All Questions
12 questions
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 10.5k
10
votes
1
answer
838
views
Is there a direct proof that affine schemes are fppf quasi-compact?
Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...
- 9,408
8
votes
0
answers
230
views
A Hartogs-type criterion for flatness
Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two.
Now, let $Y$ be a smooth quasi-affine connected variety ...
- 81
5
votes
1
answer
988
views
Spreading out flat morphisms of schemes
In EGA IV, Chapter 8, projective systems of schemes (and morphisms between them) are considered. Let $(S_{\lambda})_{\lambda \in L}$ be a projective system of schemes and let $S$ be the projective ...
- 4,746
3
votes
1
answer
419
views
Affine hulls and base change
Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...
- 6,700
3
votes
1
answer
346
views
When do generizations ("generalizations") lift uniquely?
If $f : X \to Y$ is proper, then specializations lift along $f$, and uniquely.
(This means, if $R$ is a discrete valuation ring with fraction field $K$ and I choose a factorization $\text{Spec}K \to ...
- 1,084
2
votes
1
answer
334
views
flat and finite type morphisms
Let $f:X\rightarrow Y$ a faithfully flat morphism between $k$-schemes. We assume that the fibers are locally of finite type, do we have that $f$ is locally of finite type?
- 3,472
2
votes
0
answers
198
views
Proposition 4.3.8 Qing Liu about flat morphisms of schemes
I have a problem with a detail of Qing Liu's proof of Proposition 4.3.8 (pag. 137 of "Algebraic Geometry and Arithmetic Curves").
The statement is:
Let $Y$ be a scheme having only a finite ...
- 149
1
vote
0
answers
135
views
Projection from closure of locally closed subscheme is Etale
Let $S$ an arbitrary scheme and denote by $\Delta: S \to S \times S$ the diagonal immersion and $p_i: S \times S \to S$ the both projections to first resp second factor. (in following we will wlog ...
- 6,028
0
votes
0
answers
276
views
Flatness of affine cone due to semicontinuity theorem
I would like to clarify an important aspect from the discussion in this question.
The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic
Geometry Chap. III page ...
- 6,028
0
votes
0
answers
89
views
Locally freeness of twisted direct images $\pi_* \mathcal{F}(i)$
I have a question about a step in the proof of the
Existence of Flattening Stratification I found in
Nitsure's paper here: https://arxiv.org/abs/math/0504590 This question is closely related to Local ...
- 6,028
-1
votes
1
answer
422
views
on flat morphisms
Let $j:U\rightarrow X$ an open immersion between k-schemes of finite type and $f:X\rightarrow S$ a surjective k-morphism of finite type.
We suppose that $f\circ j:U\rightarrow S$ is faithfully flat, ...
- 3,472