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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

35 votes
Accepted

Crux of Dwork's proof of rationality of the zeta function?

There is an excellent book by Neal Koblitz "p-adic numbers, p-adic analysis and zeta-functions" were the Dwork's proof is stated in a very detailed way, including all preliminaries from p-adic analysi …
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32 votes
2 answers
2k views

Etale cohomology can not be computed by Cech

It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\mat …
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17 votes

Commutative ring $R$ with no nontrivial idempotents, with a localization $R_r$ with infinite...

Let $k$ be a field and take $$R=\{(a_i)\in\prod\limits_{i\in\mathbb{N}}k[t]\mid a_i(0)=a_j(0)\text{ for all }i,j\}$$ An idempotent in this ring has to be sent to $0$ or $1$ under the map $R\xrightarr …
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15 votes
3 answers
922 views

Lower central series quotients in terms of (co)homology

Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this a …
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14 votes
0 answers
883 views

Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ i …
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12 votes
Accepted

Are "large enough" finite etale covers arithmetic?

Let's assume that $X$ admits a $K$-point $x$ and use the corresponding geometric point as the base point. The existence of a rational point is in fact necessary for a positive answer, as explained by …
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11 votes
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Printing omission in Mumford's "Lectures on Curves on an Algebraic Surface"

In the Russian edition it is $$(\xi_0,\xi_1,\xi_2,\dots)\leftrightarrow f(\xi_0)+pf(\xi_1)+p^2f(\xi_2)+\dots$$ where $f$ is the Teichmuller map.
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11 votes
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Are the Eigenvalues of the Frobenius on Crystalline cohomology bounded by degree?

I assume that you mean $H^r_{crys}(X/W(k))$ because $H^r_{crys}(X/k)$ is just the de Rham cohomology over $k$. Next, over arbitrary base field this definition of $a_i$'s(they are called slopes of Frob …
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11 votes
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Can non-split extension be isomorphic to the split one as objects

$\newcommand{\cO}{\mathcal{O}}$Consider exact sequence of trivial vector bundles $$0\to\cO\xrightarrow{\left(\begin{matrix}x \\ y\end{matrix}\right)}\cO\oplus\cO\xrightarrow{\left(\begin{matrix}y & -x …
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9 votes
0 answers
894 views

Grothendieck's motivation of crystalline cohomology

Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic red …
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9 votes
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Bundle over $\mathbb{C}^{n}\setminus{0}$

$\newcommand{tot}{\mathbb{C}^n\setminus 0}\newcommand{tan}{\mathcal{T}_{\mathbb{P}^{n-1}}}$ Since morphism $\pi$ is affine, for any quasicoherent sheaf $\mathcal{F}$ on $\mathbb{C}^n\setminus 0$ its h …
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9 votes
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Example of a connected finite group scheme which is not solvable

The connected finite kernel $H$ is not solvable, provided that $n>2$ or $p>2$, see the edit below. $\def\eps{\varepsilon} \def\m{\mathfrak{m}}$Suppose by contradiction that $H$ is solvable and the $m$ …
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9 votes
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Exactness of the Weil restriction functor $\mathrm{Res}_{X/k}$

It is not right exact. Assume that $k$ is algebraically closed. If the map $Res_{X/k}B\to Res_{X/k}C$ was surjective as a map of sheaves for the fppf topology, then in particular, the map on sections …
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9 votes
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Multiplicativity twisted Hochschild Kostant Rosenberg isomorphism

I am far from being expert in this subject, but I will try to present my understading of there this multiplicativity comes from. I wiil refer to authors you mention but only to the parts which I hope …
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9 votes
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Some basic questions on crystalline cohomology

1)Yes, such decomposition follows from the fact that Frobenius on the de Rham-Witt differential forms acts in a way that slopes on $H^i(X, W\Omega^j)[1/p]$ are in the interval $[j,j+1)$. This forces t …
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