  The aim of this paper is to study the theory of cohomology annihilators over
commutative Gorenstein rings. We adopt a triangulated category point of view
and study the annihilation of stable category of maximal Cohen-Macaulay
modules. We prove that in dimension one the cohomology annihilator ideal and
the conductor ideal coincide under mild assumptions. We present a condition on
a ring homomorphism between Gorenstein rings which allows us to carry the
cohomology annihilator of the domain to that of the codomain. As an
application, we generalize the Milnor-Jung formula for algebraic curves to
their double branched covers. We also show that the cohomology annihilator of a
Gorenstein local ring is contained in the cohomology annihilator of its
Henselization and in dimension one the cohomology annihilator of its
completion. Finally, we investigate a relation between the cohomology
annihilator of a Gorenstein ring and stable annihilators of its noncommutative
resolutions.
