###
Algebras with scalar involution revisited

We study algebras with scalar involution and, more generally, conic
algebras (formerly known as quadratic algebras) over an arbitrary base ring k
on a fixed finitely generated and projective k-module X with base point
1_{X}. By variation of the base ring, these algebras define schemes
whose structure is described. They also admit natural group actions under
which they are trivial torsors. We determine the quotients by these group
actions. This requires a new invariant of conic algebras, an alternating
trilinear map on M = X/k . 1_{X} with values in the second symmetric
power of M. An important tool is the coordinatization of conic algebras in
terms of a linear form, a cross product and a bilinear form on M, all depending
on a choice of unital linear form on X, which replaces the usual description in
terms of a vector algebra and a bilinear form in case 2 is a unit in the base
ring.
(Revised version 24 March 2011, to appear in J. Pure Appl. Algebra)

O. Loos < ottmar.loos@uibk.ac.at >