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In [[mathematics]], '''sheaf cohomology''' is the aspect of [[sheaf theory]], concerned with sheaves of [[abelian group]]s, that applies [[homological algebra]] to make possible effective calculation of the [[global section]]s of a sheaf ''F''. This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants.
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Its development was rapid in the years after 1950, when it was realised that sheaf cohomology was connected with more classical methods applied to the [[Riemann-Roch theorem]], the analysis of a [[linear system of divisors]] in [[algebraic geometry]], [[several complex variables]], and [[Hodge theory]]. The dimensions or ranks of sheaf cohomology groups became a fresh source of geometric data, or gave rise to new interpretations of older work.
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==Definitions==
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===The approach of Čech cohomology===
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{{Main| Čech cohomology}}
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The first version of sheaf cohomology to be defined was that based on [[Čech cohomology]], in which the relatively small change was made of attributing to an [[open set]] ''U'' of a [[topological space]] ''X'' an abelian group ''F''(''U'') that 'varies' with ''U'', rather than an abelian group ''A'' that is fixed ahead of time. This means that [[cochain]]s are easy to write down rather concretely; in fact the model applications, such as the [[Cousin problems]] on [[meromorphic function]]s, stay within fairly familiar mathematical territory. From the sheaf point of view, the Čech theory is the restriction to sheaves of [[locally constant function]]s with values in ''A''. Within sheaf theory it is easy to see that 'twisted' versions, with [[local coefficients]] on which the [[fundamental group]] acts, are also subsumed &mdash; along with some very different sorts of more general coefficients.
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One problem with that theory was that Čech cohomology itself fails to have good properties, unless ''X'' itself is [[well-behaved]]. This is not a difficulty in case ''X'' is something like a [[manifold]]; but embarrassing for applications to algebraic geometry, since the [[Zariski topology]] is in general not [[Hausdorff space|Hausdorff]]. The problem with the Čech theory manifests itself in the failure of the [[long exact sequence]] of [[cohomology group]]s associated to a [[short exact sequence]] of sheaves. This in practice is the basic method of attacking a calculation (i.e. to show how a given sheaf is involved with others in a short exact sequence, and draw consequences). The theory stood in this state of disarray only for a short while: [[Jean-Pierre Serre]] showed that the Čech theory worked, and on the other hand [[Alexandre Grothendieck]] proposed a more abstract definition that would build in the long exact sequence.
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===Definition by derived functors===
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The Grothendieck definition clarified the status of sheaf cohomology of a topological space ''X'' with coefficients in a sheaf <math>\mathcal F</math> as the [[derived functor|right derived functor]] of the [[Sheaf (mathematics)|global section]] functor:
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:<math>\Gamma_X: \mathcal F \mapsto \mathcal F(X).</math>
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This functor is not an [[exact functor]], a fact familiar in other terms from the theory of [[branch cut]]s (for example, in the case of the [[logarithm]] of a [[complex number]]: see [[exponential sequence]]). It is a [[left exact functor]], and therefore has a sequence of right derived [[functor]]s, denoted by
<ul>
:<math>H^i(X, \mathcal F), i \geq 0.</math>
 
