Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exist …

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 $f\circ j:U\rightarrow S$ …

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 …

I am trying to understand a particular case of this question. Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to …

It is known that $A[[X]]$ is flat if $A$ is noetherian (see for example Bourbaki, Algèbre commutative, Ch. III, §3, Cor. 3 p. 146). What happens if A is not noetherian? Is there …

Let $E$ be an elliptic curve over an algebraically closed field $k$ of characteristic $p$. Is there any nice computation for the group $H^1(E,\alpha_p)$ and $H^1(E,\mathbb{G}_a)$? …

Let $k$ be a field, $S$ and $R$ be local $k$-algebras with residue field $k$ and $\phi:S\to R$ be a local homomorphism. Then $\phi$ induces (obviously) a natural transformation of …

Let $S$ be a smooth projective curve over $\mathbf C$ and let $f:X\to S$ be a projective flat morphism with "semi-stable" fibres (i.e., the fibres are reduced and strict normal cro …

Let $f:X\rightarrow Y$ be a flat morphism of schemes of finite type over a field $k$, and assume $Y$ is irreducible. Let $X_1, \dots, X_n$ be the scheme-theoretic irreducible compo …

Let $f:X\rightarrow Y$ a $k$-morphism of finite type from a Cohen-Macaulay scheme to a normal scheme such that all the fibers have the same dimension, do we have that $f$ is flat? …

Let $\cal X$ be a DM stack of finite type over a field (if necessary, I will assume that $k=\mathbb{C}$ and $\cal X$ is a scheme, or even a variety) and $G$ be a finite group. The …

More precisely: Given a finite-type morphism $f \colon X \to Y$ of nice schemes (say, both of finite type over a field), is there a "natural" coherent sheaf $\mathcal F_f$ on $ …

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. …

Let $X \to Y$ be a family of hypersurfaces in a constant $\mathbb{P}^n$, i.e. $X \subset Y \times \mathbb{P}^n$ is locally on $Y$ given by one equation of degree $d$ in $\mathbb{P} …

(everything below is defined over an algebraically closed field) Let $D$ be a (smooth) surface, and let $X \subset T \times D$ be a flat family of curves on $D$, where $T$ is irr …