Additivity theorem for algebraic L-theory? There is an additivity theorem for algebraic K-theory. My question is is there an additivity theorem for algebraic L-theory?
 A: Let me just ignore the appearing decorations. I think the argument should work with all decorations. The $L$-groups of an additive category $\mathcal{A}$ with involution are defined as the cobordism classes or certain chain complexes over $\mathcal{A}$. The group structure is induced by direct sum of chain complexes.
If we have a additive functor between two additive categories $\mathcal{A}\rightarrow \mathcal{A}'$, it induces a group homomorphism $L_n(\mathcal{A})\rightarrow L_n(\mathcal{A}')$ by applying it degreewise. Now if we have two such functors $F,F'$  and an element $[C_*]\in L_n(\mathcal{A})$ then 
$$L_n(F\oplus F')([C_*]):=[(F(C_*)\oplus F'(C_*)]=[(F(C_*)]+[F'(C_*)]=(L_n(F)+L_n(F'))(C_*).$$
Thus additivity follows quite immediately from the definition. 
Definitions of the L-groups can be found for example in Ranicki, The algebraic theory of surgery, I Fuoundations, Prop. 3.2
I think $K$-theory is much harder, since the $K$-groups can only be defined as the homotopy groups of some space/spectrum. For $L$-theory those groups can be defined purely algebraically. However the $L$-theory spectra are still needed - for example to define homology with coefficients in L-theory.
