All Questions
Tagged with limits-colimits ring-theory
27
questions
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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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1
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69
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Is there a direct limit in the category of rings for hypercomplex numbers [closed]
I recently learned about the concept limits in categories. From R we can construct C the H etc... by iterating the Cayley-Dickson construction.
My question is: Can we construct a (non-associative)ring ...
- 27
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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 $...
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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 ...
- 410
2
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Direct limits $A_0\to A_1\to... A$ with split monomorphisms.. should the maps $A_i\to A$ be split monos also?
Let $\mathcal{C}$ be the category of $\mathbb{Z}$-graded rings. I have a sequence $A_i$ of $\mathbb{Z}$-graded rings, and split monomorphisms $\varphi_i:A_i\to A_{i+1}$ i.e. $$A_0\rightarrowtail A_1\...
- 3,440
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46
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Endofunctor of modules induced from pushout of commutative rings
I have commutative rings $A$, $B$ and $C$, and unital ring homomorphisms $f:A\to B$, $g:A\to C$. Using $f$ and $g$, we can regard $B$ and $C$ as $A$-modules, and thus define the $A$-module $B\otimes_A ...
- 3,440
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143
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To prove $ \Bbb{Z}_p$ is isomorphic to $\lim_{←}\Bbb{Z}_p/p^n \Bbb{Z}_p$ as a ring
I want to prove $ \Bbb{Z}_p$ is isomorphic to $\lim_{←}\Bbb{Z}_p/p^n \Bbb{Z}_p$ as a ring.
Here, $\Bbb{Z}_p$ is completion (as metric space) of
$\Bbb{Z}$ with p adic metric.
My try:
Let define natural ...
- 6,351
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1
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74
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Does $\lim_←$ and composite of field commutes?
Does inverse limit and composite of field commutes?
For example, let $K_1$ and $K_2$ be fields.
We take $\lim$ with respect to $p$-th Frobenius (not surjective),
then,
$\lim K_1K_2＝(\lim K_1)(\lim K_2)...
- 6,351
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1
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100
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Zero element in an inverse limit in category of rings
Let $(R,f_m)$ be the inverse limit of an inverse system $(R_m,f_{mn})$ of commutative rings with $1$. Let $x \in R$ be s.t. $f_{m}(x)=0$ for all $m$. Is it then true that $x$ must be the $0$ element ...
- 405
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36
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Embedding integers in the inverse limit
Let $S=(R_m,f_{mn})_{m\geq n\geq 0}$ be an inverse system of polynomial rings over the integers and unital ring homomorphisms between them. Let $R_S$ be the inverse limit of $S$ in the category Ring ...
- 621
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25
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Embedding the vertex of a cone inside inverse limit
Let $(R_i,f_{ij})_{i \in I}$ be an inverse system of rings and ring homomorphisms. Let $(L,p_i)_{i \in I}$ be the inverse limit of $(R_i,f_{ij})_{i \in I}$ and let $(K,f_i)_{i \in I}$ be a cone into $...
- 405
2
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72
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Inverse limit of rings of polynomials in many variables [duplicate]
I am trying to understand inverse limits and so I decided to try and compute one of such limits. We are going to work in the category of rings. Let $R$ be a ring. First, I set up an inverse system. ...
- 405
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121
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Completion of a stalk not integral domain
We consider the closed subscheme $X:= V(Y^2-X^2(x+1))$ of affine plane $\mathbb{A}^2_k= Spec \text{ } k[X,Y]$. field $k$ is arbitrary. let $x:=(0,0)$ the point representing the prime ideal $(X,Y) \...
user526728
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108
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Are (co)limits of topological rings in $\mathsf{Ring}$ with continuous legs always a (co)limit in $\mathsf{Top}$ for a suitable topology?
This question is motivated by the characterization of the $p$-adic numbers as a limit of finite fields via truncation.
Since my knowledge of the former is close to nonexistent, I will try to ask the ...
- 17.2k
2
votes
1
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321
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Projective resolution of a filtered colimit
In Cartan and Eilenberg's Homological Algebra, they claim the following
V9.5* If $A = \lim\limits_\longrightarrow A_\alpha$ then there exist projective resolutions $X_\alpha$ of $A_\alpha$ forming ...
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