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Cartier operator

Contents

Notation

For the following discussion we suppose Ο€:Xβ†’S\pi: X\to S is a smooth map of schemes. Let d:π’ͺ Xβ†’Ξ© X/S 1d:\mathcal{O}_X\to \Omega^1_{X/S} be the standard differential. It is an integrable connection on π’ͺ X\mathcal{O}_X. We’ll define the (relative) de Rham cohomology to be the hyper-derived functor pushforward applied to the de Rham complex R qΟ€ *(Ξ© X/S β€’)\mathbf{R}^q\pi_*(\Omega_{X/S}^\bullet).

We will denote the relative Frobenius map F:Xβ†’X (p)F: X\to X^{(p)} where X (p)X^{(p)} is the pullback of the structure map and the absolute Frobenius F ab:Sβ†’SF_{ab}: S\to S, i.e. X (p)=XβŠ— Ο€ βˆ’1(π’ͺ S)Ο€ βˆ’1(π’ͺ S)X^{(p)}=X\otimes_{\pi^{-1}(\mathcal{O}_S)} \pi^{-1}(\mathcal{O}_S). If π’œ β€’\mathcal{A}^\bullet is a complex of sheaves, then we denote by β„‹ q(π’œ β€’)\mathcal{H}^q(\mathcal{A}^\bullet) the sheaf that is obtained by taking cohomology with respect to the maps of the complex.

The Cartier Isomorphism

Suppose that SS is an 𝔽 p\mathbb{F}_p-scheme and X/SX/S is smooth, then there is a unique map of graded π’ͺ X (p)\mathcal{O}_{X^{(p)}}-algebras C βˆ’1:Ξ© X (p)/S iβ†’βˆΌβ„‹ i(F *Ξ© X/S β€’)C^{-1}: \Omega^i_{X^{(p)}/S} \stackrel{\sim}{\to} \mathcal{H}^i(F_*\Omega^\bullet_{X/S}) that satisfies:

C βˆ’1(1)=1C^{-1}(1)=1,

C βˆ’1(Ο‰βˆ§Ο„)=C βˆ’1(Ο‰)∧C βˆ’1(Ο„)C^{-1}(\omega\wedge \tau)=C^{-1}(\omega)\wedge C^{-1}(\tau)

and C βˆ’1(dF ab βˆ’1(f))=[f pβˆ’1df]C^{-1}(dF_{ab}^{-1}(f))=[f^{p-1}df] in β„‹ 1(F *Ξ© X/S β€’)\mathcal{H}^1(F_*\Omega^\bullet_{X/S}).

The inverse of this isomorphism is the traditional Cartier isomorphism.

The construction is quite simple. First note that we can immediately reduce to constructing C βˆ’1C^{-1} for i=1i=1. This is because if C βˆ’1(1)=1C^{-1}(1)=1 and C βˆ’1C^{-1} is π’ͺ X (p)\mathcal{O}_{X^{(p)}}-linear it is determined for i=0i=0. Likewise, if C βˆ’1C^{-1} is determined for i=1i=1, then the case iβ‰₯1i\geq 1 is determined from the second property.

Now to construct for i=1i=1 we just note that such a map is equivalent to a (Ο€ (p)) βˆ’1(\pi^{(p)})^{-1}-linear derivation π’ͺ X (p)β†’β„‹ 1(F *Ξ© X/S β€’)\mathcal{O}_{X^{(p)}}\to \mathcal{H}^1(F_*\Omega_{X/S}^\bullet). This is equivalent to defining a map on local sections Ξ΄:π’ͺ XΓ—Ο€ βˆ’1(π’ͺ S)β†’β„‹ 1(Ξ© X/S β€’)\delta: \mathcal{O}_X\times \pi^{-1}(\mathcal{O}_S)\to \mathcal{H}^1(\Omega_{X/S}^\bullet) that is biadditive and satisfies the extra three properties

Ξ΄(fs,sβ€²)=Ξ΄(f,s psβ€²)\delta(fs, s')=\delta(f, s^ps'),

Ξ΄(gf,s)=g pΞ΄(f,s)+f pΞ΄(g,s)\delta(gf, s)=g^p\delta(f,s)+f^p\delta(g,s) and

Ξ΄(f,1)=[f pβˆ’1df]\delta(f,1)=[f^{p-1}df].

Now define the map explicitly by Ξ΄(f,s)=[sf pβˆ’1df]\delta(f,s)=[sf^{p-1}df]. It can be checked that this map satisfies all the properties listed and is indeed an isomorphism. This is C βˆ’1C^{-1}, the inverse of the Cartier isomorphism.

Relation to the Hodge-de Rham Spectral Sequence

For this discussion let’s assume that X/kX/k is proper and smooth. Deligne and Illusie had an insight that the degeneration of the Hodge-de Rham spectral sequence (HdR SS) is closely related to the Cartier isomorphism. Recall that the HdR SS is formed by taking the spectral sequence associated to hypercohomology E 1 p,q=H q(X,Ξ© X/k p)β‡’H dR p+q(X/k)E_1^{p,q}=H^q(X, \Omega^p_{X/k})\Rightarrow H^{p+q}_{dR}(X/k).

Now notice that if we form the complex ⨁ iΞ© X (p) i[βˆ’i]\bigoplus_{i}\Omega^i_{X^{(p)}}[-i] which is Ξ© i\Omega^i in degree ii and d=0d=0 everywhere, then the left side of the inverse Cartier isomorphism is exactly β„‹ i(⨁ iΞ© X (p) i[βˆ’i])\mathcal{H}^i(\bigoplus_{i}\Omega^i_{X^{(p)}}[-i]). Likewise, the right side is β„‹ i\mathcal{H}^i of the complex F *Ξ© X/k β€’F_*\Omega_{X/k}^\bullet. We can think of both of these complexes as living in D(X (p)):=D qCoh b(X (p))\mathbf{D}(X^{(p)}):=\mathbf{D}^b_{qCoh}(X^{(p)}).

We can ask whether or not there is some map in the derived category Ο•:⨁ iΞ© X (p) i[βˆ’i]β†’F *Ξ© X/k β€’\phi: \bigoplus_{i}\Omega^i_{X^{(p)}}[-i] \to F_*\Omega_{X/k}^\bullet with the property that β„‹ i(Ο•)=C βˆ’1\mathcal{H}^i(\phi)=C^{-1} for all ii. It turns out this is a sufficient condition for convergence of the HdR SS. This is just because we get a string of isomorphisms

H n(X,Ξ© X β€’)=H n(X (p),F *Ξ© X β€’)≃⨁ iH nβˆ’i(X (p),Ξ© X i)\mathbf{H}^n(X, \Omega_X^\bullet)=\mathbf{H}^n(X^{(p)}, F_*\Omega_X^\bullet)\simeq \bigoplus_{i} H^{n-i}(X^{(p)}, \Omega_X^i). Thus the dimensions of the kk-vector spaces at the E 1E_1 term match the dimensions at the E ∞E_\infty term. Since everything is a kk-vector space this is all that is needed for degeneration (there can be no non-trivial quotients without dimension decreasing).

The Generalization to Non-commutative algebra

See Kaledin Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie on the arxiv

References

Nilpotent Connections and the Monodromy Theorem by Nicholas M. Katz

Relevements modulo p 2p^2 et decomposition du complexe de de Rham by Deligne and Illusie

Created on May 25, 2011 02:46:46 by hilbertthm90 (174.61.151.67)