User:John Z/drafts/Grothendieck group

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K₀ is often defined for a ring or for a ringed space. The usual construction is as follows: For a not necessarily commutative ring R, one lets the abelian category be the category of all finitely generated projective modules over the ring. For a ringed space (X,O_X), one lets the abelian category be the category of all coherent sheaves on X. This makes K₀ into a functor.

There is another Grothendieck group of a ring or a ringed space which is sometimes useful. The Grothendieck group G₀ of a ring is the Grothendieck group associated to the category of all finitely generated modules over a ring. Similarly, the Grothendieck group G₀ of a ringed space is the Grothendieck group associated to the category of all quasicoherent sheaves on the ringed space. G₀ is not a functor, but nevertheless it carries important information.

K₀ is often defined for a ring or for a ringed space. For a not necessarily commutative ring R, one lets the abelian category be the category of all finitely generated projective modules over the ring. For a ringed space (X,O_X), one lets the abelian category be the category of all coherent sheaves on X. This makes K₀ into a covariant functor from the category of rings and ring homomorphisms to abelian groups. Tensor products can be used to define the functor on ring homorphisms as they preserve projectiveness and exactness of a sequence of projective objects. If one can take tensor products of the modules or sheaves represented in K₀", as in the case of commutative rings or commutatively ringed spaces, 'K₀ is in fact a ring, the "Grothendieck ring" with this tensor product multiplication.

There is another Grothendieck group of a ring or a ringed space which is sometimes useful. The Grothendieck group G₀ of a ring is the Grothendieck group associated to the category of all finitely generated modules over a ring. Similarly, the Grothendieck group G₀ of a ringed space is the Grothendieck group associated to the category of all coherent sheaves on the ringed space. G is naturally isomorphic to K when one can finitely resolve arbitrary modules or coherent sheaves into projective, locally free ones, as in the case of regular, finite dimensional schemes, or smooth varieties over a field in particular. G₀ is a module over the "Grothendieck ring" 'K₀ .

G₀ is a functor of the same variance as K - covariant for rings, contravariant for ringed spaces for flat morphisms of rings or ringed spaces, as flatness is by definition the property of preserving exactness of all sequences of objects. One can also consider G a covariant functor of ringed spaces by defining it on ringed space morphisms as the alternating sum of higher direct images if this make sense - i.e. when these eventually vanish, as for proper morphisms of schemes or analytic spaces.