All Questions
17 questions
2
votes
1
answer
250
views
Images of smooth schemes under lci morphisms
Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say ...
user505967
2
votes
1
answer
290
views
Flat scheme-theoretic closure
Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...
user501361
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
1
vote
1
answer
193
views
Lifts of smooth algebras
Let $(R, I)$ be a Henselian pair, with $I$ a finitely generated ideal.
We know that for any smooth $R/I$-algebra $A_0$, there exists a smooth $R$-algebra $A$ such that $A/I\simeq A_0$.
We also know ...
- 180
4
votes
1
answer
218
views
Henselianizations over countable index sets
Let $A$ be a ring, $I\subset A$ a finitely generated ideal.
The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...
user132229
7
votes
1
answer
531
views
Vector space objects in schemes - confusion
Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...
user128448
1
vote
0
answers
99
views
Special formal lifts of smooth algebras
Let $A$ be a smooth algebra over $k$ a finite field.
Say $B$ is a $p$-adically complete smooth algebra over the Witt ring $W(k)$, lifting $A$.
Assume $B$ is of the form $W(k)\langle t_1,\ldots, t_n\...
user124171
6
votes
1
answer
417
views
Smooth algebras always lift
Let $k$ be a finite field, $A$ a smooth $k$-algebra.
Does there exists a smooth algebra $B$ over the Witt vectors $W(k)$, such that $B/p\simeq A$? How is it constructed?
user92332
3
votes
1
answer
117
views
Liftings and closed immersions
Let $A$ be a flat $\mathbf{Z}_p$-algebra, $\overline{I}\subset A/p$ an ideal.
Can we find an ideal $I\subset A$ such that
$I$ mod $p$ = $\overline{I}$
$I$ does not contain $p$.
It's harder than it ...
user95222
4
votes
1
answer
461
views
Coherent modules over complete adic rings: counterexamples
Let $A$ be a coherent ring, complete with respect to the adic topology generated by a finitely generated ideal $I$.
Define the category $Coh(A,I)$ whose objects are inverse systems $\{M_n\}$ of $A$-...
user122630
1
vote
0
answers
607
views
Push-forward along closed immersion
Let $X$ be a scheme, $p : Z\to X$ a closed immersion, $\mathcal{F}$ a locally free sheaf of modules on $Z$ of finite rank.
Assume both $\mathcal{O}_Z$ and $\mathcal{O}_X$ are coherent sheaves of $\...
user120812
4
votes
1
answer
786
views
Descent of étale torsors
Let $X$ be a scheme over a field $k$, $G$ a finite abelian group of size invertible on $X$. Suppose $K/k$ is a Galois field extension and let $Y\to X_K$ be an étale $G$-torsor.
For what field ...
user124171
0
votes
1
answer
305
views
Integral morphism between universally closed and separated schemes
Let $f : X\to Y$ be a morphism between schemes over $\text{Spec}(\mathbf{Z}_p)$.
Assume:
$f$ is integral
both $X$ and $Y$ are universally closed and separated over $\mathbf{Z}_p$
$f$ mod $p^n$ is an ...
user113393
1
vote
1
answer
181
views
Relative approximation of morphisms
Let $S$ a qcqs scheme, and let $f : X := \varprojlim_j X_j \to S$ be an inverse limit of qcqs schemes $f_j : X_j \to S$ with affine transition maps.
Suppose $f$ is (P). Is $f_j$ also (P) for all $j$ ...
user95222
8
votes
1
answer
787
views
Commutative algebra counterexample
Let $M$ be an $R[x]$-module, such that $M$ is finitely generated as an $R$-module.
Does there exist one such $M$, such that $M\otimes_{R[x]}R[x,x^{-1}]$ is not finitely generated as an $R$-module?
user120812
4
votes
1
answer
358
views
Gluing finitely presented quasi coherent sheaves
Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A_i)\subset X\}_{i=1,\ldots, n}$ a finite affine open cover.
Suppose a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is such ...
user119470
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