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head 1.1; access; symbols; locks; strict; comment @% @; 1.1 date 2007.10.13.16.01.41; author mellit; state Exp; branches; next ; desc @@ 1.1 log @first addition @ text @\input commons.tex \begin{document} \bibliographystyle{alpha} \section{De Rham complex} Complex, filtrations. \section{The cycle class} \subsection{The construction of the cycle class map in \cite{bloch:semireg}.} Here Bloch refers to \cite{groth:fga}, Expos\'e 149 and \cite{harts:rd}, p.176. Let $f:X\rightarrow S$ be smooth and projective. Let $Z\subset X$ be a local complete intersection of codimension $k$ (e.g. $X$ is nonsingular over a field and $Z$ is also nonsingular). Then there exists a canonical cycle class of $Z$ in \[ \{Z\}\in\HC_Z^{2k}(X, F^p\Omega_{X/S}^\bullet). \] The theorem in \cite{harts:rd} is the following: \begin{thm}[Fundamental local isomorphism] Let $i:Z\rightarrow X$ be a closed immersion of preschemes, where $Z$ is locally a complete intersection in $X$ of codimension $k$, and let $F$ be a sheaf of $O_X$-modules on $X$. Then there is a natural functorial isomorphism \[ \iExt_{O_X}^k(O_Z,F)\xrightarrow{\sim} F\otimes_{O_X} \omega_{Z/X}. \] Furthermore, if $F$ is $i^*$-acyclic, then \[ \iExt_{O_X}^j(O_Z,F)=0 \qquad\text{for $j\neq k$.} \] \end{thm} Recall that by definition $\omega_{Z/X}=(\bigwedge^k(J/J^2))^\vee$, where $J$ is the sheaf of ideals defining $Z$. Substituting $F=\bigwedge^k(J/J^2)$ we obtain a canonical isomorphism \[ \iExt_{O_X}^k(O_Z,\bigwedge^k(J/J^2))\xrightarrow{\sim} O_Z, \] which gives a canonical section \[ \alpha_Z\in\Gamma(X, \iExt_{O_X}^k(O_Z,\bigwedge^k(J/J^2))). \] There is a spectral sequence \[ E^{pq}_2=H^p(X, \iExt_{O_X}^q(O_Z,\bigwedge^k(J/J^2))) \] \subsection{The construction of the cycle class map in \cite{elzein:compdual}.} \subsection{Cycle class map via D-modules} We try to follow the construction in \cite{saitom:IMHM}, 1.15. For a separated and reduced algebraic variety $X$ over $\C$ we denote by $D^b_h(\D_X)$ the bounded derived category which consists of complexes of $\D_X$-modules with holonomic homologies. We have the usual functors $f_*$, $f_!$, $f^*$, $f^!$, $\DD$, $\boxtimes$, $\otimes$, inner hom $\ihom$. There are the adjoint relations: \[ \Hom(f^*M, N)=\Hom(M, f_*N),\qquad \Hom(f_!M, N)=\Hom(M, f^!N). \] Let $k$ be a field. Then we have the standard $\D_{\spec k}$-module $k$. For each $X$ over $k$ with $a_X:X\rightarrow \spec k$ we define \[ k_X = a_X^* k. \] For $X$ smooth we obtain \[ k_X = a_X^* k = a_X^![-2d_X] = O_X[-2d_X]. \] Note that \[ \DD k_X = \DD a_X^* k = a_X^! \DD k = O_X = k_X[2d_X]. \] The intersection complex is defined as \[ IC_X = \image(j_! k_U[d_X], j_* k_U[d_X]) \] for any $U\subset X$ which is smooth and open. If $X$ is itself smooth we obtain $IC_X=k_X[d_X]=O_X[-d_X]$. Note that \[ \DD IC_X = \DD(k_X[d_X]) = (\DD k_X)[-d_X] = k_X[d_X] = IC_X. \] Suppose $i:Z\hookrightarrow X$ is a closed immersion. Suppose $Z$ is smooth. We have $k_Z=i^* k_X$. There is the identity morphism in $\Hom(k_Z, k_Z)$ which induces a morphism \[ i^\# :k_X\rightarrow i_* k_Z. \] We apply duality to obtain \[ \DD i^\#:\DD i_* k_Z \rightarrow \DD k_X. \] Consider $\DD i_* k_Z$: \[ \DD i_* k_Z = i_! k_Z[2d_Z] = i_* k_Z[2d_Z] \] since $i$ is a closed immersion hence proper. Therefore the dual morphism acts as \[ \DD i^\#:i_* k_Z[2d_Z] \rightarrow k_X[2d_X] \] and we can make a composite \[ \DD i^\# [-2d_Z] \circ i^\#: k_X \rightarrow k_X[2d_X-2d_Z]. \] This gives an element of \[ \Hom(k_X, k_X[2d_X-2d_Z]) = \Hom(k, a_{X*}k_X[2d_X-2d_Z]) = H^{2d_X-2d_Z}(a_{X*} k_X). \] One could also take the identity in $\Hom(i^! k_X, i^! k_X)$ and obtain an element \[ i^\flat:\Hom(i_!i^! k_X, k_X) = \Hom(i_*i^! k_X, k_X). \] \[ \Hom(i^* k_X, i^! k_X) = \Hom(k_X, i_* i^! k_X) \] \bibliography{refs} \end{document}@
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