Link homology and Frobenius extensions

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant. AMS Subject Classification: 57M27 Frobenius systems. Suppose ι : R −→ A is an inclusion of commutative rings, and ι(1) = 1. The restriction functor Res : A−mod −→ R−mod has left and right adjoint functors: the induction functor Ind(M) = A⊗R M and the coinduction functor CoInd(M) = HomR(A,M). Following Kadison [Ka] and others, we say that ι is a Frobenius extension if the induction and coinduction functors are isomorphic. Equivalently, ι is Frobenius if the restriction functor has a biadjoint (two-sided adjoint). We note that Kadison [Ka] treats the more general case of not necessarily commutative R and A. In this paper we consider only commutative rings. The following proposition is well-known (or see [Ka, Section 4]). Proposition 1 ι is a Frobenius extension iff there exists an A-bimodule map ∆ : A −→ A ⊗R A and an R-module map ǫ : A −→ R such that ∆ is coassociative and cocommutative, and (ǫ⊗ Id)∆ = Id. A Frobenius extension, together with a choice of ǫ and ∆, will be denoted F = (R,A, ǫ,∆) and called a Frobenius system (as in [Ka]). A Frobenius system defines a 2-dimensional TQFT, a tensor functor from oriented (1 + 1)-cobordisms to R-modules, by assigning R to the empty 1manifold, A to the circle, A⊗RA to the disjoint union of two circles, etc. The