Weak homological dimensions and biflat K\"othe algebras

A. Yu. Pirkovskii

Sbornik. Mathematics, Tome 199 (2008) no. 5, pp. 673-705

Résumé

The homological properties of metrizable Köthe algebras $\lambda(P)$ are studied. A criterion for an algebra $A=\lambda(P)$ 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 $A\mathbin{\widehat\otimes}A\to A$. The weak homological dimensions (the weak global dimension $\operatorname{w{.}dg}$ and the weak bidimension $\operatorname{w{.}db}$) of biflat Köthe algebras are calculated. Namely, it is shown that the conditions $\operatorname{w{.}db}\lambda(P)\le1$ and $\operatorname{w{.}dg}\lambda(P)\le1$ are equivalent to the nuclearity of $\lambda(P)$; and if $\lambda(P)$ is non-nuclear, then $\operatorname{w{.}dg}\lambda(P)=\operatorname{w{.}db}\lambda(P)=2$. It is established that the nuclearity of a biflat Köthe algebra $\lambda(P)$, under certain additional conditions on the Köthe set $P$, implies the stronger estimate $\operatorname{db}\lambda(P)\le1$, where $\operatorname{db}$ is the (projective) bidimension. On the other hand, an example is constructed of a nuclear biflat Köthe algebra $\lambda(P)$ such that $\operatorname{db}\lambda(P)=2$ (while $\operatorname{w{.}db}\lambda(P)=1$). 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.