Koszul differential graded algebras and BGG correspondence

In this paper, we take a different point of view. Let k be a field. A connected DG algebra over k is a positively graded k-algebra A = ⊕ n≥0 A n with A 0 = k such that there is a differential d : A → A of degree 1 which is also a graded derivation. A connected DG algebra A is said to be a Koszul DG algebra if the minimal semifree resolution of the trivial DG module A k has a semifree basis consisting of homogeneous elements of degree zero (Definition 2.1). Our definition of Koszul DG algebra is a natural generalization of the usual Koszul algebra. As we will see in Section 2, a connected graded algebra regarded as a DG algebra with zero differential is a Koszul DG algebra if and only if it is a Koszul algebra in the usual sense. Examples of Koszul DG algebras can be found in various fields. For example, let M be a connected n-dimensional C ∞ manifold, and let (A * (M ) = a minimal model A ( [KM]) or Sullivan model ( [FHT2]), which is certainly a connected DG algebra. If the manifold M has some further properties (e.g., M = T n the ndimensional torus), then the de Rham cohomology algebra H(A * (M )) is a Koszul algebra. Hence the cohomology algebra of its minimal model (or Sullivan model) A is Koszul as A is quasi-isomorphic to A * (M ). Then A is a Koszul DG algebra by Proposition 2.3. More examples of Koszul DG algebra will be given in Section 2. In fact, we will see that any Koszul algebra can be viewed as the cohomology algebra of some Koszul DG algebra.

Bernstein-Gelfand-Gelfand in [BGG] established an equivalence between the stable category of finitely generated graded modules over the exterior algebra V with V = kx 0 ⊕ kx 1 ⊕ • • • ⊕ kx n , and the bounded derived category of coherent sheaves on the projective space P n . This equivalence is now called the BGG correspondence. BGG correspondence has been generalized to noncommutative projective geometry by several authors. Let R be a (noncommutative) Koszul algebra. If R is AS-regular, Jørgensen proved in [Jo] that there is an equivalence between the stable category over the graded Frobenius algebra E(R) = Ext * R (k, k) and the derived category of the noncommutative analogue QGr(R) of the quasi-coherent sheaves over R; Martínez Villa-Saorín proved in [MS] that the stable category of the finite dimensional modules over E(R) is equivalent to the bounded derived category of the noncommutative analogue qgrR of the coherent sheaves over R. Mori in [Mo] proved a similar version under a more general condition. One of our purposes in this paper is to establish a DG version of the BGG correspondence. In some special case, the DG version of the BGG correspondence coincides with the classical one as established in [BGG] and [MS].

The paper is organized as follows.

In Section 1, we give some preliminaries and fix some notations for the paper.

In Section 2, we first propose a definition for Koszul DG algebras (Definition 2.1), then give some examples and discuss some basic properties of Koszul DG algebras. For any connected DG algebra A, we prove that if the cohomology algebra H(A) is Koszul in the usual sense, then A is a Koszul DG algebra (Proposition 2.3). The converse is not true in general.

In Section 3, we discuss the structure of the Ext-algebras of Koszul DG algebras. For any Koszul DG algebra A, we prove that the Ext-algebra E = Ext * A ( A k, A k) of A is an augmented, filtered algebra. Moreover, if H(A) is a Koszul algebra, then the associated graded algebra gr(E) is isomorphic to the dual Koszul algebra (H(A)) ! (Theorem 3.3). If further, A k is compact, then E is a finite dimensional local algebra; when H(A) is Koszul, the filtration on E is exactly the Jacobson radical filtration (Theorem 3.5). Using bar and cobar constructions, we prove the following version of the Koszul duality on the Ext-algebras (Theorem 3.8):

Theorem [Koszul Duality on Ext-algebra]. Let A be a Koszul DG algebra and E be its Ext-algebra.

As a corollary, we show that the Ext-algebra of a Koszul DG algebra A with A k compact is strongly quasi-Koszul ( [GM]) if and only if its cohomology algebra H(A) is a Koszul algebra.

In Section 4, by using Lefèvre-Hasegawa’s theorem in [Le,Ch.2] (see Theorem 4.1), we establish a DG version of Koszul equivalence and duality (Theorems 4.4 and 4.7).

Theorem [Koszul equivalence and duality]. Let A be a Koszul DG algebra and E be its Ext-algebra. Suppose A k is compact. Then there is an equivalence of triangulated categories between D + (E) and D + dg (A op ); and there is a duality of triangulated categories between D b (mod-E op ) and D c (A op ).

Here D + (E) is the th

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