The Monoid Structure on Homotopy Obstructions

Abstract

Let $A$ be a commutative noetherian ring, containing a field $k$, with $1/2\in k$, $\dim A=d$, and let $P$ be a projective $A$-module or $rank(P)=n$. In continuation of \cite{MM}, we study Homotopy obstructions for $P$ to split off a free direct summand. Let ${\mathcal LO}(P)$ be the set of all pairs $(I, \omega)$, where $I$ is an ideal of $A$ and $\omega: P\rightarrow I/I^2$ is a surjective map. The homotopy relations on ${\mathcal LO}(P)$, induced by ${\mathcal LO}(P[T])$, leads to a set $\pi_0\left({\mathcal LO}(P)\right)$ of equivalence classes in ${\mathcal LO}(P)$. There are two distinguished elements ${\bf e}_0, {\bf e}_1\in \pi_0\left({\mathcal LO}(P)\right)$, respectively, the images of $(0, 0)$ and $(A, 0)$. Define the obstruction class $e(P)={\bf e}_0\in \pi_0\left({\mathcal LO}(P)\right)$. The following results are under suitable smoothness or regularity hypotheses. When $2n\geq d+3$, we prove $e(P)={\bf e}_1 \Leftrightarrow P\cong Q\oplus A$. We prove, if $2n\geq d+2$, then $\pi_0\left({\mathcal LO}(P)\right)$ has a natural structure of a monoid, which is a group if $P\cong Q\oplus A$. Further, we give a definition of a Euler class group $E(P)$. Under suitable smoothness hypotheses, we prove, if $P\cong Q\oplus A$ and $2n\geq d+3$, then there is natural isomorphism $E(P) \rightarrow \pi_0\left({\mathcal LO}(P)\right)$ of groups.