The concept of Koszul differential graded (DG for short) algebra is introduced in [8].Let A be a Koszul DG algebra.If the Ext-algebra of A is finite-dimensional,i.e.,the trivial module A k is a compact object in the derived category of DG A-modules,then it is shown in [8] that A has many nice properties.However,if the Ext-algebra is infinitedimensional,little is known about A.As shown in [15] (see also Proposition 2.2),A k is not compact if H(A) is finite-dimensional.In this paper,it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality.A DG version of the BGG correspondence is deduced from the Koszul duality theorem.
The (singular) orthogonal graph O(2ν + δ,q) over a field with q elements and of characteristic 2 (where ν 1, and δ = 0,1 or 2) is introduced. When ν = 1, O(2 · 1,q), O(2 · 1 + 1,q) and O(2 · 1 + 2,q) are complete graphs with 2, q + 1 and q2 + 1 vertices, respectively. When ν 2, O(2ν + δ,q) is strongly regular and its parameters are computed. O(2ν + 1,q) is isomorphic to the symplectic graph Sp(2ν,q). The chromatic number of O(2ν + δ,q) except when δ = 0 and ν is odd is computed and the group of graph automorphisms of O(2ν + δ,q) is determined.
