The Grothendieck Groups of Module Categories over Coherent Algebras

Thesis (Ph.D.)--University of Washington, 2014Let k be a field and B either a finitely generated free k-algebra, or a regular k-algebra of global dimension two with at least three generators, generated in arbitrary positive degrees. Let qgr B be the quotient category of finitely presented graded right B-modules modulo those that are finite dimensional. We compute the Grothendieck group K0(qgr B). In particular, if the inverse of the Hilbert series of B (which is a polynomial) is irreducible, thenK0(qgr B) is isomorphic to Z[α] as ordered abelian groups where α is the smallest positive real pole of the Hilbert series of B and where Z[α] inherits its order structure fromR. We also obtain general conditions on an algebra B under which our computation of K0(qgr B) applies