All Questions
Tagged with limits-colimits abelian-categories
25
questions
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votes
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answer
80
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Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives
Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result.
I have no idea on ...
- 3,800
1
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0
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57
views
Non trivial colimit for rings in a finite diagram
I’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a ...
0
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0
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73
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Is the category of graded modules over a graded-commutative ring an AB5 category?
Is the category of $\mathbb{Z}$-graded modules over a graded-commutative ring an AB5 category? It is abelian, the subobjects of each object form a set, and it admits arbitrary coproducts. But I don't ...
- 3,440
6
votes
1
answer
517
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Is taking (co)limits exact in an Abelian category?
Let $\mathcal{A}$ be a complete and cocomplete Abelian category.
Let $J$ a small category, and let $F_1 \xrightarrow{r} F_2 \xrightarrow{s} F_3$ be a sequence of diagrams/functors $F_i : J\rightarrow \...
- 2,461
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128
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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
2
votes
0
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95
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Objects in Ind-category are filtered colimits of compact subobjects
In his paper 'Categories Tensorielles' in section 2.2 Deligne states that if in a tensor category $\mathcal{A}$ all objects are of finite length, then every object of the Ind-category $\text{Ind}\...
- 1,190
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249
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Calculating finite inverse limits of Abelian groups
If we are given an infinite system of abelian groups $(\dots \xrightarrow{\varphi_2} A_2 \xrightarrow{\varphi_1} A_1 \xrightarrow{\varphi_0} A_0)$ then I know its inverse limit can be found by
$$\lim_{...
- 265
2
votes
1
answer
145
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Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$
I have the following
(a) Define a sketch $s_{\mathbf{Grp}}$ and a equivalence functor $$E: \mathbf{Grp}\to \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$$ (b) Knowing that finite limits commute with ...
- 1,382
1
vote
1
answer
100
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Cocomplete $R$-linear categories are tensored : adjoint functor theorem?
Let $B$ be an abelian category which is actually $Mod_R$-enriched for some ring $R$ (say unital commutative ring).
For $b\in B$, we have a functor $\hom(b,-) : B\to Mod_R$ which preserves limits, so ...
- 43.9k
1
vote
1
answer
279
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On covariant linear functors $T: R$-Mod $\to R$-Mod which preserves direct-limits or inverse-limits
Let $R$ be a commutative ring with unity. Let $R$-Mod denote the category of $R$-modules, and $Ab$ denote the category of Abelian groups. Now, it is known that a covariant additive functor
$T: R$-...
- 1,443
1
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1
answer
148
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Does every covariant, additive, faithfully-exact functor $T:R$-Mod $\to Ab$ preserve either direct sum or direct product?
Let $R$ be a commutative Noetherian ring. Let $Ab$ denote the category of abelian groups.
Let $T:R$-Mod $\to Ab$ be a covariant, additive functor such that for any sequence of $R$-modules, $A \...
- 1,443
3
votes
1
answer
212
views
Constructing limits in an additive category given the existence of products and kernels
The title says it all really. Given an additive category $\mathcal{A}$, is having all kernels and arbitrary products sufficient to conclude that it has all limits? Dually, is having all cokernels and ...
- 3,653
4
votes
1
answer
652
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Vanishing of $\varprojlim^1$ on Mittag-Leffler sequences story.
I'm trying to clarify myself on some points about the story of the first derived functor $\varprojlim^1$ of the projective limit functor vanishing on some kind of filtered inverse systems in arbitrary ...
- 342
5
votes
1
answer
2k
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Limits and $Hom(-,Y)$-functor in abelian categories
Let $\cal C$ be an abelian category and let $(\{x_i\} , \{\phi_{ij}:x_i\rightarrow y_j\}_{i\leq j})$ be a direct system with direct limit $(\varinjlim x_i, \phi_i: x_i \rightarrow \varinjlim x_i)$. ...
- 155
1
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1
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269
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Colimit of submodules
I was going through a proof in the paper "Local unit versus local projectivity", where I came across the fact that for an $R$ module $P$ if $P = \operatorname{colim}\limits_{i\in I} P_i$ where $I$ is ...
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