Recent results on finiteness conditions in Jordan pairs

Abstract

This article collects a number of recent results related to various types of finiteness conditions in Jordan theory. With a few exceptions, the conditions are at least as strong as the descending chain condition on principal inner ideals; in particular, they imply regularity in the nondegenerate case. Thus finiteness conditions of Noetherian type will not be considered, nor conditions like finite generation, algebraicity, or polynomial identities.1991 Mathematics Subject Classification: 17C10, 17C27, 17C30, 17C55. Finite dimensionThis is the condition that the underlying module of a Jordan system J (algebra, triple system or pair) over a commutative ring of scalars k be a finite-dimensional vector space; in particular, that k be a field. From a purely algebraic point of view the condition is not very satisfactory because it is not tied to the Jordan structure. However, finite-dimensionality allows methods from algebraic geometry and algebraic group theory to be used; see, e.g., the books by Braun-Koecher [5] and Springer [41] and papers too numerous to mention, starting with the original paper by Jordan, von Neumann and Wigner.A slight weakening of finite-dimensionality but in the same spirit is the requirement that, for k an arbitrary commutative ring, J be finitely generated and projective as a Axmodule. Jordan systems of this type allow again the use of algebraicgeometric methods [20,19], but classical algebraic geometry has to be supplanted by scheme theory. There are the beginnings of a theory of separable Jordan systems over commutative rings [18], in analogy to the theory of Azumaya algebras, but a full theory remains to be worked out.The logical continuation of this train of thought is to globalize the theory, i.e. to replace the commutative base ring k, or rather its prime spectrum, by an arbitrary base scheme X and let J be a locally free Οχ-module of finite rank. H.P. Petersson [40] has recently started to develop a theory of composition algebras in this setup.

Properly algebraic and spectrum-finite ideals in Jordan systems

“…F = F/^B(F) is nondegenerate. Indeed, F contains no algebraic ideals 4= 0 and thus even has trivial nil radical [8, [4][5][6][7][8][9][10][11][12][13][14][15].…”

“…In the remainder of this section we prove an analogue of Amitsur's result for general Jordan pairs (2-9) and Jordan algebras (2-10) which is slightly more complicated. We begin with the Jordan pair version of Amitsur's Resolvent Trick [1, theorem 1], see also [15,[4][5][6][7][8][9][10][11][12] in the Jordan algebra case. For (x, y)eV let y) = k\S-p (x,y) be the resolvent set of (x, y).…”