For every positively graded
algebra A, we show that its categories of linear complexes of projectives and almost
injectives (see definition below) are both naturally equivalent to the category of
graded modules over the quadratic dual algebra A!. In case A = Λ is a graded factor
of a path algebra with Yoneda algebra Γ, we show that the category ℒcΓ of linear
complexes of finitely generated right projectives over Γ is dual to the category of
locally finite graded left modules over the quadratic algebra Λ associated to Λ.
When Λ is Koszul and Γ is graded right coherent, we also prove that the
suspended category grΛ has a (triangulated) stabilization S(grΛ) which is
triangle-equivalent to the bounded derived category of the ‘category of tails’
fpgrΓ∕LΓ.
