manuscripta mathematica manuscript No. (will be inserted by the editor) Jean Fasel The excess intersection formula for Grothendieck-Witt groups Received: date / Revised version: date Abstract. We prove the excess intersection and self intersection formulae for Grothendieck-Witt groups. 1. Introduction Let X ′ j  g  Y ′ f  X i  Y be a fibre product of (reasonable) schemes. Suppose that i and j are reg- ular embeddings and that f and g are of finite Tor-dimension. The excess intersection formula for K-theory computes the defect of commutativity of the square K 0 (X ′ ) j∗  K 0 (Y ′ ) K 0 (X) g ∗  i∗  K 0 (Y ) f ∗  where K 0 (_) denotes either the Grothendieck group of the category of co- herent sheaves, either the Grothendieck group of the category of (coherent) locally free sheaves on X. If N Y X denotes the normal cone to X in Y and N Y ′ X ′ is the normal cone of X ′ in Y ′ then N Y ′ X ′ is an admissible sub- bundle of g ∗ N Y X. If E denotes the quotient bundle, the excess intersection formula states that for any α ∈ K 0 (X) f ∗ i ∗ (α)= j ∗ (e(E) · g ∗ α) where e(E) is the Euler class of E. This formula makes sense in various other contexts than K-theory, such as Chow groups and motivic cohomology, equivariant K-theory and others (see [6], [4] and [12]). Jean Fasel: 1984 Mathematics Road, UBC, V6T 1Z2 Vancouver BC, Canada