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