F.J. Bloore and T.J. Harding

Isomorphism of de Rham cohomology and relative Hochschild cohomology 
of differential operators


Abstract:  Any de Rham $p$-form $\alpha$ on a manifold $M$ may be extended to 
           become a Hochschild $p$-cochain $\alpha_S$ on the associative algebra
           $\cal D$ of differential operators on $C^\infty (M,\Bbb R)$. The map
           $\alpha\mapsto\alpha_S$ depends on a choice of ``alocation'', $S$, which 
           is a rule for filling in any $(p+1)$-tuple of sufficiently nearby points 
           $x_0,\dots,x_p$ of $M$ with a $p$-simplex $S(x_0,\dots, x_p)$ having
           these points as vertices. We show that $(d\alpha)_S= \delta(\alpha_S)$
           so that the map $\alpha\to\alpha_S$ passes to cohomology. We indicate
           the pattern of the proof that the map $[\alpha] \to [\alpha_S]$ sending
           $H^p_{DR}(M,\Bbb R)\to H^p(\cal D,C^\infty (M,\Bbb R); \cal D)$ is           
           independent of $S$ and is in fact an isomorphism. Here the latter group
           is the relative Hochschild cohomology group of $\cal D$ relative to
           $C^\infty (M,\Bbb R)$, with coefficients in $\cal D$.

Keywords:  De Rham cohomology, Hochschild cohomology, differential operators.

MS classification:  16E40, 16S32, 53C99.          


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Bibliographic information: Differential Geometry and its Applications
           Proc. Conf., August 24-28, 1992, Opava, Czechoslovakia
           Silesian University, Opava, 1993, 65-70

Received:  2 November 1992
           
Figures:   Two figures in PostScript format are available as Fig1 and Fig2