  This paper studies the homology and cohomology of the Temperley-Lieb algebra
TL_n(a), interpreted as appropriate Tor and Ext groups. Our main result applies
under the common assumption that a=v+v^{-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: TL_n(a) is not flat over TL_m(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 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.
