In this article, we use the 2-ring structure of Algebraic K-Theory with supports to prove three results on intersection theory. The first result (Theorem A) is a vanishing theorem for intersection multiplicities which was conjectured by Serre [25], who proved it in several cases (loc. cit.); another proof was obtained by Roberts [24](see 7.4). The second result (Theorem B) describes, in positive characteristic, the action of the Frobenius endomorphism on the Euler characteristic of a complex; it is a special case of a conjecture by Szpiro [28]. Finally, in Theorem C, we introduce a multiplicative structure on the Chow groups of any noetherian regular scheme, after tensoring these by Q. To state A and B more precisely, let R be a local noetherian ring of finite dimension d which is a complete intersection, ie the quotient of a regular local ring by a regular sequence. Let M, N be two R-modules of finite type and finite projective dimension. Assume that the R-module M| N has finite