1,034 research outputs found

    Relative cyclic homology of square zero extensions

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    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

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    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

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    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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