Weak homological dimensions and biflat Köthe algebras

The homological properties of metrizable Köthe algebras λ(P) are studied. A criterion for an algebra A = λ(P) to be biflat in terms of the Köthe set P is obtained, which implies, in particular, that for such algebras the properties of being biprojective, biflat, and flat on the left are equivalent to the surjectivity of the multiplication operator A ⊗̂ A → A. The weak homological dimensions (the weak global dimension w. dg and the weak bidimension w. db) of biflat Köthe algebras are calculated. Namely, it is shown that the conditions w. dbλ(P) ≤ 1 and w. dgλ(P) ≤ 1 are equivalent to the nuclearity of λ(P); and if λ(P) is non-nuclear, then w. dg λ(P) = w. db λ(P) = 2. It is established that the nuclearity of a biflat Köthe algebra λ(P), under certain additional conditions on the Köthe set P, implies the stronger estimate db λ(P) ≤ 1, where db is the (projective) bidimension. On the other hand, an example is constructed of a nuclear biflat Köthe algebra λ(P) such that db λ(P) = 2 (while w.db λ(P) = 1). Finally, it is shown that many biflat Köthe algebras, while not being amenable, have trivial Hochschild homology groups in positive degrees (with arbitrary coefficients). Bibliography: 37 titles. © 2008 RAS(DoM) and LMS.