i}$ is a sheaf for $F_{i}$ sheaves. I know that if ...
- 697
3
votes
0
answers
624
views
Direct limit, inverse limit and Spec
The set-up
$k$ is a field
$T_i = \operatorname{Spec} A_i$ is an inverse system of affine $k$-schemes, where $i<j$ if $\operatorname{Spec} A_j \subset \operatorname{Spec} A_i$ (inclusion).
$X$ is a ...
- 1,207