This article is cited in
9 papers
Weak homological dimensions and biflat Köthe algebras
A. Yu. Pirkovskii Peoples Friendship University of Russia
Abstract:
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.
UDC:
517.98.2
MSC: Primary
46M18; Secondary
46H25,
46A45,
18G20 Received: 07.09.2007 and 06.11.2007
DOI:
10.4213/sm3940
Fulltext:
PDF file (757 kB)
