Let X be a quasi-projective complex variety. It follows from the work of Voevodsky that the motivic cohomology of X, denoted as $H^{p,q}(X)$ where q and p are integers with q nonnegative, can be represented in the triangulated category of motives over the field of complex numbers, denoted as $DM^{eff,-}_{Nis}$. That is, there exists an object $\wp_{mot}(q)$ in $DM^{eff,-}_{Nis}$ such that $$H^{p,q}(X)=Hom_{DM^{eff,-}_{Nis}}(M(X), \wp_{mot}(q)[p-2q])$$ where $M(X)$ is the motive of X. We construct objects $\wp_{mor}(q)$ and $\wp_{Sing}(q)$ in $DM^{eff,-}_{Nis}$ to represent the morphic cohomology $L^qH^p(X)$ and the singular cohomology $H^p_{Sing}(X^{an})$ of $X$. More precisely, $$L^qH^p(X)=Hom_{DM^{eff,-}_{Nis}}(M(X), \wp_{mor}(q)[p-2q])$$, $$H^p_{Sing}(X^{an})=Hom_{DM^{eff,-}_{Nis}}(M(X), \wp_{Sing}(q)[p-2q])$$ where X is smooth. If X is singular, we define the morphic cohomology of X by the above formula. As an application, we show that Friedlander's comparison result $L^qH^p(X) \cong H^p_{Sing}(X^{an})$, where X is smooth of pure dimension d and $q\geq d$, can be generalized to singular varieties. As a second application, the morphic cohomology operations are considered.

Last modified

• 08/31/2018

Creator

• Chenghao Chu

DOI

• https://doi.org/10.21985/N2072P

Subject

• Mathematics

Keyword

• Mathematics

Date created

• 2008-05-02

Resource type

• Dissertation

Rights statement

• In Copyright