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.
