The big Chern classes and the Chern character

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dc.creatorRamadoss, Ajay C.
dc.date2005-12-05
dc.date2008-06-05
dc.date.accessioned2026-07-07T09:51:35Z
dc.date.available2026-07-07T09:51:35Z
dc.descriptionLet $X$ be a smooth scheme over a field of characteristic 0. Let $\dd^{\bullet}(X)$ be the complex of polydifferential operators on $X$ equipped with Hochschild co-boundary. Let $L(\dd^1(X))$ be the free Lie algebra generated over $\strc$ by $\dd^1(X)$ concentrated in degree 1 equipped with Hochschild co-boundary. We have a symmetrization map $I: \oplus_k \sss^k(L(\dd^1(X))) \rar \dd^{\bullet}(X)$. Theorem 1 of this paper measures how the map $I$ fails to commute with multiplication. A consequence of Theorem 1 and Theorem 2 is Corollary 1, a result "dual" to Theorem 1 of Markarian [3] that measures how the Hochschild-Kostant-Rosenberg quasi-isomorphism fails to commute with multiplication. In order to understand Theorem 1 conceptually, we prove a theorem (Theorem 3) stating that $\dd^{\bullet}(X)$ is the universal enveloping algebra of $T_X[-1]$ in $\dcat$. An easy consequence of Theorem 3 is Theorem 4, which interprets the Chern character $E$ as the "character of the representation $E$ of $T_X[-1]$" and gives a description of the big Chern classes of $E$. Finally, Theorem 4 along with Theorem 1 is used to give an explicit formula (Theorem 5) expressing the big Chern classes of $E$ in terms of the components of the Chern character of $E$.
dc.descriptionFinal version. To appear in International Journal of Mathematics
dc.identifierhttps://arxiv.org/abs/math/0512104
dc.identifierhttp://arxiv.org/abs/math/0512104
dc.identifierInternational Journal of Mathematics (IJM) Volume:19 Issue:6 (July 2008) (699 - 746)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/165298
dc.subjectAlgebraic Geometry
dc.subject14F99, 16E40
dc.titleThe big Chern classes and the Chern character
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