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Let   $K$   be  the fraction field of a two-dimensional henselian, excellent, and equi-characteristic local domain. We prove a local–global principle for Galois cohomology with certain finite coefficients over   $K$  . We use classical machinery from étale cohomology theory, drawing upon an idea in Saito’s work on two-dimensional local class field theory. This approach works equally well over the function field of a curve over an equi-characteristic henselian discrete valuation field, thereby giving a different proof of (a slightly generalized version of) a recent result of Harbater, Hartmann, and Krashen. We also present two applications. One is the Hasse principle for torsors under quasi-split semisimple simply connected groups without   $E_8$   factor. The other gives an explicit upper bound for the Pythagoras number of a Laurent series field in three variables. This bound is sharper than earlier estimates.

A Cohomological Hasse Principle Over Two-dimensional Local Rings