In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes.[1][2] It combines several good properties possessed by its related "sub"topologies, such as the qfh and cdh topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmanian, Huber and Jörder's study of differential forms, etc.

Definition

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Voevodsky defined the h topology to be the topology associated to finite families   of morphisms of finite type such that   is a universal topological epimorphism (i.e., a set of points in the target is an open subset if and only if its preimage is open, and any base change also has this property[3][4]). Voevodsky worked with this topology exclusively on categories   of schemes of finite type over a Noetherian base scheme S.

Bhatt-Scholze define the h topology on the category   of schemes of finite presentation over a qcqs base scheme   to be generated by  -covers of finite presentation. They show (generalising results of Voevodsky) that the h topology is generated by:

  1. fppf-coverings, and
  2. families of the form   where
    1.   is a proper morphism of finite presentation,
    2.   is a closed immersion of finite presentation, and
    3.   is an isomorphism over  .

Note that   is allowed in an abstract blowup, in which case Z is a nilimmersion of finite presentation.

Examples

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The h-topology is not subcanonical, so representable presheaves are almost never h-sheaves. However, the h-sheafification of representable sheaves are interesting and useful objects; while presheaves of relative cycles are not representable, their associated h-sheaves are representable in the sense that there exists a disjoint union of quasi-projective schemes whose h-sheafifications agree with these h-sheaves of relative cycles.[5]

Any h-sheaf in positive characteristic satisfies   where we interpret   as the colimit   over the Frobenii (if the Frobenius is of finite presentation, and if not, use an analogous colimit consisting of morphisms of finite presentation). In fact, (in positive characteristic) the h-sheafification   of the structure sheaf   is given by  . So the structure sheaf "is an h-sheaf on the category of perfect schemes" (although this sentence doesn't really make sense mathematically since morphisms between perfect schemes are almost never of finite presentation). In characteristic zero similar results hold with perfection replaced by semi-normalisation.

Huber-Jörder study the h-sheafification   of the presheaf   of Kähler differentials on categories of schemes of finite type over a characteristic zero base field  . They show that if X is smooth, then  , and for various nice non-smooth X, the sheaf   recovers objects such as reflexive differentials and torsion-free differentials. Since the Frobenius is an h-covering, in positive characteristic we get   for  , but analogous results are true if we replace the h-topology with the cdh-topology.

By the Nullstellensatz, a morphism of finite presentation   towards the spectrum of a field   admits a section up to finite extension. That is, there exists a finite field extension   and a factorisation  . Consequently, for any presheaf   and field   we have   where  , resp.  , denotes the h-sheafification, resp. etale sheafification.

Properties

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As mentioned above, in positive characteristic, any h-sheaf satisfies  . In characteristic zero, we have   where   is the semi-normalisation (the scheme with the same underlying topological space, but the structure sheaf is replaced with its termwise seminormalisation).

Since the h-topology is finer than the Zariski topology, every scheme admits an h-covering by affine schemes.

Using abstract blowups and Noetherian induction, if   is a field admitting resolution of singularities (e.g., a characteristic zero field) then any scheme of finite type over   admits an h-covering by smooth  -schemes. More generally, in any situation where de Jong's theorem on alterations is valid we can find h-coverings by regular schemes.

Since finite morphisms are h-coverings, algebraic correspondences are finite sums of morphisms.[2]

cdh topology

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The cdh topology on the category   of schemes of finite presentation over a qcqs base scheme   is generated by:

  1. Nisnevich coverings, and
  2. families of the form   where
    1.   is a proper morphism of finite presentation,
    2.   is a closed immersion of finite presentation, and
    3.   is an isomorphism over  .

The cd stands for completely decomposed (in the same sense it is used for the Nisnevich topology). As mentioned in the examples section, over a field admitting resolution of singularities, any variety admits a cdh-covering by smooth varieties. This topology is heavily used in the study of Voevodsky motives with integral coefficients (with rational coefficients the h-topology together with de Jong alterations is used).

Since the Frobenius is not a cdh-covering, the cdh-topology is also a useful replacement for the h-topology in the study of differentials in positive characteristic.

Rather confusingly, there are completely decomposed h-coverings, which are not cdh-coverings, for example the completely decomposed family of flat morphisms  .

Relation to v-topology and arc-topology

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The v-topology (or universally subtrusive topology) is equivalent to the h-topology on the category   of schemes of finite type over a Noetherian base scheme S. Indeed, a morphism in   is universally subtrusive if and only if it is universally submersive Rydh (2010, Cor.2.10). In other words,

 

More generally, on the category   of all qcqs schemes, neither of the v- nor the h- topologies are finer than the other:   and  . There are v-covers which are not h-covers (e.g.,  ) and h-covers which are not v-covers (e.g.,   where R is a valuation ring of rank 2 and   is the non-open, non-closed prime Rydh (2010, Example 4.3)).

However, we could define an h-analogue of the fpqc topology by saying that an hqc-covering is a family   such that for each affine open   there exists a finite set K, a map   and affine opens   such that   is universally submersive (with no finiteness conditions). Then every v-covering is an hqc-covering.

 

Indeed, any subtrusive morphism is submersive (this is an easy exercise using Rydh (2010, Cor.1.5 and Def.2.2)).

By a theorem of Rydh, for a map   of qcqs schemes with   Noetherian,   is a v-cover if and only if it is an arc-cover (for the statement in this form see Bhatt & Mathew (2018, Prop.2.6)). That is, in the Noetherian setting everything said above for the v-topology is valid for the arc-topology.

Notes

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  1. ^ Voevodsky, V. (1996), "Homology of schemes", Selecta Mathematica, New Series, 2 (1): 111–153, doi:10.1007/BF01587941, MR 1403354
  2. ^ a b Suslin, Andrei; Voevodsky, Vladimir (1996), "Singular homology of abstract algebraic varieties", Inventiones Mathematicae, 123 (1): 61–94, doi:10.1007/BF01232367, MR 1376246
  3. ^ SGA I, Exposé IX, définition 2.1
  4. ^ Suslin and Voevodsky, 4.1
  5. ^ Suslin, Andrei; Voevodsky, Vladimir (2000), "Relative cycles and Chow sheaves", Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143, Princeton University Press, pp. 10–86, ISBN 0-691-04814-2, MR 1764199

Further reading

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