  Let $Q$ be a commutative, Noetherian ring and $Z \subseteq
\operatorname{Spec}(Q)$ a closed subset. Define $K_0^Z(Q)$ to be the
Grothendieck group of those bounded complexes of finitely generated projective
$Q$-modules that have homology supported on $Z$. We develop "cyclic" Adams
operations on $K_0^Z(Q)$ and we prove these operations satisfy the four axioms
used by Gillet and Soul\'e in their paper "Intersection Theory Using Adams
Operations". From this we recover a shorter proof of Serre's Vanishing
Conjecture. We also show our cyclic Adams operations agree with the Adams
operations defined by Gillet and Soul\'e in certain cases.
