On a homology of algebras with unit
Volume 110 / 2014
Annales Polonici Mathematici 110 (2014), 189-208 MSC: Primary 55N35; Secondary 13D03. DOI: 10.4064/ap110-2-6
We present a very general construction of a chain complex for an arbitrary (even non-associative and non-commutative) algebra with unit and with any topology over a field with a suitable topology. We prove that for the algebra of smooth functions on a smooth manifold with the weak topology the homology vector spaces of this chain complex coincide with the classical singular homology groups of the manifold with real coefficients. We also show that for an associative and commutative algebra with unit endowed with the discrete topology this chain complex is dual to the de Rham complex.