zbMATH — the first resource for mathematics

The radical of the homotopy Lie algebra. (English) Zbl 0654.55011
This paper represents a refinement and extension of work by various subsets of the authors on the structure of the respective homotopy Lie algebras of spaces and local rings. While the graded Lie algebra \(\pi_*(\Omega X)\otimes {\mathbb{Q}}\) is a familiar object of study, it is only in recent times that an analogous structure \(\pi\) *(A) has been associated to a noetherian local ring A. Indeed, the parallels between rational homotopy theory and local algebra have become extensive [see L. Avramov and S. Halperin, Lect. Notes Math. 1183, 1-27 (1986; Zbl 0588.13010)]. The present work focuses on the radical R \((=\) sum of all solvable ideals) of the homotopy Lie algebras above. The main result states that, under the hypothesis of finite L.S. category m for the minimal models of a space or (the Koszul complex of) a local ring, the radical R is finite dimensional and \(\dim (R_{even})\leq m\). (Of course, for a local ring, the category hypothesis has a completely ring theoretic translation via the homology of the Koszul complex). The proof relates the L.S. category of a minimal model to standard invariants of the associated homotopy Lie algebra: global dimension and depth. In particular, it is shown that category is an upper bound for depth. The conclusion of the main result is then obtained by proving the general theorem that the radical of any graded Lie algebra of finite depth has the enunciated properties.
Reviewer: J.Oprea

55P62 Rational homotopy theory
17B70 Graded Lie (super)algebras
13H99 Local rings and semilocal rings
Full Text: DOI