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I heard fleetingly (but forgot the concrete context) that schemes over henselian local rings have generally a rich divisor theory that on one hand somehow allows in certain way to "reduce" the analysis …

The latter means that if $X,Y$ are schemes, $h_X,h_Y$ their Yoneda representations and we have natural maps $h_X \to \mathcal{X}, h_Y \to \mathcal{X}$ then the fiber product $h_X \times_{\mathcal{X}} … Recall then we talking about schemes a scheme $S$ is called separated if the diagonal map $S→S×S$ is a closed immersion. …

Since abelian varieties are required to be irreducible, they give rise to integral schemes. … Also let us mention that for $K$-group schemes of finite type smooth is equivalent to geometrically reduced, which means that all stalks of the structure sheaf of $A×_K \bar{K}$ are reduced. …

It is a well known fact that proper scheme $X$ over $k$ has a up to isomorphism unique dualizing sheaf (EGA I, Hartshorne). This dualizing sheaf $\omega_X$ comes with two striking properties: (i) T …

My question deals with Michael Artin's paper "On isolated rational singularities of surfaces"; more precisely the proof of Theorem 4 on page 133. Here the relevant excerpt: The Setting: Let $\bar{V}= …

schemes in usual sense). … What is the philosophy of taking formal completions of usual schemes? …

This question deals with a concrete exercise from Geomerty of Schemes by Eisenbud and Harris but also moreover the general philosophy attacking typical problems in algebraic geometry of following relative … nature: say we have a "nice" enough map (having here sloppy said something "fibration like" in mind) $f:X \to Y$ of schemes over base field $K$, and assume $Y$ (wlog we can assume it to be affine) has …

According to Wikipedia there are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by étale equivalence relations, or as sheaves on a big étale site that are … locally isomorphic to schemes. …

One can always assume that $R$ and $U$ are affine schemes. … one may always assume that $R$ and $U$ are affine schemes. …

I study Nitin Nitsure's paper Construction of Hilbert and Quot Schemes (arXiv:math/0504590) and have some problems with the content of imposed universal property (F) in the section "Use of Flattening Stratification …

Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) consist …

I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “dévissage of st …

Let $C$ be a connected, nodal (I'm working with definition from Alper's notes on Stacks & Moduli, see p 210), projective curve over an alg closed field $k$, beeing everywhere smooth except at a node $ …

I'm reading Mumford's & Oda's Algebraic Geometry II and I'm confused about explanations on geometric intuition of sections $H^0(\Theta_X, X)$ of the tangent sheaf on page 287: Let $X$ a geometricall …

There is a well known result that states that there exist an exact sequence of finite flat group schemes over $R$: $$\tag{$*$} 1 \to G^0 \to G \to G^{\text{ét}} \to 1$$ where $G^0$ is the neutral component …