Cohomological dimension
The cohomological dimension of a topological space
relative to the group of coefficients
is the maximum integer
for which there exists closed subsets
of
such that the cohomology groups
are non-zero. The homological dimension
is similarly defined (cf. Homological dimension of a space). Finite Lebesgue dimension (covering dimension) is the same as
(or
) if
is the subgroup of the integers (or real numbers modulo 1). In Euclidean space
the equation
is equivalent to the property that
is locally linked by
-dimensional cycles (with coefficients in
). For paracompact spaces
, the inequality
is equivalent to the existence of soft resolutions (cf. Soft sheaf and Resolution) for
of length
. Since soft sheaves are acyclic, in this way a connection is established with the general definition of dimension in homological algebra; for example, the injective (or projective) dimension of a module is
if it has an injective (or projective) resolution of length
; the global dimension of a ring is the maximum of the injective (or projective) dimensions of the modules over the ring and is the analogue of the Lebesgue dimension of
.
References
[1] | P.S. Aleksandrov, Ann. of Math. (1929) pp. 101–187 , 30 |
[2] | P.S. Aleksandrov, "Dimensionstheorie: Ein Beitrag zur Geometrie der abgeschlossenen Mengen" Math. Ann. , 106 (1932) pp. 161–238 |
[3] | P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian) |
[4] | A.E. [A.E. Kharlap] Harlap, "Local homology and cohomology, homology dimension and generalized manifolds" Math. USSR Sb. , 25 : 3 (1975) pp. 323–349 Mat. Sb. , 96 : 3 (1975) pp. 347–373 |
[5] | V.I. Kuz'minov, "Homological dimension theory" Russian Math. Surveys , 29 : 5 (1968) pp. 1–45 Uspekhi Mat. Nauk , 23 : 5 (1968) pp. 3–49 |
[6] | G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967) |
[7] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) |
E.G. Sklyarenko
The cohomological dimension of a scheme is the analogue of the notion of the cohomological dimension of a topological space for an algebraic variety or a scheme with a selected cohomology theory. Let be an algebraic variety or a Noetherian scheme of dimension
. The cohomological dimension of
is defined to be the integer
equal to the infimum of all those
for which
for all Abelian sheaves
on the topological space
when
. The inequality
![]() |
holds. The coherent cohomological dimension of the scheme is the number
equal to the infimum of those
for which
for all coherent algebraic sheaves
(cf. Coherent algebraic sheaf) on
when
. By definition,
. By Serre's theorem,
if and only if
is an affine scheme. On the other hand, if
is an algebraic variety over a field
, then
if and only if
is proper over
(Lichtenbaum's theorem, see [3]).
Let be a proper scheme over a field
, let
be a closed subscheme of
of codimension
and let
. Then the following statements hold ([2]–[4]).
If is the set-theoretic complete intersection of ample divisors on
, then
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If is a projective Cohen–Macaulay variety (for instance, a non-singular projective variety) and
is zero-dimensional, then
. The condition
is equivalent to
being connected. If
is a projective space and
is connected and has dimension
, then
![]() |
If is a complex algebraic variety, then one can consider the cohomological dimension of the corresponding topological space
. In the general case when
is a Noetherian scheme, the analogue of cohomological dimension is the notion of the étale cohomological dimension of the scheme
. More precisely, let
be the étale topology of the Grothendieck scheme
and let
be a prime number. By the cohomological
-dimension of the scheme
(or the étale cohomological dimension) one means the number
equal to the infimum of those
for which
for all
-torsion Abelian sheaves
on
when
. If
is an affine scheme, then
is also called the cohomological dimension of the ring
. In particular, if
is a field, then the notion of
is the same as that of the cohomological dimension of a field as studied in the theory of Galois cohomology.
If is an algebraic variety of dimension
over a field
and if
, then
. In particular, if
is a separably closed field, then
. If
is an affine algebraic variety over the separably closed field
, then
.
Let be a field of finite characteristic
; then for any Noetherian scheme
over
, the inequality
![]() |
holds. In particular, for any Noetherian commutative ring ,
![]() |
If is a quasi-projective algebraic variety over the separably closed field
, then
, where
is the characteristic of
.
References
[1] | A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohôku Math. J. , 9 (1957) pp. 119–221 |
[2] | R. Hartshorne, "Cohomological dimension of algebraic varieties" Ann. of Math. , 88 (1968) pp. 403–450 |
[3] | R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) |
[4] | R. Hartshorne, "Cohomology of non-complete algebraic varieties" Compositio Math. (1971) pp. 257–264 |
[5] | M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Théorie des topos et cohomologie étale des schemas (SGA 4, vol. II, III) , Lect. notes in math. , 270; 305 , Springer (1972–1973) |
I.V. Dolgachev
Comments
References
[a1] | B. Iversen, "Cohomology of sheaves" , Springer (1986) |
Cohomological dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomological_dimension&oldid=24054