Cohomological dimension

The cohomological dimension $ ( \mathop{\rm dim} _ {G} X ) $ of a topological space $ X $ relative to the group of coefficients $ G $ is the maximum integer $ p $ for which there exists closed subsets $ A $ of $ X $ such that the cohomology groups $ H ^ {p} ( X , A ; G ) $ are non-zero. The homological dimension $ h \mathop{\rm dim} _ {G} X $ is similarly defined (cf. Homological dimension of a space). Finite Lebesgue dimension (covering dimension) is the same as $ \mathop{\rm dim} _ {G} $( or $ h \mathop{\rm dim} _ {G} $) if $ G $ is the subgroup of the integers (or real numbers modulo 1). In Euclidean space $ X \subset \mathbf R ^ {n} $ the equation $ \mathop{\rm dim} _ {G} X = p $ is equivalent to the property that $ X $ is locally linked by $ ( n - p - 1 ) $- dimensional cycles (with coefficients in $ G $). For paracompact spaces $ X $, the inequality $ \mathop{\rm dim} _ {G} X \leq p $ is equivalent to the existence of soft resolutions (cf. Soft sheaf and Resolution) for $ G $ of length $ p $. 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 $ \leq p $ if it has an injective (or projective) resolution of length $ p $; 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 $ X $.

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 Zbl 0325.57002
[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 Zbl 0187.20103
[6]	G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967) MR0221500 Zbl 0158.20505
[7]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305

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 $ X $ be an algebraic variety or a Noetherian scheme of dimension $ n $. The cohomological dimension of $ X $ is defined to be the integer $ \mathop{\rm cd} ( X) $ equal to the infimum of all those $ i $ for which $ H ^ {j} ( X , {\mathcal F} ) = 0 $ for all Abelian sheaves $ {\mathcal F} $ on the topological space $ X $ when $ j > i $. The inequality

$$ \mathop{\rm cd} ( X) \leq n $$

holds. The coherent cohomological dimension of the scheme $ X $ is the number $ \mathop{\rm cohcd} ( X) $ equal to the infimum of those $ i $ for which $ H ^ {j} ( X , {\mathcal F} ) = 0 $ for all coherent algebraic sheaves $ {\mathcal F} $( cf. Coherent algebraic sheaf) on $ X $ when $ j > i $. By definition, $ \mathop{\rm cohcd} ( X) \leq \mathop{\rm cd} ( X) $. By Serre's theorem, $ \mathop{\rm cohcd} ( X) = 0 $ if and only if $ X $ is an affine scheme. On the other hand, if $ X $ is an algebraic variety over a field $ k $, then $ \mathop{\rm cohcd} ( X) = n $ if and only if $ X $ is proper over $ k $( Lichtenbaum's theorem, see [3]).

Let $ X $ be a proper scheme over a field $ k $, let $ Y $ be a closed subscheme of $ X $ of codimension $ d $ and let $ U = X \setminus Y $. Then the following statements hold ([2]–[4]).

If $ Y $ is the set-theoretic complete intersection of ample divisors on $ X $, then

$$ \mathop{\rm cohcd} ( U) \leq d - 1 . $$

If $ X $ is a projective Cohen–Macaulay variety (for instance, a non-singular projective variety) and $ Y $ is zero-dimensional, then $ \mathop{\rm cohcd} ( U) = n - 1 $. The condition $ \mathop{\rm cohcd} ( U) \leq n - 2 $ is equivalent to $ Y $ being connected. If $ X = P ^ {n} $ is a projective space and $ Y $ is connected and has dimension $ \geq 1 $, then

$$ \mathop{\rm cohcd} ( U) < n - 1 . $$

If $ X $ is a complex algebraic variety, then one can consider the cohomological dimension of the corresponding topological space $ X ( C) $. In the general case when $ X $ is a Noetherian scheme, the analogue of cohomological dimension is the notion of the étale cohomological dimension of the scheme $ X $. More precisely, let $ X _ {\textrm{ et } } $ be the étale topology of the Grothendieck scheme $ X $ and let $ l $ be a prime number. By the cohomological $ l $- dimension of the scheme $ X $( or the étale cohomological dimension) one means the number $ \mathop{\rm cd} _ {l} ( X) $ equal to the infimum of those $ i $ for which $ H ^ {j} ( X _ {\textrm{ et } } , {\mathcal F} ) = 0 $ for all $ l $- torsion Abelian sheaves $ {\mathcal F} $ on $ X _ {\textrm{ et } } $ when $ j > i $. If $ X = \mathop{\rm Spec} A $ is an affine scheme, then $ \mathop{\rm cd} _ {l} ( \mathop{\rm Spec} A ) $ is also called the cohomological dimension of the ring $ A $. In particular, if $ A $ is a field, then the notion of $ \mathop{\rm cd} _ {l} ( A) $ is the same as that of the cohomological dimension of a field as studied in the theory of Galois cohomology.

If $ X $ is an algebraic variety of dimension $ n $ over a field $ k $ and if $ l \neq \mathop{\rm char} k $, then $ \mathop{\rm cd} _ {l} ( X) \leq 2 n + \mathop{\rm cd} _ {l} ( k) $. In particular, if $ k $ is a separably closed field, then $ \mathop{\rm cd} _ {l} ( X) \leq 2 n $. If $ X $ is an affine algebraic variety over the separably closed field $ k $, then $ \mathop{\rm cd} _ {l} ( X) \leq \mathop{\rm dim} X $.

Let $ k $ be a field of finite characteristic $ p $; then for any Noetherian scheme $ X $ over $ k $, the inequality

$$ \mathop{\rm cd} _ {p} ( X) \leq \mathop{\rm cohcd} ( X) + 1 $$

holds. In particular, for any Noetherian commutative ring $ A $,

$$ \mathop{\rm cd} _ {p} ( A) \leq 1 . $$

If $ X $ is a quasi-projective algebraic variety over the separably closed field $ k $, then $ \mathop{\rm cd} _ {p} ( X) \leq \mathop{\rm dim} X $, where $ p $ is the characteristic of $ k $.

References

[1]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohôku Math. J. , 9 (1957) pp. 119–221 MR0102537
[2]	R. Hartshorne, "Cohomological dimension of algebraic varieties" Ann. of Math. , 88 (1968) pp. 403–450 MR0232780 Zbl 0169.23302
[3]	R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) MR0282977 Zbl 0208.48901
[4]	R. Hartshorne, "Cohomology of non-complete algebraic varieties" Compositio Math. (1971) pp. 257–264 MR0302649 Zbl 0221.14014
[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) MR1080173 MR1080172 MR0717602 MR0717586 MR0505104 MR0505101

I.V. Dolgachev

References

[a1]	B. Iversen, "Cohomology of sheaves" , Springer (1986) MR0842190 Zbl 0559.55001