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      Questions tagged [etale-cohomology]

      for questions about etale cohomology of schemes, including foundational material and applications.

      6
      votes
      0answers
      94 views

      Extension of sheaves of $\infty$-algebras

      Let $(\mathcal{C},\tau)$ be a site, whose Grothendieck topology is $\tau$ $F : \mathcal{C}\to D(k)$ a sheaf of $\infty$-$k$-algebras, with $k$ a ring and $D(k)$ the derived category of $k$-modules. ...
      3
      votes
      0answers
      121 views

      Functoriality of base change morphisms

      Consider a commutative diagram of morphisms of schemes: $$\begin{array}{ccccccccc} X_2 & \xrightarrow{j_2} & Y_2 \\ f'\downarrow & & \downarrow f \\ X_1 & \xrightarrow{j_1} &...
      3
      votes
      0answers
      97 views

      On the existence of nice hypercovers

      Let $\mathcal{C}$ be a site and $X$ a sheaf of sets on $\mathcal{C}$. Then there exists a hypercover $K_{\bullet}$ of $X$ such that $K_n$ is a coproduct of representable presheaves on $\mathcal{C}$ ...
      9
      votes
      2answers
      1k views

      Étale cohomology of morphism whose fibers are vector spaces

      Let $X\rightarrow Y$ be a morphism (may not be smooth) of varieties such that the fibres are vector spaces. Are the $l$-adic cohomologies of $X$ and $Y$ equal? If not, under what condition (other ...
      3
      votes
      0answers
      156 views

      Does there exist trace maps between $\ell$-adic cohomology groups for finite flat morphisms?

      Let $\operatorname{ProjSmooth}_k$ be the category of smooth projective varieties over a field $k$, and consider the $\ell$ cohomology theory $H^*(-)$ ($l \not = \operatorname{char} k$), how to define ...
      3
      votes
      1answer
      196 views

      Nearby cycles and extension by zero

      Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$. Call $i_s ...
      2
      votes
      0answers
      124 views

      Berthelot’s comparison theorem and functoriality

      Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$. Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
      4
      votes
      1answer
      143 views

      Morphism of sites and abelian sheaf cohomology

      Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi $$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$ By assumption, $f^{-1}$ is ...
      14
      votes
      1answer
      548 views

      GAGA for henselian schemes

      In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes. Let $I$ be a finitely generated ideal in a ...
      6
      votes
      1answer
      171 views

      Vanishing of higher direct image of finite morphisms relative to the fppf topology

      Let $f:X \to Y$ be a finite morphism of schemes. Let $\mathcal{F}$ be a sheaf of abelian groups on the the etale site of $X$ then we know that $R^{i}f_{*} \mathcal{F} = 0$. Is this statement also true ...
      1
      vote
      0answers
      131 views

      Cohomology of constant sheaves

      Let $X= spec(k)$ where $k$ is an algebraically closed field. Consider the constant sheaf $\mathbb{Z}$ on the fppf site of $X$. I'm interested in computing $H^1_{fppf}(X, \mathbb{Z})$. I know that $H^...
      1
      vote
      1answer
      177 views

      Swan-conductor and base change

      Let $C$ be a proper smooth curve over a perfect field $K$ of positive characteristic $p$, $u: U \hookrightarrow C$ strictly open and $\mathfrak{F}$ a lisse (lcc) $\mathbb{F}_l $-sheaf $(l \neq p)$ on $...
      1
      vote
      0answers
      63 views

      Definition for equivariant $l$-adic sheaves

      What is the definition of equivariant $l$-adic or ($\mathbb{Z}_l$-) sheaves? Suppose $G$ acts on $X$, I could pick a $G$-equivariant etale sheaf of $\mathbb{Z}/l^n$ module on $X$ for each $n$, and ...
      2
      votes
      1answer
      157 views

      Vector bundles on henselian schemes

      Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$. We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose ...
      12
      votes
      1answer
      341 views

      Some basic questions on crystalline cohomology

      Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$. Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...

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