The Grothendieck Groups of Module Categories over Coherent Algebras

Abstract

Let <italic>k</italic> be a field and <italic>B</italic> either a finitely generated free <italic>k</italic>-algebra, or a regular <italic>k</italic>-algebra of global dimension two with at least three generators, generated in arbitrary positive degrees. Let qgr <italic>B</italic> be the quotient category of finitely presented graded right <italic>B</italic>-modules modulo those that are finite dimensional. We compute the Grothendieck group <italic>K<italic>₀(qgr <italic>B</italic>). In particular, if the inverse of the Hilbert series of <italic>B</italic> (which is a polynomial) is irreducible, then<italic>K<italic>₀(qgr <italic>B</italic>) is isomorphic to <bold>Z</bold>[α] as ordered abelian groups where α is the smallest positive real pole of the Hilbert series of <italic>B</italic> and where <bold>Z</bold>[α] inherits its order structure from<bold>R</bold>. We also obtain general conditions on an algebra <italic>B</italic> under which our computation of <italic>K<italic>₀(qgr <italic>B</italic>) applies.