All Questions
Tagged with limits-colimits commutative-algebra
46
questions
2
votes
0
answers
119
views
Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory?
I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around.
(1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$...
- 1,632
2
votes
0
answers
93
views
Spec of an infinite intersection of ideals, Spec of a colimit
This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning.
Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). ...
- 447
2
votes
1
answer
79
views
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 ...
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
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
2
votes
0
answers
99
views
Spec commutes with cofiltered limits in concrete example
Currently I am learning some alg. geometry and I would like to show the following claim:
Let $\mathfrak{p}$ be some prime ideal of $A$. Then
$$ \varprojlim_{f\notin \mathfrak{p}} \operatorname{Spec}...
- 944
3
votes
0
answers
624
views
Direct limit, inverse limit and Spec
The set-up
$k$ is a field
$T_i = \operatorname{Spec} A_i$ is an inverse system of affine $k$-schemes, where $i<j$ if $\operatorname{Spec} A_j \subset \operatorname{Spec} A_i$ (inclusion).
$X$ is a ...
- 1,207
0
votes
1
answer
127
views
Direct Limit of Polynomials
This is probably a very stupid question, but what is the direct limit $\lim k[x]/x^n$, where the map $k[x]/x^n \to k[x]/x^{n+1}$ is given by multiplication by $x$. For $\mathbb{Z}/p$ this is the ...
- 970
1
vote
0
answers
47
views
Understanding $\varprojlim_{n}R/I^n$
I've leaned the $I$-adic topology and completion of a ring. I wonder if I'm understand the inverse limit correctly. Let $R$ be a commutative ring with unity and $I\subset R$ be an ideal. Then
$$\...
- 5,071
1
vote
0
answers
942
views
(Co)kernels commute with direct sums (and limits)?
Let $R$ be a commutative ring and $(\varphi_i: (M_i \to N_i))_{i \in I}$ a
family of $R$-module morphisms. Take
$$\varphi= \bigoplus \varphi: \bigoplus_i M_i \to \bigoplus_i N_i$$
Questions:
Does the ...
- 7,551
4
votes
1
answer
85
views
Right exactness of projective systems
Suppose that we have a systems of exact sequences $(A_{n}\rightarrow B_{n}\rightarrow C_{n}\rightarrow 0)_{n\in \mathbb{N}}$ together with transitions maps $(A_{n+1}\rightarrow A_{n})_{n\in \mathbb{N}}...
- 105
1
vote
1
answer
86
views
Non-mechanical proof in direct limit (or colimit)
Let $I$ be a directed set and for the sake of simplicity, let us work with a directed system of modules over some commutative ring $A$ (although I am looking for an answer which can be extended to a ...
- 1,445
1
vote
1
answer
203
views
Is direct limit created by the forgetful functor from an Eilenberg-Moore algebra
Proposition. Let $T$ be a monad on $C$ and consider the
forgetful functor
$$
R^T \colon C^T \to C
$$
from the category of Eilenberg-Moore algebras to $C$. This functor
creates limits;
creates ...
user716004
2
votes
1
answer
194
views
Direct product of a family of groups/modules is the direct limit of a directed system formed by the family of groups/modules?
Can the direct product of a family of groups/modules be regarded as the direct limit of a directed system? If yes, than can someone elaborate the desired directed system with it's objects and maps?
- 1,544
2
votes
0
answers
53
views
Power series ring over a noetherian complete local ring of positive residue characteristic
Let $R$ be a noetherian complete local ring of positive residue characteristic $p>0$. A note I am reading states without proof that we have an isomorphism
$$\varprojlim_{\nu}R[X]/\left((X+1)^{p^{\...
- 5,571