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ON THE CYCLIC HOMOLOGY OF SUPERMATRICES
, 905
"... Abstract. The aim of this note is to show that the generalized supertrace, constructed in another paper of the author, inducing an isomorphism between the Hochschild homology of a superalgebra and that of the superalgebra of square supermatrices of a given type over A, induces, also, an isomorphism ..."
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Abstract. The aim of this note is to show that the generalized supertrace, constructed in another paper of the author, inducing an isomorphism between the Hochschild homology of a superalgebra and that of the superalgebra of square supermatrices of a given type over A, induces, also, an isomorphism between the cyclic homologies of the two superalgebras. 1. Superalgebras and supermatrices Superalgebras (i.e. Z2graded algebras) are a very important particular case of graded algebras that play an important role in modern mathematics and theoretical physics. They are, particularly, central objects, as basic algebraic machinery in the theory of supermanifolds, which are objects similar to the ordinary manifolds, regarded as ringed spaces, but the sheaf of smooth functions on the manifolds is replaced by a suitable sheaf of superalgebras. Their different kind of homologies can be defined in a way quite similar to the ungraded case, but we have to take into account the grading, which results in extra signs added in suitable places. We are not going to say much about superalgebras, they are described in details,
THE DENNIS ’ SUPERTRACE AND THE HOCHSCHILD HOMOLOGY OF SUPERMATRICES
, 905
"... Abstract. We construct, in this paper, a generalization of the Dennis trace (for matrices) to the case of the supermatrices over an arbitrary (not necessarily commutative) superalgebra with unit. By analogy with the ungraded case, we show how it is possible to use this map to construct an isomorphis ..."
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Abstract. We construct, in this paper, a generalization of the Dennis trace (for matrices) to the case of the supermatrices over an arbitrary (not necessarily commutative) superalgebra with unit. By analogy with the ungraded case, we show how it is possible to use this map to construct an isomorphism from the Hochschild homology of the superalgebra to the Hochschild homology of the supermatrix algebra. 1. The supertrace and the supercommutator We remind first a couple of things related to the Z2grading of the supermatrix algebra Mp,q(R). For the general material regarding the superalgebra the reader should consult the classical books of Bartocci, Bruzzo and HernandezRuiperez ([3]) and Manin ([6]). A matrix A ∈ Mp,q(R) is considered to be homogeneous if it can be decomposed into blocks