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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 is the number of vertices minus the number of edges. (Olaf Post calls this a "well-known formula".)

More generally, one can define the Euler characteristic of any chain complex to be the alternating sum of the ranks of the homology groups of the chain complex, assuming that all these ranks are finite.

A version of Euler characteristic used in algebraic geometry is as follows. For any coherent sheaf F {\displaystyle {\mathcal {F}}} on a proper scheme X, one defines its Euler characteristic to be

where h i (X, F) {\displaystyle \ h^{i}(X,{\mathcal {F}})\ } is the dimension of the i -th sheaf cohomology group of F {\displaystyle {\mathcal {F}}}. In this case, the dimensions are all finite by 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 F {\displaystyle \ {\mathcal {F}}\ } by acyclic sheaves.

Another generalization of the concept of Euler characteristic on manifolds comes from orbifolds (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 1 + 1 / p , where p is a prime number corresponding to the cone angle 2 π / p.

The concept of Euler characteristic of the reduced homology of a bounded finite poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and largest elements; call them 0 and 1. The Euler characteristic of such a poset is defined as the integer μ (0,1), where μ is the Möbius function in that poset's incidence algebra.

This can be further generalized by defining a rational valued Euler characteristic for certain finite 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 or monoid G is 1 / | G | , and the Euler characteristic of a finite groupoid is the sum of 1 / | G |, where we picked one representative group G for each connected component of the groupoid.