Crystal (mathematics)

In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by , who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the crystalline site are analogous to quasicoherent modules over a scheme.

An isocrystal is a crystal up to isogeny. They are $p$-adic analogues of $\mathbf_l$-adic étale sheaves, introduced by and (though the definition of isocrystal only appears in part II of this paper by ). Convergent isocrystals are a variation of isocrystals that work better over non-perfect fields, and overconvergent isocrystals are another variation related to overconvergent cohomology theories.

A Dieudonné crystal is a crystal with Verschiebung and Frobenius maps. An F-crystal is a structure in semilinear algebra somewhat related to crystals.

Crystals over the infinitesimal and crystalline sites

The infinitesimal site $\text(X/S)$ has as objects the infinitesimal extensions of open sets of $X$.
If $X$ is a scheme over $S$ then the sheaf $O_$ is defined by
$O_(T)$ = coordinate ring of $T$, where we write $T$ as an abbreviation for
an object $U\to T$ of $\text(X/S)$. Sheaves on this site grow in the sense that they can be extended from open sets to infinitesimal extensions of open sets.

A crystal on the site $\text(X/S)$ is a sheaf $F$ of $O_$ modules that is rigid in the following sense:
:for any map $f$ between objects $T$, $T'$; of $\text(X/S)$, the natural map from $f^* F(T)$ to $F(T')$ is an isomorphism.
This is similar to the definition of a quasicoherent sheaf of modules in the Zariski topology.

An example of a crystal is the sheaf $O_$.

Crystals on the crystalline site
are defined in a similar way.

Crystals in fibered categories

In general, if $E$ is a fibered category over $F$, then a crystal is a cartesian section of the fibered category. In the special case when $F$ is the category of infinitesimal extensions of a scheme $X$ and $E$ the category of quasicoherent modules over objects of $F$, then crystals of this fibered category are the same as crystals of the infinitesimal site.

References

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* (letter to Atiyah, Oct. 14 1963)
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*{{Citation , last1=Kedlaya , first1=Kiran S. , editor1-last=Abramovich , editor1-first=Dan , author-link=Kiran Kedlaya , editor2-last=Bertram , editor2-first=A. , editor3-last=Katzarkov , editor3-first=L. , editor4-last=Pandharipande , editor4-first=Rahul , editor5-last=Thaddeus. , editor5-first=M. , title=Algebraic geometry---Seattle 2005. Part 2 , publisher=Amer. Math. Soc. , location=Providence, R.I. , series=Proc. Sympos. Pure Math. , isbn=978-0-8218-4703-9 , mr=2483951 , year=2009 , volume=80 , chapter=p-adic cohomology , arxiv=math/0601507 , pages=667–684 , bibcode=2006math......1507K

Algebraic geometry