 also works in the algebraic context without considering only quasi-projective varieties, if one uses the more sophisticated definition of $\bK_{alg}(X \xrightarrow {f} Y)=K_0(D^b_{f-perf}(X))$ as the Grothendieck goup of the triangulated category of $f$-perfect complexes.  And a similar definition can also be used in the context of compact complex analytic varieties (c