All Questions
Tagged with limits-colimits abstract-algebra
129
questions
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65
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If a module is a limit of two inverse systems, then the two systems are isomorphic.
The original problem comes from corollary (10.10.6), chapter 10, Volumn I, EGA.
I state it in the language of modules here for convenience.
Claim. If an $R$-module $F$ is a limit of two inverse(or ...
- 1,211
2
votes
1
answer
79
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Interpretation of closure in inverse limit
Can one interpret the closure of a set inside an inverse limit as the closure of its individual components? I have not been able to find a source confirming or denying this claim. I have only been ...
1
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0
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143
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Does profinite completion preserve injectivity?
Let $G$ be an abelian group.
Let $\widehat{G}$ be a profinite completion of $G$.
Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are ...
- 6,351
1
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57
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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 ...
1
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0
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67
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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
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1
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35
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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
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2
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68
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Direct limit of a system: computation
I am considering a homomorphism from $\mathbb{Z}_4\rightarrow \mathbb{Z}_6$ given by $\bar{1}\mapsto \bar{3}$. My question is:
What is the direct limit of this system in the category of abelian ...
- 3,047
1
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1
answer
63
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$F : I → C$ is a functor from a filtered category with any colimit. $ι : J → I$ is the embedding with $J$ cofinal. Then colim $F \cong$ colim($F ◦ ι$)
Let $I$ be a filtered category. We say a full subcategory $J$ is cofinal if for
every object $A \in I$, there is an object $B \in J$ so that $\text{Hom}(A, B) \neq \emptyset$. Let $F : I \to \mathcal ...
- 313
2
votes
1
answer
53
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Reference for construction of coproduct of Boolean algebras
I was recently trying to understand how coproducts of Boolean algebras work, since I need them for my research. I came across a StackExchange question with a great constructive answer from "Math ...
- 21
1
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70
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Completion with respect to the topology defined by the subgroups of finite index
Let $M$ be an toplogical abelian group.
I heard we can define $\hat{M}$ by completion of $M$ with respect to the topology defined by the subgroups of finite index.
For example, if $M=\Bbb{Z}$, then ...
- 6,351
0
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1
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74
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Does profinite integer multiplication have integer "eigenvalues" in general?
Suppose $a \in \widehat{\Bbb{Z}}$, say $a = (\overline{a_1}, \overline{a_2}, \overline{a_3}, \overline{a_4}, \dots)$.
I'm wondering if for any $b \in \widehat{\Bbb{Z}}$ whether there exists a number $...
2
votes
1
answer
91
views
Two definitions of categorical limits
For $C$ a locally small category, $J$ an essentially small category and $F\colon J\rightarrow C$ a functor, the limit of $F$, if it exists, can be defined as a representation of the functor
$\...
- 1,769
3
votes
2
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161
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Size-issues in the definition of categorical limits
I know (at least) two definitions of categorical limits. The setting is always as follows. Let $F:J\rightarrow C$ be a functor between not necessarily small nor locally small categories.
For the first ...
- 1,769
1
vote
1
answer
48
views
Terminology: (Co)equalizer of a family of morphisms?
Let $A,B\in \mathsf{C}$ be two objects in a category $\mathsf{C}$. Let $(f_i\colon A\rightarrow B)_{i\in I}$ be a family of parallel morphisms in $\mathsf{C}$. Consider the corresponding diagram in $\...
- 1,769
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1
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105
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Ideals of direct limit of rings
I am currently studying the direct limit of rings and I am stuck in the following question. Please help me.
Let $\{R_{i}\}_{i \in I}$ be a nonempty family of commutative rings with unity and $\langle ...
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