From Leibniz Homology to Cyclic Homology

For an algebra R over a commutative ring k, a natural homomorphism _*: HL_*+1(R) HH_* (R) from Leibniz to Hochschild homology is constructed that is induced by an antisymmetrization map on the chain level. The map _* is surjective when R = gl(A), A an algebra over a characteristic zero field. If f: A B is an algebra homomorophism, the relative groups HL_* (gl(f)) are studied, where gl(f): gl(A) gl(B) is the induced map on matrices. Letting HC_* denote cyclic homology, if f is surjective with nilpotent kernel, there is a natural surjection HL_*+1(gl(f)) HC_* (f) in the characteristic zero setting.