throughout the 1970s and early 1980s, "k"-theory on singular varieties still lacked adequate foundations. while it was believed that quillen's "k"-theory gave the correct groups, it was not known that these groups had all of the envisaged properties. for this, algebraic "k"-theory had to be reformulated. this was done by thomason in a lengthy monograph which he co-credited to his dead friend thomas trobaugh, who he said gave him a key idea in a dream. thomason combined waldhausen's construction of "k"-theory with the foundations of intersection theory described in volume six of grothendieck's séminaire de géométrie algébrique du bois marie. there, "k" was described in terms of complexes of sheaves on algebraic varieties. thomason discovered that if one worked with in derived category of sheaves, there was a simple description of when a complex of sheaves could be extended from an open subset of a variety to the whole variety. by applying waldhausen's construction of "k"-theory to derived categories, thomason was able to prove that algebraic "k"-theory had all the expected properties of a cohomology theory.