Hebrew University Amitsur Algebra Seminar ---------------------------------- Time: Thursday, Apr 25, 12:00-13:15 Place: math 209 Title: General Bilinear Forms and Applications Speaker: Uriya First (HebrewU) Abstract: We introduce the new notion of general bilinear forms and present several of its applications. In particular, we give new proofs to theorems of Saltman and Osborn. In more detail: Let F be a field, and let V be a finite dimensional vector space. There is a well-known one-to-one correspondence between the regular (=nondegenerate) bilinear forms on V, considered up to multiplication by an element of F*, and the set of F-anti-automorphisms of End(V). This result is the foundation of the widely studied connection between hermitian forms and involutions. We present the new notion of general bilinear forms, and use it to prove a generalization of the correspondence to modules over arbitrary rings: If R is a ring (not necessarily commutative, possibly without involution) and M is a finitely generated projective R-module, then there is a one-to-one correspondence between the regular general bilinear forms over M, considered up to similarity, and the set of anti-automorphisms of End_R(M). In particular, we obtain a new proof of the classical correspondence (which does not use the Skolem-Noether theorem). The generalized correspondence is used to obtain new short proofs of: (1) A theorem of Saltman: An Azumaya algebra is Brauer equivalent to its opposite iff it is Brauer equivanet to an algebra with an involution of the first kind. (The notions will be defined and explained in the lecture.) (2) A result of Osborn classifying rings with involution in which all elements invariant under the involution are invertible or nilpotet. Both results are in fact slightly generalized. Other applications of the correspondence will be presented if time permits. You are cordially invited! <http://math.huji.ac.il/amitsur.html> --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Konstantin Golubev <kgolubev@gmail.com>