THE SHORT EXACT SEQUENCE IN DEFINABLE GALOIS COHOMOLOGY

In [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let M be an atomic and strongly $\omega $ -homogeneous structure over a set of parameters A. Let B be a normal extension of A in M. We show that a short exact sequence of automorphism groups $1 \to \operatorname {\mathrm {Aut}}(M/B) \to \operatorname {\mathrm {Aut}}(M/A) \to \operatorname {\mathrm {Aut}}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. We also discuss compatibilities with [3]. Our result complements the long exact sequence in definable Galois cohomology developed in [4].