Is the orthogonal complement of an admissible subcategory admissible itself?

Question

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?

Answer 1

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.