Search Results

If $f$ is quasi-finite, you can follow chapter IV of Berthelot's thesis (Springer Lecture notes 407). Otherwise, you will need to enter the derived world, much like in the case of D-modules. Gaitsgo …

It's not clear what you mean by "exactly same with" in your theorem, but even if we generously interpret it to mean that the map $\bar{Y} \to \bar{X}$ factors through $\tilde{\bar{X}}$, the claim does …

It's an ind-affine ind-algebraic group - the usual choice of increasing union gives a diagram of closed embeddings of affine algebraic groups. It is a sheaf in any topology where $GL_n$ is a sheaf, e …

Over an algebraically closed field of characteristic zero, a choice of complex analytification gives you an equivalence between finite étale covers of the moduli scheme $\mathcal{M}_{0,n}$ and finite …

I assume you're asking about quasicoherent sheaves. Since the inclusion of $U$ into $X$ is an open immersion, $j_!$ is exact. You can find this in Tag 03DJ. Since $j$ is an affine morphism, $j_*$ i …

To show that the scheme isomorphism is an isomorphism of group schemes, it suffices to check that the multiplication maps coincide, i.e., $$m_{(X \times_S Y)_{S'}} = \phi^{-1} \circ (m_{X_{S'}} \times …

It is common to use the name "the $\mathbb{G}_m$-torsor associated to $\mathscr{L}$", and it is concisely written as $L = \underline{\operatorname{Isom}}_X(\mathscr{O}_X, \mathscr{L})$. However, an a …

Willy Liu answered this in a comment. By EGA 4v4 Corollary 17.11.4, a necessary and sufficient condition that $X \to Y$ be smooth at $x \in X$ is that there be an open neighborhood $U \subset X$ of $ …

For your first question, I think the answer depends on whether you think representability means "in schemes" or "in algebraic spaces". For schemes, you really want a projective morphism $Y \to S$ for …

Following Niels's suggestion, I'm turning my comments into an answer. The first point I want to make is the observation that if $G$ is an infinite group, then the constant group scheme $G_k$ is not a …

If you view a $\mathbb{Z}$-valued Cartier divisor (on say, an integral separated scheme X) as a $\mathbb{G}\_m$-torsor on X with generic trivialization, then for any torus T with character group $X^\a …

This paper by Kedlaya might be what you want, since it contains some rearrangement of the words you used, but I can't really tell from the question. If you want a proper F-scheme to have an etale map …

There appears to be some confusion in the question: The circle $S^1$ is not a complex manifold, so it does not admit a meaningful notion of holomorphic line bundle. If you try to construct a complex …

Keerthi Madapusi Sampath already answered the question in the comments, so I'll just add some remarks. The definitions in question are given in section 2.3.1, on page 80 of Beilinson and Drinfeld's …

The question was answered in the comments, so I'm adding this to knock the question off the "unanswered" list. The literal answer to your first question is "Yes if and only if $n \leq 1$".