Forms of higher degree permitting composition (Q717824)

Results of R. D. Schafer on \(k\)-algebras \(A\) with nondegenerate composition forms of degree \(d\) are generalized to \(R\)-algebras where \(R\) is a commutative associative ring with \(d!\in R^{\times}\). For \(d=3\) and \(d=4\) the possible ranks of \(A\) and further properties of \(A\) are stated. In \S3 these results are extended to locally ringed spaces \(X\) with structure sheaf \({\mathcal O}_X\) such that \(d!\in H^0(X,{\mathcal O}^{\times}_X)\). For a curve \(X\) of genus \(0\) or \(1\) and \({\mathcal O}_X\)-algebras \({\mathcal A}\) permitting cubic composition \((d=3)\) many more details and examples are presented. The last \S4 contains examples of degenerate cubic forms permitting composition on the projective space \(X=\mathbb P^m\) over \(R\).