Grothendieck groups for categories of complexes

Abstract The new intersection theorem states that, over a Noetherian local ring R , for any non-exact complex concentrated in degrees n ,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R . One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if P d (length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d ,…0, the inclusion P d (length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of P d (length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.