Reducers and K0 with support

Let R be a commutative noetherian ring, L be a Serre subcategory of the category of finitely generated R-modules and P be the category of finitely generated projective R-modules. We define invariants based on the categories of chain complexes from P homologically supported in L with support between 0 and n ({ChL[0,n](P)}n is an element of N) and the K-group with support K0(R on L) similar to the stable range in classical K-theory. We introduce a notion called a reducer which we use to express the class of a complex in terms of classes of complexes of smaller amplitude and use these to study and bound the above defined invariants by standard invariants like arithmetic rank, grade and projective dimension. We also give conditions for K0(R on L) to be isomorphic to K0(modules in L withfiniteprojective dimension).