Homological bidimension of biprojective topological algebras and nuclearity

The paper computes the homological dimension of certain biprojective topological algebras, in particular Köthe sequence spaces. It is a well-known result of Khelemskiĭ that biprojective Banach algebras have homological bi-dimension at most 2. For Banach algebras this often can be combined with the result of Khelemskiĭ which often forbids the homological bidimension being 1; see the results of Khelemskiĭ and Selivanov. Thus we see in the Banach algebra case the bidimension is usually 2. For Fréchet algebras, we still have the estimate that the dimension is less than 2, but the dimension 2, is not forbidden. par The main result of the paper is to show that the Köthe sequence space lambda(P) has homological bidimension 1 if it is a nuclear, metrizable and smooth sequence space of infinite type. That these algebras are biprojective follows from the usual construction of a diagonal map rho(sum_i a_i e_i) = sum_i a_i e_iotimes e_i. The proof that the algebras have a global bidimension of 1, (and many other results in this paper) follows from a theorem quoted from another paper by the same author [J. Math. Sci. (New York) {bf 111} (2002), no.~2, 3476--3495; [msn] MR1847554 (2003h:46071) [/msn]].