Isomorphisms of discriminant algebras

For each integer [Formula: see text], we define a category whose objects are discriminant algebra functors in rank [Formula: see text], namely, choices of how to attach functorially to each rank-[Formula: see text] algebra a quadratic algebra with the same discriminant. We show that the discriminant algebra functors defined by Loos, Rost, and the present authors are all isomorphic in this category, and prove furthermore that in ranks [Formula: see text] discriminant algebra functors are unique up to unique isomorphism.

Stability of Asai local factors for GL(2)

Let [Formula: see text] be a non-archimedean local field of characteristic not equal to 2 and let [Formula: see text] be a quadratic algebra. We prove the stability of local factors attached to irreducible admissible (complex) representations of [Formula: see text] via the Rankin–Selberg method under highly ramified twists. This includes both the Asai as well as the Rankin–Selberg local factors attached to pairs. Our method relies on expressing the gamma factor as a Mellin transform using Bessel functions.

KOSZUL CALCULUS

We present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved foranyquadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.