the generic point.

Symplectic modules {#symplectic}
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The group $\Ko AL$
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Let $A$ be a commutative ring and let $L$ be a line bundle over $A$. Let $GW^2(A,L)$ be the Grothendieck-Witt group of the (exact) category of projective $A$-modules endowed with the duality $\homm A{\_}L$ and the usual canonical isomorphism $\omega:Id\to \homm A{\homm A{\_}L}L$. Explicitly, $GW^2(A,L)$ is the Grothendieck group of the monoid (with the orthogonal sum $\bot$ as operation) of isometry classes of pairs $(P,\phi)$, where $P$ is a projective $A$-module and $\phi:P\to \homm APL$ is an anti-symmetric isomorphism.

If $Q$ is a projective $A$-module, we define $H(Q)$ to be the pair $(Q\oplus \homm AQL,h)$, where $$h:Q\oplus \homm APL\to \homm AQL\oplus \homm A{\homm APL}L$$ is defined by $h(q,f)(q^\prime,f^\prime)=f(q^\prime)-f^\prime(q)$. This induces a homomorphism $$H:K_0(A)\to GW^2(A,L)$$ whose cokernel is the Witt group $W^2(A,L)$.

We denote by $\Ko AL$ the quotient of $GW^2(A,L)$ by the subgroup generated by $H(A)$. If $L=A$, we simply denote by $\K A$ the group $\Ko AA$.

Euler classes
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Let $P$ be a projective module of rank $2$. Then $P$ is endowed with an anti-symmetric form $\chi:P\to \homm AP{\det P}$ defined by $\chi(p)(q)=p\wedge q$. We denote by $e(P)$ the class of $(P,\chi)$ in $\Ko A{\det P}$. 