Algebras with scalar involution revisited

Abstract

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 in detail. 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 k .