Stable Projective Homotopy Theory of Modules, Tails, and Koszul Duality

Abstract

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.

Key Words:

• Koszul duality
• Stable homotopy theory of modules
• Tails
• Totally linear complex
• Weakly Koszul

2000 Mathematics Subject Classification:

• Primary 16S37
• Secondary 16E05, 18E30, 18E35

ACKNOWLEDGMENTS

The authors thank Northeastern University in Boston, MA, USA, and the program PAPIIT from Universidad Nacional Autonoma de Mexico for funding this research project.

This article was conceived and the main results were obtained in July 2003 when the second author was visiting the Institute of Mathematics, UNAM, Morelia. The authors thank the Institute for its hospitality and support. The results of this article were announced at the Joint Meeting of the American Mathematical Society and the Mathematical Society of Mexico in Houston in May 2004 and at the Oberwolfach Meeting on Representation Theory of Finite-Dimensional Algebras, [9], in February 2005.

Notes

The projective homotopy is in the sense of Eckmann–Hilton. The stable homotopy is meant to be the stabilization of the homotopy by the loop space, again in the sense of Eckmann–Hilton, functor.

Using the fact that the exterior algebra is self-injective, one immediately recovers an equivalence between the original categories by composing our duality and the self-duality with respect to the base field on the stable category of finite modules over the exterior algebra.

All of our complexes are homological.

To unambiguously refer to the dexterity of Γ-modules, we adhere to the convention that the composition fg of two elements in Γ is given by first performing a chain map in the class of g, followed by a chain map in the class of f.

The inverse of this map is just restriction to Λ₁.

The symbol Ω here stands for a syzygy module in a minimal projective resolution. See Lemma 23.

Thus if F inverts φ, then so does any other morphism between the same pairs.

Because of our lame notation, caution should be exercised with such representatives as the same 𝒜-morphism can “represent” many morphisms in 𝒜[φ⁻¹]; for example, f _n represents a morphism φ ^m M → φ ^m N for each integer m.

Communicated by D. Zacharia.

To Kent Fuller, a friend and a teacher.