Author: Teimuraz Pirashvili

Affiliation: Razmadze Mathematical Institute

Summary:

Call a closed model category very strict if every object
is fibrant and cofibrant.

A Frobenius structure on an abelian category is a class
of its objects forming both an injective and a projective
structure in sense of Maranda.

A one-to-one correspondence between very strict model
category structures and Frobenius structures is constructed.

It is proved that the projective (resp. injective)
Eckmann-Hilton homotopy theory on the category of modules
over a ring gives rise to a very strict model category structure
if and only if the ring is left perfect and right coherent
(resp. left Noetherian).

This is the english translation of a paper
originally published in Georgian in

Bulletin of the Academy of Sciences of the Georgian SSR
vol. 124 (1986), No. 2, 205--208.

In almost twenty years after after this paper, much more in this
direction has been done, e. g. by M. Hovey.