 
Summary: ON THE GALOIS STRUCTURE OF EQUIVARIANT LINE BUNDLES
ON CURVES
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 Gstable 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 Gcovers" of X). In this paper we study the structure of
Gstable 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 Gstable 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
