Sur l'homologie des groupes unitaires à coefficients polynomiaux

Abstract

AbstractLetAbe a ring with anti-involution andFa nice functor (tensor or symmetric power, for example) from finitely-generated projectiveA-modules to abelian groups. We show that the homology of the hyperbolic unitary groupsUn,n(A) with coefficients inF(A2n) can be expressed stably (i.e. after taking the colimit overn) by the homology of these groups with untwisted coefficients and functor homology groups that we can compute in suitable cases (for example, whenAis a field of characteristic 0 or a ring without ℤ-torsion andFa tensor power). This extends the result whereAis a finite field, which was dealt with previously by C. Vespa and the author (Ann. Sci. ENS, 2010).The proof begins by relating, without any assumption onF, our homology groups to the homology of a category of hermitian spaces with coefficients twisted byF. Then, whenFis polynomial, we establish — following a method due to Scorichenko — an isomorphism between this homology and the homology of another category of (possibly degenerate) hermitian spaces, which is computable (in good cases) by standard methods of homological algebra in functor categories (using adjunctions, Künneth formula…). We give some examples.Finally, we deal with the analogous problem for non-hyperbolic unitary groups in some special cases, for example euclidean orthogonal groupsOn(A) (the ringAbeing here commutative). The isomorphism between functor homology and group homology with twisted coefficients does not hold in full generality; nevertheless we succeed to get it whenAis a field or, for example, a subring of ℚ containing ℤ[1/2]. The method, which is similar to that in the previous case, uses a general result of symmetrisation in functor homology proved at the beginning of the article.