Notions of category in differential algebra. (English) Zbl 0656.55003

Halperin, S.; Lemaire, J.-M.

Algebraic topology, rational homotopy, Proc. Conf., Louvain-la- Neuve/Belg. 1986, Lect. Notes Math. 1318, 138-154 (1988).

[For the entire collection see Zbl 0652.00011.]
In this paper, the authors introduce new invariants, M-cat(A,d), A- cat(A,d), defined for associative differential algebras over an arbitrary field \({\mathbb{K}}\). (\({\mathbb{K}}^ D.\)G.A. for short.) Applied to the D.G.A., \(C^*(S;{\mathbb{K}})\) of singular cochains on a simply connected space S they obtain A-cat(S;\({\mathbb{K}})\) and M-cat(S;\({\mathbb{K}})\) and prove that for all \({\mathbb{K}}\), M-cat(S;\({\mathbb{K}})\leq A-cat(S;{\mathbb{K}})\leq cat S\) (cat S means the Lusternik-Schnirelman category of the space S). Applied to the minimal model of Sullivan \(\Lambda\) X,d of the space S, they prove: \(A- cat(\Lambda X,d)\leq cat(S_ 0)\) where \(S_ 0\) denotes the space S localised at 0. In a short appendix the authors define free models of a D.G.A over any field \({\mathbb{K}}\). This notion is the main tool of the study of M-cat and A-cat. Aside from its topological interpretation, these invariants have had applications both in the topological context and the study of local rings.

Reviewer: J.C.Thomas

MSC:

55M30	Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55P62	Rational homotopy theory
13H10	Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
16W50	Graded rings and modules (associative rings and algebras)

Keywords:

associative differential algebras; singular cochains; Lusternik- Schnirelman category; minimal model; free models; local rings

Citations:

Zbl 0652.00011