# 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 $X$.

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#### 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