Let $F$ be a field of zero characteristic. All groups are taken modulo torsion.

Consider a residue map from the exterior algebra of the multiplicative group of the function field of the projective line to the sum over all points except $\infty$ of exterior algebras of the multiplicative groups of the closed points.This gives a complex, which is exact in the first and the last term.

$0 \longrightarrow \bigwedge F^* \longrightarrow\bigwedge F(t)^* \stackrel{\oplus \partial_P}{\longrightarrow}\bigoplus \bigwedge F(P)^*\longrightarrow 0.$

After factorization over an ideal, generated by elements $a \wedge (1-a) \wedge u_1 \wedge ... \wedge u_k,$ this complex becomes exact, by Milnor's theorem:

$0 \longrightarrow K^M(F^*) \longrightarrow K^M(F(t)^*) \stackrel{\oplus \partial_P}{\longrightarrow}\bigoplus K^M(F(P)^*)\longrightarrow 0.$

I am interested in the middle homology group of the complex, written above. Since residue map satisfies graduate Leibniz rule, middle homology group is an algebra. What other natural structures it carries? I have found for it a natural presentation by generators and relations, but only viewed as a vector space. I am also interested in understanding better the ring structure.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.