Editing Euler characteristic (section)

==Generalizations==
For every combinatorial [[cell complex]], one defines the Euler characteristic as the number of 0-cells, minus the number of 1-cells, plus the number of 2-cells, etc., if this alternating sum is finite. In particular, the Euler characteristic of a finite set is simply its cardinality, and the Euler characteristic of a [[Graph (discrete mathematics)|graph]] is the number of vertices minus the number of edges. (Olaf Post calls this a "well-known formula".<ref>{{cite book |first=Olaf |last=Post |year=2009 |section=Spectral analysis of metric graphs and related spaces |title=Limits of graphs in group theory and computer science |place=Lausanne, CH |publisher=[[EPFL Press]] |pages=109–140 |arxiv=0712.1507 |bibcode=2007arXiv0712.1507P}}</ref>)

More generally, one can define the Euler characteristic of any [[chain complex]] to be the alternating sum of the [[rank of an abelian group|ranks]] of the homology groups of the chain complex, assuming that all these ranks are finite.<ref>{{nlab|id=Euler+characteristic|title=Euler characteristic}}</ref>

A version of Euler characteristic used in [[algebraic geometry]] is as follows. For any [[coherent sheaf]] <math>\mathcal{F}</math> on a proper [[Scheme (mathematics)|scheme]] {{mvar|X}}, one defines its Euler characteristic to be
:<math> \chi ( \mathcal{F})= \sum_i (-1)^i h^i(X,\mathcal{F})\ ,</math>
where <math>\ h^i(X, \mathcal{F})\ </math> is the dimension of the {{mvar|i}}-th [[sheaf cohomology]] group of <math>\mathcal{F}</math>. In this case, the dimensions are all finite by [[Coherent_sheaf_cohomology#Finite-dimensionality_of_cohomology|Grothendieck's finiteness theorem]]. This is an instance of the Euler characteristic of a chain complex, where the chain complex is a finite resolution of <math>\ \mathcal{F}\ </math> by acyclic sheaves.

Another generalization of the concept of Euler characteristic on manifolds comes from [[orbifold]]s (see [[Euler characteristic of an orbifold]]). While every manifold has an integer Euler characteristic, an orbifold can have a fractional Euler characteristic. For example, the teardrop orbifold has Euler characteristic {{nobr|1 + {{sfrac|1| {{mvar|p}} }} ,}} where {{mvar|p}} is a prime number corresponding to the cone angle {{sfrac| 2{{pi}} | {{mvar|p}} }}.

The concept of Euler characteristic of the [[reduced homology]] of a bounded finite [[partially ordered set|poset]] is another generalization, important in [[combinatorics]]. A poset is "bounded" if it has smallest and largest elements; call them&nbsp;0 and&nbsp;1. The Euler characteristic of such a poset is defined as the integer {{math|''μ''(0,1)}}, where {{mvar|μ}} is the [[Incidence algebra|Möbius function]] in that poset's [[incidence algebra]].

This can be further generalized by defining a [[rational number|rational valued]] Euler characteristic for certain finite [[category (mathematics)|categories]], a notion compatible with the Euler characteristics of graphs, orbifolds and posets mentioned above. In this setting, the Euler characteristic of a finite [[group (mathematics)|group]] or [[monoid]] {{mvar|G}} is {{sfrac|1| {{abs| {{mvar|G}} }} }}, and the Euler characteristic of a finite [[groupoid]] is the sum of {{sfrac|1| {{abs|{{mvar|G}}{{sub|{{mvar|i}}}} }}}}, where we picked one representative group {{mvar|G}}{{sub|{{mvar|i}}}} for each connected component of the groupoid.<ref>{{cite journal |first=Tom |last=Leinster |year=2008 |title=The Euler characteristic of a category |journal=Documenta Mathematica |volume=13 |pages=21–49 |doi=10.4171/dm/240 |doi-access=free |s2cid=1046313 |url=http://www.math.uiuc.edu/documenta/vol-13/02.pdf |via=[[University of Illinois, Urbana-Champaign|U. Illinois, Urbana-Champaign]] |archive-url=https://web.archive.org/web/20140606220358/http://www.math.uiuc.edu/documenta/vol-13/02.pdf |archive-date=2014-06-06}}</ref>