All Questions
7 questions
4
votes
1
answer
259
views
On inverse limits of $\pi$-adically complete algebras
Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...
- 541
2
votes
2
answers
417
views
Transition maps in trivial direct limit
If $\{X_i, i\in I\}$ is a directed system of abelian groups such that we have
$$\varinjlim_{i\in I}X_i = 0$$
is it true that for every $i$ and large enough $j\ge i$ the transition map $f_{i,j} : X_i\...
user290895
2
votes
1
answer
231
views
Lifting of flat lci maps
Suppose $R$ is a Noetherian ring and $I$ a nontrivial ideal of $R$, and $A_0\to B_0$ a finite faithfully flat lci map of smooth $R_0 := R/I$-algebras.
We fix a smooth $R$-algebra $A$ lifting $A_0$ and ...
user315884
5
votes
1
answer
408
views
On universally closed morphisms of reduced schemes
In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...
user315884
4
votes
1
answer
327
views
Detecting closed immersions on fibers
Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes.
Assume $X$ and $S$ are $R$-flat and universally closed.
If the special fiber of $X\to S$ is a closed immersion, is $X\...
user315884
4
votes
1
answer
694
views
Proper mapping theorem
Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules.
In particular, the coherent ...
user120812
2
votes
0
answers
325
views
A question on direct limits of rings, and descent of ideals
Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer.
Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...
user92332