4
$\begingroup$

I´m studying Huybrechts book "Fourier-Mukai transforms in algebraic geometry" and I came up with the following: as an example of semi orthogonal decomposition of a triangulated category $\mathcal{D}$ it is given $\mathcal{D}_1=\mathcal{D}´^{\perp}$, $\mathcal{D}_2=\mathcal{D}´$ where $\mathcal{D}´$ is an admissible (hence full and triangulated) subcategory of $\mathcal{D}$. According to the definition all subcategories in a semi orthogonal decomposition are to be admissible (i.e. there exists $\pi:\mathcal{D}\longrightarrow\mathcal{D}´$ a right adjoint functor to the inclusion). My problem is that I cannot see how $\mathcal{D}´^\perp$ is admissible for an admissible $\mathcal{D}´$.

Any idea?

$\endgroup$
3
$\begingroup$

If $D$ is arbitrary then $D_1$ is not necessarily admissible. However, if $D$ is saturated, then $D_1$ is admissible. See the paper of Bondal and Kapranov for a proof.

$\endgroup$
2

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.