K-theory of non-additive functors of finite degree

Abstract

If R is a ring with unit let ~ R denote the category of finitely generated projective left R-modules. Algebraic K-theory (of projective modules) is a functor K : ~--,ag~, where stay is the category of abelian groups and homomorphisms, and ~ is the category whose objects are rings R, S . . . . and whose morphisms f ~ ( R , S ) are linear (additive) functors f : ~R---+~S. In the present paper we extend this functoriality to K : ~ , , d f ¢ , , where ~a,, .~,~, , have the same objects as ~a, ~4~, but more morphisms: Morphisms of~a, are polynomial functors F : ~R---+~S, or functors of finite degree; and morphisms in ~ , ( A , B) are mappings A-+B of finite degree (linear, quadratic, cubic . . . . ). The assignment F~K(F) is degree-preserving (or rather, filtration-preserving). More generally, we establish the same kind of functoriality for K-theory of functor-categories (~aR}~, and for relative K-theory of surjective ringhomomorphisms ~o : R---~/~. The main labor consists in finding a description of the sets KR resp. K~o which does not use direct sums of objects or sums of morphisms in ~aR. The Atiyah-description of topological K-theory in terms of chain complexes of vector-bundles and chain-homotopy (essentially) avoids direct sums, and Sections 1 and 2 of the paper consist in adapting his procedure to the algebraic situation. Since chain homotopy still involves addition of morphisms we pass from chain complexes to semisimplicial modules (Section 3). To these structures arbitrary nonadditive functors F apply and they preserve homotopy; if their degree is finite they also preserve finite-dimensionality. Using this semi-simplicial technique we obtain the polynomial functoriality of K in Section 4. The last Section 5 is essentially an application to algebraic topology. It shows that for symmetric powers and many other functors • of compact C W-spaces the Euler-characteristic e of • Y, or the characteristic rational function X of ~/'g : ~b Y---+ q~ Y has the form e(4~ Y) = ~p(e Y), X(cb9) = ~(X9), where ~p :~---+]g, 7' :Q{t}*---,11~{t}* are functions whose degrees equal the degree of 4,. For symmetric powers, • = SP", we determine explicitely an expression for the Lefschetz-number of SP"9 in terms of X 9.