Recognizing Dualizing Complexes

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves 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 follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra. Let A be a noetherian local commutative ring and let M be a complex of A-modules with homology HM non-zero and finitely generated and HiM = 0 for i < 0. Theorem 2.2 shows that M is a dualizing complex for A if and only if the trivial extension A ⋉M is a Gorenstein Differential Graded Algebra (DGA). Phrased as a slogan: DGAs can be used to recognize dualizing complexes. In corollary 2.3 this is used to show that A has a dualizing complex if and only if it is a quotient of a Gorenstein local DGA. The notion of Gorenstein DGA I shall use is the one from [5]; it is recalled in definition 1.3. But note that for the DGAs in theorem 2.2 and corollary 2.3, the condition of being Gorenstein can also be expressed by the familar equation diml ExtR(l, R) = 1, see remark 2.4. DGAs satisfying this equation were considered at length in [1]. A brief introduction to the theory of DGAs is in [5]. 1. Definitions When A is a noetherian commutative ring, D(A) denotes the derived category of complexes of A-modules, and D(A) denotes the full subcategory of complexes M such that HM is a finitely generated module over A. The following definition is due to [6, def., p. 258]. Definition 1.1 (Dualizing complexes). Let A be a noetherian local commutative ring. The complex D in D(A) is called a dualizing complex for A if the canonical morphism A −→ RHomA(D,D) 2000 Mathematics Subject Classification. 13D25, 16E45.