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 $\mathcal{L}O(P)$ be the set of all pairs $(I,\omega )$, where $I$ is an ideal of $A$ and $\omega :P\to I/I^2$ is a surjective map. The homotopy relations on $\mathcal{L}O(P)$, induced by $\mathcal{L}O(P[T])$, leads to a set ${\pi }_0\left(\mathcal{L}O(P)\right)$ of equivalence classes in $\mathcal{L}O(P)$. There are two distinguished elements ${\mathbf{e}}_0,{\mathbf{e}}_1\in {\pi }_0\left(\mathcal{L}O(P)\right)$, respectively, the images of $(0,0)$ and $(A,0)$. Define the obstruction class $e(P)={\mathbf{e}}_0\in {\pi }_0\left(\mathcal{L}O(P)\right)$. The following results are under suitable smoothness or regularity hypotheses. When $2n\ge d+3$, we prove $e(P)={\mathbf{e}}_1\iff P\cong Q\oplus A$. We prove, if $2n\ge d+2$, then ${\pi }_0\left(\mathcal{L}O(P)\right)$ 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 $2n\ge d+3$, then there is natural isomorphism $E(P)\to {\pi }_0\left(\mathcal{L}O(P)\right)$ of groups.

Simplified and naturalized the proof of additive structure in section 6. In fact, in the earlier version, the proof went to an unintended track