View source for Cohomological dimension

The cohomological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230501.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230502.png" /> relative to the group of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230503.png" /> is the maximum integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230504.png" /> for which there exists closed subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230506.png" /> such that the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230507.png" /> are non-zero. The homological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230508.png" /> is similarly defined (cf. [[Homological dimension of a space|Homological dimension of a space]]). Finite Lebesgue dimension (covering dimension) is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c0230509.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305010.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305011.png" /> is the subgroup of the integers (or real numbers modulo 1). In Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305012.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305013.png" /> is equivalent to the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305014.png" /> is locally linked by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305015.png" />-dimensional cycles (with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305016.png" />). For paracompact spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305017.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305018.png" /> is equivalent to the existence of soft resolutions (cf. [[Soft sheaf|Soft sheaf]] and [[Resolution|Resolution]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305019.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305020.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305021.png" /> if it has an injective (or projective) resolution of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305022.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305023.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   ''Ann. of Math.''  (1929)  pp. 101–187 , '''30'''</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Aleksandrov,   "Dimensionstheorie: Ein Beitrag zur Geometrie der abgeschlossenen Mengen"  ''Math. Ann.'' , '''106'''  (1932)  pp. 161–238</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,   "An introduction to homological dimension theory and general combinatorial topology" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.E. Bredon,   "Sheaf theory" , McGraw-Hill  (1967)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR></table>

''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305024.png" /> be an algebraic variety or a Noetherian scheme of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305025.png" />. The cohomological dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305026.png" /> is defined to be the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305027.png" /> equal to the infimum of all those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305028.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305029.png" /> for all Abelian sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305030.png" /> on the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305031.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305032.png" />. The inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305033.png" /></td> </tr></table>

holds. The coherent cohomological dimension of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305034.png" /> is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305035.png" /> equal to the infimum of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305036.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305037.png" /> for all coherent algebraic sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305038.png" /> (cf. [[Coherent algebraic sheaf|Coherent algebraic sheaf]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305039.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305040.png" />. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305041.png" />. By Serre's theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305042.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305043.png" /> is an affine scheme. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305044.png" /> is an algebraic variety over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305046.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305047.png" /> is proper over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305048.png" /> (Lichtenbaum's theorem, see [[#References|[3]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305049.png" /> be a proper scheme over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305050.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305051.png" /> be a closed subscheme of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305052.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305053.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305054.png" />. Then the following statements hold ([[#References|[2]]]–[[#References|[4]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305055.png" /> is the set-theoretic complete intersection of ample divisors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305056.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305057.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305058.png" /> is a projective Cohen–Macaulay variety (for instance, a non-singular projective variety) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305059.png" /> is zero-dimensional, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305060.png" />. The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305061.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305062.png" /> being connected. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305063.png" /> is a projective space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305064.png" /> is connected and has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305065.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305066.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305067.png" /> is a complex algebraic variety, then one can consider the cohomological dimension of the corresponding topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305068.png" />. In the general case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305069.png" /> is a Noetherian scheme, the analogue of cohomological dimension is the notion of the étale cohomological dimension of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305070.png" />. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305071.png" /> be the [[Etale topology|étale topology]] of the Grothendieck scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305072.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305073.png" /> be a prime number. By the cohomological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305075.png" />-dimension of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305076.png" /> (or the étale cohomological dimension) one means the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305077.png" /> equal to the infimum of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305078.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305079.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305080.png" />-torsion Abelian sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305081.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305082.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305083.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305084.png" /> is an affine scheme, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305085.png" /> is also called the cohomological dimension of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305086.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305087.png" /> is a field, then the notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305088.png" /> is the same as that of the cohomological dimension of a field as studied in the theory of [[Galois cohomology|Galois cohomology]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305089.png" /> is an algebraic variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305090.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305091.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305092.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305093.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305094.png" /> is a separably closed field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305095.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305096.png" /> is an affine algebraic variety over the separably closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305097.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305098.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c02305099.png" /> be a field of finite characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050100.png" />; then for any Noetherian scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050101.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050102.png" />, the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050103.png" /></td> </tr></table>

holds. In particular, for any Noetherian commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050104.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050105.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050106.png" /> is a quasi-projective algebraic variety over the separably closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050107.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050109.png" /> is the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023050/c023050110.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "Sur quelques points d'algèbre homologique"  ''Tohôku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Hartshorne,   "Cohomological dimension of algebraic varieties"  ''Ann. of Math.'' , '''88'''  (1968)  pp. 403–450</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Hartshorne,   "Ample subvarieties of algebraic varieties" , Springer  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Hartshorne,   "Cohomology of non-complete algebraic varieties"  ''Compositio Math.''  (1971)  pp. 257–264</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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)</TD></TR></table>

''I.V. Dolgachev''

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Iversen,   "Cohomology of sheaves" , Springer  (1986)</TD></TR></table>