Algebras of homological dimension 1

V. E. Govorov

Sbornik. Mathematics, Tome 44 (1983) no. 1, pp. 97-107

Résumé

Augmented algebras over a field of homological dimension 1 ($\operatorname{hd}R=1$) are studied. It is proved that if $\operatorname{hd}R=1$, then the associated graded algebra $E(R)$ is free. If the filtration of the algebra $R$ defined by the powers of the augmentation ideal is separated, then the following conditions are equivalent: 1) $\operatorname{hd}R=1$, 2) $E(R)$ is free, 3) $\operatorname{w.g.dim}R=1$. Some properties of groups of homological dimension 1 are presented. It is proved that, in the category of graded algebras, the functor that produces homology groups carries a direct sum into a free product and a free product into a direct sum. Bibliography: 6 titles.

Bibliographie

[1] Kartan L., Eilenberg S., Gomologicheskaya algebra, IL, M., 1960

[2] Maklein S., Gomologiya, Mir, M., 1966

[3] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[4] Govorov V. E., “Razmernost i kratnost graduirovannykh algebr”, Sib. matem. zh., XIV:6 (1973), 1200–1206 | MR

[5] Stambach V., Homology in Group Theory, Lecture Notes in Math. 359, Springer-Verlag, Berlin a. o, 1973 | MR

[6] Sjödin G., “On the filtered modules and their associated graded modules”, Math. Scand., 33 (1973), 229–249 | MR