Can we check smoothness of a morphism after base change to the algebraic...
I know that smoothness is fppf local on the base, but this is not enough because taking algebraic closures is not finitely presented. The reason I'm asking this is because I want an easy/quick argument...
View ArticleFlatness over regular local rings of dimension 3
Let $R$ be a regular local ring of dimension $n$. Let $i:\text{Flat}\to\text{Fin}$ be the fully faithful inclusion of the category of flat finitely generated type $R$-modules into all finitely...
View ArticleFlatness of certain subrings
The following question appears, more or less, here:Let $k$ be an algebraically closed field of characteristic zero and let $S$ be a commutative $k$-algebra(I do not mind to further assume that $S$ is...
View ArticleCan a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"The answer is yes: take $J=0$. For a...
View ArticleWhen flat base change is reduced
I am sorry, if this a very standard fact.Let $f\colon X\to S$ be a morphism between varieties over field of characteristic zero. Let $\psi \colon S'\to S$ be a flat morphism. Is it true that $X\times_S...
View ArticleFaithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then...
View ArticleProposition 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 number of...
View ArticleDoes smoothness descend along flat morphisms?
Suppose $f:X\to Y$ is a flat morphism of schemes. If $X$ is smooth at $x$, must $Y$ be smooth at $f(x)$?If $f$ is locally finitely presented, then it is open (using EGA IV 1.10.4), so after replacing...
View ArticleWhat is like "flat" but for based connected CW-complexes?
Let $X$ be a based connected CW-complex $X$.Say that $X$ is CW-flat if, for each map of based connected CW-complexes $f \colon Y \to X$ such that $\pi_n(f)$ is injective, $\pi_n X \wedge f$ is also...
View ArticleExistence of functorial (K-)flat resolutions?
I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe that...
View ArticleVanishing of all higher direct images for a non-flat morphism
Let $f: X \to Y$ be a proper morphism of algebraic varieties which is not flat. Then for a line bundle $L$ on $X$, is the vanishing of all higher direct images ($R^i f_* L = 0$ for all $i \geq 0$)...
View ArticleDirect product of direct sum of a flat module
In the book "Rings and Categories of Modules" by Anderson & Fuller, this problem is given: If $V^A$, i.e. the direct product of the module $V$ by the index set $A$, is flat for all sets $A$, then...
View ArticleFlatness of "derived local system sheaves"
Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of...
View ArticleIs there a notion of „flatness” in point-set topology?
In algebraic geometry, flat morphisms are usually associated with the intuition that they have „continuously varying fibers”. Is there a notion in topology formalizing the same intuition? Consider for...
View ArticleFinite flat maps
Let $f : A \to B$ be a finite, finitely presented, flat map of (commutative) rings. It is a known consequence of Chevalley's theorem (on constructible sets) that the induced map $Spec B \to Spec A$ is...
View ArticleZariski's main theorem + Raynaud-Gruson’s platification
If $X\to Y$ is a quasi-finite map of finite presentation between qcqs schemes and $U\subseteq X$ is open such that $U\to Y$ is flat, then we have the following two results:Raynaud-Gruson’s...
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