Summary: ON THE GALOIS STRUCTURE OF EQUIVARIANT LINE BUNDLES
By A. AGBOOLA and D. BURNS
Abstract. Let k be a finite field, and let X be a smooth, projective curve over k with structure sheaf
O. Let G be a finite group, and write Cl (O[G]) for the reduced Grothendieck group of the category
of O[G]-vector bundles. In this paper we describe explicitly the subgroup of Cl (O[G]) which is
generated by the classes arising from G-stable invertible sheaves on tame Galois covers of X which
have Galois group G.
Introduction. Let k be a finite field of characteristic p, and let G be a finite
abelian group. Suppose that f: Y ,! X is a tamely ramified Galois covering
of smooth projective curves over k, with Galois group G. (We shall refer to
such coverings as "tame G-covers" of X). In this paper we study the structure of
G-stable line bundles on such curves Y.
In order to be more precise we let Cl (OX[G]), respectively Cl (k[G]), denote
the reduced Grothendieck group of OX[G]-vector bundles, respectively of finitely
generated k[G]-modules which are cohomologically trivial for G. Suppose that A
is a G-stable line bundle on Y. Then fA is an OX[G]-vector bundle, and it gives
rise to a class ( fA) 2 Cl (OX[G]). We shall say that an element of Cl (OX[G])
is realizable if it may be obtained in this manner for some choice of Y and A.
We shall combine techniques of [C], [M] and [Bu] to obtain an explicit