|
This article is cited in 9 scientific papers (total 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.
Received: 07.09.2007 and 06.11.2007
Citation:
A. Yu. Pirkovskii, “Weak homological dimensions and biflat Köthe algebras”, Mat. Sb., 199:5 (2008), 45–80; Sb. Math., 199:5 (2008), 673–705
Linking options:
https://www.mathnet.ru/eng/sm3940https://doi.org/10.1070/SM2008v199n05ABEH003939 https://www.mathnet.ru/eng/sm/v199/i5/p45
|
Statistics & downloads: |
Abstract page: | 543 | Russian version PDF: | 229 | English version PDF: | 12 | References: | 67 | First page: | 4 |
|