SPLITTING FAMILIES IN GALOIS COHOMOLOGY

Let k be a field, with absolute Galois group Gamma. Let A/k be a finite etale group scheme of multiplicative type, i.e., a finite discrete Gamma-module. Let n >= 2 be an integer, and let x is an element of H-n (k, A) be a cohomology class. We show that there exists a countable set I, and a family (X-i)(i is an element of I) of (smooth, geometrically integral) k-varieties, such that the following holds: for any field extension l/k, the restriction of x vanishes in H-n (l, A) if and only if (at least) one of the X-i has an l-point. In addition, we show that the X-i can be made into an ind-variety. In the case n = 2, we note that one variety is enough.