All Questions
Tagged with limits-and-colimits homological-algebra
20 questions
5
votes
0
answers
361
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On a simple alternative correction to Roos' theorem on $\varprojlim^1$
Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
- 1,021
1
vote
0
answers
54
views
contravariant finiteness and limit closure: is there dual to a result of Crawley-Boevey?
Let $\mathcal A$ be a locally finitely presented category. Theorem 4.2 of https://doi.org/10.1080/00927879408824927 says that given a full additive subcategory $\mathcal D$ of finitely presented ...
- 480
6
votes
1
answer
233
views
Comparing stabilization of stable category modulo injectives and a Verdier localization
Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
- 413
3
votes
1
answer
123
views
Vanishing of self-hom in Spanier–Whitehead stabilization category
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
- 413
0
votes
2
answers
156
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Essentially zero inverse system of abelian groups
I am learning local cohomology from Hartshorne’s Local Cohomology book.
My question is about the notion of essentially zero inverse system of abelian groups, which is defined to be an inverse system ...
- 639
6
votes
1
answer
397
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Vanishing of higher limits
Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
- 2,998
0
votes
1
answer
176
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The direct limit of invertible functions on a variety
(I asked this question a couple of days back on Stackexchange but with no success, it seems elementary, but I am struggling to go about attempting it.)
Let $X$ be a smooth geometrically integral ...
- 895
5
votes
0
answers
146
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Applications of $FP_\infty$ groups preserving direct systems
In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are ...
6
votes
1
answer
348
views
Weibel's H-book, Milnor's exact sequence for spectral sequence of filtered complex, Theorem 5.5.5
This is a question which I asked on StackExchange first, but might be more suited here.
I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the ...
- 1,071
9
votes
2
answers
988
views
Reference for homotopy colimit = total complex
I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
- 15.8k
5
votes
1
answer
173
views
Projective module which splits off sequence of submodules, but not the sum
Does there exist an example of a module $X$ over some ring $R$ together with submodules $T_i$ such that:
$X$ is projective,
$X$ splits as an internal direct sum $X\cong T_1\oplus T_2\oplus \ldots \...
- 858
4
votes
1
answer
291
views
Limit of split short exact sequences
Let $X$ be a module over some ring which splits as $$X\cong M_1\oplus S_1\cong M_1\oplus M_2 \oplus S_2 \cong M_1\oplus M_2 \oplus M_3\oplus S_3\cong \ldots$$
where the isomorphisms come from ...
- 858
1
vote
0
answers
213
views
Zero in colimit of sheaves category
This question is motivated by showing that the category $\mathbf{Sheaves} (X)$ from the open subset excluding the empty set category to the category of abelian group $\mathbf{Ab}$ has enough injective ...
- 1,168
2
votes
1
answer
140
views
Need of filtered indexed categories
Similar questions have already been asked here and here but not exactly in the direction I need.
I have a (small) index category $\mathcal{I}$ which is not cofiltered, and I need to consider ...
- 5,670
8
votes
1
answer
1k
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Surjectivity of a map on inverse limits
(The following is crossposted from Math.SE, where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...
- 1,042
7
votes
1
answer
197
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Homology of a limit of semidirect products
Suppose I have two families of groups $A_k$ and $B_k$ indexed by the natural numbers and suppose $B_k$ acts on $A_k$. Suppose there are groups homomorphisms $A_{k+1} \rtimes B_{k+1} \to A_k \rtimes ...
- 111
1
vote
1
answer
606
views
Does the inverse limit of complexes with bounded cohomology have a bounded cohomology?
Let $A$ be a ring (commutative and noetherian if it helps).
Suppose we are given an inverse system $M_i$ of complexes of $A$-modules (where $i$ is a natural number),
and integers $a<b$
such that ...
- 13
11
votes
1
answer
502
views
Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?
Consider a diagram D: I→ChR of real connective chain complexes.
In the example I have in mind all chain complexes are concentrated in some fixed degree n.
There is a canonical map lim D → holim D ...
- 37.8k
5
votes
1
answer
918
views
colimits of spectral sequences
I'm looking for some references about colimits of spectral sequences.
More precisely: let $X : I \longrightarrow \cal{C}$ be a functor from a filtered category $I$ to the category of double cochain ...
- 1,975
1
vote
2
answers
708
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Tot and colimits
This must be a well-known exercise with spectral sequences, but I don't know a reference for it. I'm trying to figure out when does $Tot$ commute with colimits.
More precisely, let $X$ be a double ...
- 1,975