In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. 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.
Voevodsky defined the h topology to be the topology associated to finite families
\{pi:Ui\toX\}
\amalgUi\toX
ft | |
Sch | |
/S |
Bhatt-Scholze define the h topology on the category
fp | |
Sch | |
/S |
S
v
\{X'\toX,Z\toX\}
X'\toX
Z\toX
X'\toX
X\setminusZ
Note that
X'=\varnothing
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.
Any h-sheaf in positive characteristic satisfies
F(X)=F(Xperf)
Xperf
\operatorname{colim}(F(X)\stackrel{Frob
l{O}h
l{O}
l{O}h(X)=l{O}(Xperf)
Huber-Jörder study the h-sheafification
n | |
\Omega | |
h |
X\mapsto\Gamma(X,
n | |
\Omega | |
X/k |
)
k
n(X) | |
\Omega | |
h |
=\Gamma(X,
n) | |
\Omega | |
X/k |
n | |
\Omega | |
h |
n | |
\Omega | |
h |
=0
n>0
By the Nullstellensatz, a morphism of finite presentation
X\to\operatorname{Spec}(k)
k
L/k
\operatorname{Spec}(L)\toX\to\operatorname{Spec}(k)
F
k
Fh(k)=Fet(kperf)
Fh
Fet
As mentioned above, in positive characteristic, any h-sheaf satisfies
F(X)=F(Xperf)
F(X)=F(Xsn)
Xsn
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
k
k
k
Since finite morphisms are h-coverings, algebraic correspondences are finite sums of morphisms.
The cdh topology on the category
fp | |
Sch | |
/S |
S
\{X'\toX,Z\toX\}
X'\toX
Z\toX
X'\toX
X\setminusZ
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
\{A1\stackrel{x\mapstox2}{\to}A1,A1\setminus\{0\}\stackrel{x\mapstox}{\to}A1\}
The v-topology (or universally subtrusive topology) is equivalent to the h-topology on the category
ft | |
Sch | |
S |
ft | |
Sch | |
S |
More generally, on the category
Sch
Shvh(Sch)\not\subsetShvv(Sch)
Shvv(Sch)\not\subsetShvh(Sch)
Spec(C(x))\toSpec(C)
Spec(R/ak{p})\sqcupSpec(Rak{p})\toSpec(R)
ak{p}
However, we could define an h-analogue of the fpqc topology by saying that an hqc-covering is a family
\{Ti\toT\}i
U\subseteqT
i:K\toI
Ui(k)\subseteqTi(k) x TU
\sqcupkUi(k)\toU
Indeed, any subtrusive morphism is submersive (this is an easy exercise using).
By a theorem of Rydh, for a map
f:Y\toX
X
f