Recollement and Tilting Complexes

First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P^{centerdot} to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements \{\cat{D}_{A/AeA}(A), \cat{D}(A), \cat{D}(eAe)} and \{\cat{D}_{B/BfB}(B), \cat{D}(B), \cat{D}(fBf)}, where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex $\varDelta^{\centerdot}_{T}$ which induces an equivalence between \cat{D}_{A/AeA}(A) and \cat{D}_{B/BfB}(B). Third, we apply the above to a symmetric algebra A. We show that a partial tilting complex $P^{\centerdot}$ for A of length 2 extends to a tilting complex, and that $P^{\centerdot}$ is a tilting complex if and only if the number of indecomposable types of $P^{\centerdot}$ is one of A. Finally, we show that for an idempotent e of A, a tilting complex for eAe extends to a recollement tilting complex for A, and that its standard equivalence induces an equivalence between \cat{Mod}A/AeA and \cat{Mod}B/BfB.Comment: 24 page

Extensions of rings and tilting complexes

AbstractWe give conditions that extensions of rings make tilting complexes. Moreover, we show that Frobenius extensions are invariant under derived equivalences which are induced by these tilting complexes

Adaptability and selectivity of human peroxisome proliferator-activated receptor (PPAR) pan agonists revealed from crystal structures

The structures of the ligand-binding domains (LBDs) of human peroxisome proliferator-activated receptors (PPARα, PPARγ and PPARδ) in complexes with a pan agonist, an α/δ dual agonist and a PPARδ-specific agonist were determined. The results explain how each ligand is recognized by the PPAR LBDs at an atomic level