Stable homology isomorphisms for the partition and Jones annular algebras (Q6638164)

Let us recall that ``a family \((A_n)\) of augmented R-algebras exhibits homological stability'' means that \N\[\N\mathrm{Tor}_*^{A_n}(1,1)\mapsto \mathrm{Tor}_*^{A_{n+1}}(1,1)\N\]\Nare isomorphisms. Stability results have been proved by several authors, for Temperley-Lieb algebras, Brauer algebras, partition algebras, Iwahori-Hecke algebras of types A and B.\N\NIn this paper, the author shows that a way to deal with such questions is to fix \(n\) and to resolve an well-known algebra \(B_n\) over \(A_n\). More precisely, the author proves that the homology of the Jones annular algebras is isomorphic to that of the cyclic groups below a line of gradient 1/2. He also proves that the homology of the partition algebras and that of the symmetric groups below a line of gradient 1 are isomorphic. As a byproduct, the author proves the usual odd-strand and invertible parameter results for the Jones annular algebras.