The homology of the Temperley-Lieb algebras

We study the homology and cohomology of the Temperley-Lieb algebra TLn.a/, interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a D vCv 1 for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree n- 2. To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even. Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of "planar injective words" that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TLn(a) is not flat over TLm(a) for m < n, so that Shapiro's lemma is unavailable. We resolve this difficulty by constructing what we call "inductive resolutions" of the relevant modules. Vanishing results for the homology and cohomology of Temperley-Lieb algebras can also be obtained from the existence of the Jones-Wenzl projector. Our own vanishing results are in general far stronger than these, but in a restricted case we are able to obtain additional vanishing results via the existence of the Jones-Wenzl projector. We believe that these results, together with the second author's work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.