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Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. It is proved that M is a dualizing complex for A if and only if the trivial extension A ⋉ M is a Gorenstein differential graded algebra. As a corollary, A has a dualizing complex if and only if it is a quotient of a Gorenstein local differential graded algebra.
Existence of proper Gorenstein projective resolutions and Tate cohomology is proved
over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.
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