The [[existence theorem|existence]] of these derived functors is supplied by [[homological algebra]] of the [[abelian category]] of sheaves (and indeed this was a main reason to set up that theory). It depends on having [[injective resolution]]s; that is, ''in theory'' calculations can be done with injective resolutions, though ''in practice'' short and long exact sequences may be a better idea.
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Because the derived functor can be computed by applying the functor to any acyclic resolution and keeping the cohomology of the complex, there are a number of other ways to compute cohomology groups. Depending on the concrete situation, fine, flasque, soft or acyclic sheaves are used to calculate concrete cohomology groups -- see [[injective sheaf|injective sheaves]].
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Subsequently there were further technical extensions (for example in [[Godement]]'s book), and areas of application. For example, sheaves were applied to [[transformation group]]s; as an inspiration to [[homology theory]] in the form of [[Borel-Moore homology]] for [[locally compact space]]s; to [[representation theory]] in the [[Borel-Bott-Weil theorem]]; as well as becoming standard in algebraic geometry and [[complex manifold]]s.
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The particular needs of [[étale cohomology]] were more about reinterpreting ''sheaf'' in ''sheaf cohomology'', than ''cohomology'', given that the derived functor approach applied. [[Flat cohomology]], [[crystalline cohomology]] and successors are also applications of the basic model.
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==Euler characteristics==
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The Euler characteristic <math>\chi(\mathcal{F})</math> of a sheaf <math>\mathcal{F}</math> is defined by
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:<math> \chi(\mathcal{F}) := \sum_{i \in \mathbf{Z}_0^+} (-1)^i \,{\rm rank}\, (H^{i}(X, \mathcal F)). </math>
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To make sense of this expression, which generalises the [[Euler characteristic]] as [[alternating sum]] of [[Betti number]]s, two conditions must be fulfilled. Firstly the summands must be [[almost all]] zero, i.e. zero for <math> i \geq N </math> for some <math> N </math>. Further, ''rank'' must be some well-defined function from [[module theory]], such as [[rank of an abelian group]] or [[vector space dimension]], that yields finite values on the cohomology groups in question. Therefore [[finiteness theorem]]s of ''two'' kinds are required.  
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In theories such as [[coherent cohomology]], where such theorems exist, the value of &chi;(''F'') is typically easier to compute, from other considerations (for example the [[Hirzebruch-Riemann-Roch theorem]] or [[Grothendieck-Riemann-Roch theorem]]), than the individual ranks separately. In practice it is often ''H''<sup>0</sup>(''X'',''F'') that is of most interest; one way to compute its rank is then by means of a [[vanishing theorem]] on the other ''H''<sup>''i''</sup>(''X'',''F''). This is a standard ''indirect'' method of sheaf theory to produce numerical results.
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For a [[locally contractible]] topological space, the [[singular cohomology]] groups with coefficients in ''A'' agree with the sheaf cohomology groups with the constant sheaf of ''A'', for any abelian group ''A''.
<ul>
 
 
==References==
  <li>[http://www.garanhunsnegocios.com/index.php?page=item&id=83082 http://www.garanhunsnegocios.com/index.php?page=item&id=83082]</li>
Almost any reference on sheaves treats sheaf cohomology, for example:
 
* {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | id={{MathSciNet | id = 1288523}} | year=1994}}, emphasizing the theory in the context of [[complex manifold]]s
  <li>[http://www.vc1000.com/news/html/?95676.html http://www.vc1000.com/news/html/?95676.html]</li>
* {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | id={{MathSciNet | id = 0463157}} | year=1977}}, in the algebraic-geometric setting, i.e. referring to the [[Zariski topology]]
 
* {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | id={{MathSciNet | id = 842190}} | year=1986}}, in the topological setting
  <li>[http://grhdx.site02.51eway.com/news/html/?99953.html http://grhdx.site02.51eway.com/news/html/?99953.html]</li>
* The [http://mathoverflow.net/questions/1151/sheaf-cohomology-and-injective-resolutions thread] "Sheaf cohomology and injective resolutions" on [[MathOverflow]]
 
[[Category:Algebraic topology]]
  <li>[http://jinxi.org/forum.php?mod=viewthread&tid=272584 http://jinxi.org/forum.php?mod=viewthread&tid=272584]</li>
[[Category:Cohomology theories]]
 
[[Category:Homological algebra]]
  <li>[http://www.wlbnw.com/forum.php?mod=viewthread&tid=203329 http://www.wlbnw.com/forum.php?mod=viewthread&tid=203329]</li>
[[Category:Sheaf theory]]
 
[[Category:Topological methods of algebraic geometry]]
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