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Let $f=(\varphi,\theta):X\longrightarrow S$ a morphism of preschemes whith $\varphi$ surjective. Let $\theta(S):\Gamma(S,O_S)\longrightarrow \Gamma(S,f_* O_X)=\Gamma(\varphi^{-1}(S),O_X)=\Gamma(X,O_X) …

In EGAIV$_3$ 8.9.1 it is written: "Si $\mathcal F$ est un $O_X$-Module quasi-cohérent de présentation finie[...]" Is "$\mathcal F$ de présentation finie" not the same as "$\mathcal F$ admet une prése …

Let $f:Z\to X$ be an immersion of schemes. Let $E$ be a vector bundle on $X$(coherent and locally free of finite type and say constant rank $n$). Suppose that $O_X$ is not necessarily coherent. It se …

Let $S$ be a scheme. Let $\mathcal X$ be a algebraic $S$-stack and be $Y$ a $S$-scheme. Let $f:\mathcal X\longrightarrow Y$ be a $S$-morphism of algebraic stacks which is an open embedding (resp. a cl …

Let $f:X\longrightarrow S$ be a morphism of schemes. What is the link between sheaf-sections of $O_X$ over an open set of $X$ and morphism-sections of $f$. Is there a kind of correspondence?

Let $f:X\to S$ be a projective flat morphism with connected and reduced geometric fibers. Why do one have that $O_S\to f_* O_X$ is an isomorphism?

Let $f:X\longrightarrow S$ be a morphism of preschemes which is smooth of pure relative dimension 1. Let $\sigma:S\longrightarrow X$ be a section of $X$. Let $D$ be the (positive) divisor associated t …

Where can I find a clear exposé of the so called "standard reduction to the local artinian (with algebraically closed residue field", a sentence I read everywhere but that is never completely unfold? …

Let $f:X_1\to X_2$ be a closed submanifold. Let $\rho:G_1\to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $\rho$-equivariant. I would like to get a morp …