Grothendieck Group

The Grothendieck group of a commutative monoid , written additively, is the universal Abelian group receiving a homomorphism preserving addition and zero. Every such homomorphism from to an Abelian group factors uniquely through a group homomorphism (Weibel 2013, p. 63). Its elements are pairs modulo the equivalence relation iff for some . The group law is , and . The class represents the formal difference . For the monoid of isomorphism classes of complex vector bundles over a compact Hausdorff space under Whitney sum, the resulting group is the zeroth topological -group (Atiyah 1989; Weibel 2013, p. 82).

For an exact category whose isomorphism classes of objects form a set, is generated by those classes , with the relation for each chosen short exact sequence (Weibel 2013, p. 129). The choice of exact structure matters. For a triangulated category with a set of isomorphism classes, the corresponding relation is imposed for each distinguished triangle (Weibel 2013, p. 173).

For a smooth projective variety over , let be its bounded derived category of coherent sheaves, and write . The Euler pairing on this group is given by

where is the th shift and the hom-sets are finite-dimensional vector spaces that vanish for all but finitely many . This pairing is additive in each argument. The numerical Grothendieck group is the quotient group of by the classes satisfying for every . It has finite rank (Bridgeland 2007, pp. 319-320).

A numerical Bridgeland stability condition has a central charge that factors through . Its support property requires a norm on the real vector space and a constant such that

for every nonzero semistable object , where is its numerical class (Bayer et al. 2016, p. 922). The norm is defined on the entire real vector space, but the inequality is required only for classes of semistable objects.