4 research outputs found
The weak commutativity construction for Lie algebras
We study the analogue of Sidki's weak commutativity construction, defined
originally for groups, in the category of Lie algebras. This is the quotient
of the Lie algebra freely generated by two isomorphic
copies and of a fixed Lie algebra by the
ideal generated by the brackets , for all . We exhibit an
abelian ideal of whose associated quotient is a subdirect
sum in and we give
conditions for this ideal to be finite dimensional. We show that
has a subquotient that is isomorphic to the Schur
multiplier of . We prove that is finitely
presentable or of homological type if and only if has the
same property, but is not of type if
is a non-abelian free Lie algebra.Comment: Incorporated referee's suggestions, results unchange
Galois subcovers of the Hermitian curve in characteristic with respect to subgroups of order
A (projective, geometrically irreducible, non-singular) curve
defined over a finite field is maximal if the number
of its -rational points attains the Hasse-Weil
upper bound, that is where is the
genus of . An important question, also motivated by applications
to algebraic-geometry codes, is to find explicit equations for maximal curves.
For a few curves which are Galois covered of the Hermitian curve, this has been
done so far ad hoc, in particular in the cases where the Galois group has prime
order. In this paper we obtain explicit equations of all Galois covers of the
Hermitian curve with Galois group of order where is the
characteristic of . Doing so we also determine the
-isomorphism classes of such curves and describe their full
-automorphism groups
Ramification filtration by moduli in higher-dimensional global class field theory
In their approach to higher-dimensional global class field theory, Kato and Saito define the class group of a proper arithmetic scheme \bar{X} as an inverse limit C_{KS}(\bar{X}) = \varprojlim_{\mathcal{I}} C_{\mathcal{I}}(\bar{X}) of certain Nisnevich cohomology groups C_{\mathcal{I}}(\bar{X}) taken over all non-zero coherent ideal sheaves \mathcal{I} of \mathcal{O}_{\bar{X}}.
The ideal sheaves \mathcal{I} should be regarded as higher-dimensional analogues of the classical moduli \mathfrak{m} on a global field K, which induce a filtration of the idele class group C_K by the ray class groups C_K/C_K^{\mathfrak{m}}.
In higher dimensions however, it is not clear how the induced filtration of the abelian fundamental group can be interpreted in terms of ramification.
In view of Wiesend's class field theory, we define an easier notion of moduli in higher dimensions only involving curves on the scheme. We then show that both notions agree for moduli that correspond to tame ramification