1,034 research outputs found
Relative cyclic homology of square zero extensions
Let k be a characteristic zero field, C a k-algebra and M a square zero two
sided ideal of C. We obtain a new mixed complex, simpler that the canonical
one, giving the Hochschild and cyclic homologies of C relative to M. This
complex resembles the canonical reduced mixed complex of an augmented algebra.
We begin the study of our complex showing that it has a harmonic decomposition
like to the one considered by Cuntz and Quillen for the normalized mixed
complex of an algebra. We also give new proofs of two theorems of Goodwillie,
obtaining an improvement of one of them.Comment: 24 pages. Definitive version, to appear in Crelle Journa
Hochschild cohomology of Frobenius algebras
Let k be a field and let A be a Frobenius algebra over k. Assume that the
Nakayama automorphism of A associated to a Frobenius homomorphism of A has
finite order m, and k has a m-th primitive root of unity. Then, A has a natural
Z/mZ-gradation. Consider the decomposition of the Hochschild cohomology HH*(A),
of A with coefficients in A, induced by this gradation. We prove that just the
0-degree component of HH*(A) is non trivial. Moreover, we prove that if A is a
strongly Z/mZ-graded algebra, then Z/mZ acts on the Hochschild cohomology
HH*(A_0), of the 0-degree component of A, and HH*(A) is the set of invariants
of this action.Comment: 10 page
Hochschild (co)homology of Hopf crossed products
Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M
an E-bimodule. We obtain a complex simpler than the canonical one, giving the
Hochschild homology of E with coefficients in M. This complex is eqquiped with
a natural filtration. We prove that the associated spectral sequence coincides
with that obtained by either, the Hochschild-Serre direct method or the
Cartan-Leray-Grothendieck method. We also get similar results for the
cohomology.Comment: 26 page
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