Questions tagged [flatness]

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 ...

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 $\...

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 $...

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 $i \geq 0$) ...

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 ...

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 ...