# Questions tagged [sheaf-theory]

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### How to model (affine) schemes with a large sketch?

Guitart states in "Toute theorie est algebrique et topologique" (Proposition 17) without proof (at least, I don't understand the hints there) that the category $\mathbf{Sch}$ of schemes is ...
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### Sheaves on sites given by a (regular) cd-structure

Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see https://ncatlab.org/nlab/show/cd-structure or Voevodsky's paper Homotopy theory of simplicial presheaves in ...
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### How to motivate constructible sheaves

I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
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### Example of an Algebraic Space (“false” affine line with different tangents at origin)

I have a question about the following example from the Algebraic spaces and quotients by equivalence relation of schemes by Roy Mikael Skjelnes (page 12) of a presheaf quotient, which has associated ...
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### Refinement of hypercovers by ordinary covers

I am asking for references and discussions of statements of the form Every bounded hypercover can be refined by an ordinary cover E.g., are there conditions for a site making this statement true? My ...
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### Functorial isomorphisms

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\PSh{PSh}$We know that a presheaf $\mathcal{F}$ on category $\mathcal{C}$ gives a colimit preasheaf $\mathcal{F}^{+}$ ...
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### Is the restriction of an injective sheaf on a closed subscheme still injective?

Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$. Question. Is $i^*\mathcal{I}$ still an ...
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### Schur's lemma for sheaves with different reduced Hilbert polynomials

Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement: Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
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### Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence

I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263. Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
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### Sections of the structure sheaf of a partial flag variety on big cell

Let $G$ be a connected split reductive group over a (non-archimedean) field $K$ of char 0 with split maximal torus $T$ and a standard parabolic $P$. Denote by $W$ the Weyl group of $G$ and by $W_P$ ...
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By https://arxiv.org/abs/1406.4419 (The fundamental groupoid as a terminal costack, Ilia Pirashvili), we know that for a topological space $X$, the $2$-functor $$\text{Top}(X)\rightarrow \text{Gpd}, \... 1answer 122 views ### Connecting homomorphism in Cech cohomology Let M be a smooth manifold and \mathcal{U} be a good open cover of M. If I have an exact sequence of sheaves$$0 \longrightarrow A \stackrel{f}\longrightarrow B \stackrel{g}\longrightarrow C \...
I'm reading the paper "High direct image of dualizing sheaf" of professor Kollar. I summarizing my questions as follows: Let $f:X\rightarrow Y$ be surjective projective morphism between ...