Hochschild Homology - Definition of Hochschild Homology of Algebras - Hochschild Complex

Hochschild Complex

Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write A⊗n for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by

with boundary operator di defined by

Here ai is in A for all 1 ≤ i ≤ n and m ∈ M. If we let

then b ° b = 0, so (Cn(A,M), b) is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M.

Read more about this topic:  Hochschild Homology, Definition of Hochschild Homology of Algebras

Famous quotes containing the words hochschild and/or complex:

    “As long as the “woman’s work” that some men do is socially devalued, as long as it is defined as woman’s work, as long as it’s tacked onto a “regular” work day, men who share it are likely to develop the same jagged mouth and frazzled hair as the coffee-mug mom. The image of the new man is like the image of the supermom: it obscures the strain.”
    —Arlie Hochschild (20th century)

    “We must open our eyes and see that modern civilization has become so complex and the lives of civilized men so interwoven with the lives of other men in other countries as to make it impossible to be in this world and out of it.”
    —Franklin D. Roosevelt (1882–1945)