arXiv subjects

Nicole Lemire

Publications and source records attributed to Nicole Lemire.

22 records · Page 2Linked to original sources

arXiv2005

Cohomological dimension and Schreier's formula in Galois cohomology

Let p be a prime and F a field containing a primitive pth root of unity. Then for n in N, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is <=n if and only if the corestriction maps H^n(H,Fp) -> H^n(G,Fp) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_Fp H^n(H,Fp) of Schreier's formula for dim_Fp H^1(H,Fp).

math.NT↗

arXiv2005

When is Galois cohomology free or trivial?

Let p be a prime and F a field containing a primitive pth root of unity. Let E/F be a cyclic extension of degree p and G_E < G_F the associated absolute Galois groups. We determine precise conditions for the cohomology group H^n(E)=H^n(G_E,Fp) to be free or trivial as an Fp[Gal(E/F)]-module. We examine when these properties for H^n(E) are inherited by H^k(E), k>n, and, by analogy with cohomological dimension, we introduce notions of cohomological freeness and cohomological triviality. We give examples of H^n(E) free or trivial for each n in N with prescribed cohomological dimension.

math.NT↗

arXiv2002

Equivariant Euler characteristics of discriminants of reflection groups

Let G be a finite, complex reflection group and f its discriminant polynomial. The fibers of f admit commuting actions of G and a cyclic group. The virtual $G\times C_m$ character given by the Euler characteristic of the fiber is a refinement of the zeta function of the geometric monodromy, calculated in a paper of Denef and Loeser. We compute the virtual character explicitly, in terms of the poset of normalizers of centralizers of regular elements of G, and of the subspace arrangement given by proper eigenspaces of elements of G. As a consequence, we compute orbifold Euler characteristics and find some new "case-free" information about the discriminant.

math.GR↗

arXiv1999

On certain lattices associated with generic division algebras

Let S_n denote the symmetric group on n letters. We consider the S_n-root lattice A_{n-1} = {(z1,...,zn) in Z^n | z1+...+zn = 0}, where S_n acts on Z^n by permuting the coordinates, and its tensor, symmetric, and exterior squares. For odd values of n, we show that the tensor square is equivalent, in the sense of Colliot-Thelene and Sansuc, to the exterior square. Consequently, the rationality problem for generic division algebras, for odd values of n, amounts to proving stable rationality of the multiplicative S_n-invariant field of the exterior square of A_{n-1}. Furthermore, confirming a conjecture of Le Bruyn, we show that n=2 and n=3 are the only cases where the tensor square of A_{n-1} is equivalent to a permutation S_n-lattice. In the course of the proof of this result, we construct subgroups H of S_n, for all n that are not prime, so that the algebra of multiplicative H-invariants of A_{n-1} has a non-trivial Picard group.

math.RA↗