Tensor weight structures and \(t\)-structures on the derived categories of schemes (Q6117115)

The paper studies, for certain classes of rigidly compactly generated tensor triangulated categories, the relation between variations of some classical constructions on triangulated categories, namely t-structures (see [\textit{A. A. Beilinson} et al., Astérisque 100, None (1982; Zbl 0536.14011)]), weight structures (as defined by \textit{M. V. Bondarko} [J. \(K\)-Theory 6, No. 3, 387--504 (2010; Zbl 1303.18019)], preaisles and copreaisles [\textit{B. Keller} and \textit{D. Vossieck}, Bull. Soc. Math. Belg., Sér. A 40, No. 2, 239--253 (1988; Zbl 0671.18003)], by imposing compatibilty with the tensor structures in a natural fashion. They prove a result classifying thick \(\otimes\)-preaisles on tensor triangulated subcategory of compact objects, by showing a bijective correspondence between such preaisles and the set of compactly generated tensor t-structures. Dually, they get a classification result thick \(\otimes\)-copreaisles in the tensor triangulated category of compact objects, by showing they are in bijection with the set for compactly generated \(\otimes^c\)-weight structures. They go on to prove a classification theorem for weight structures on the derived category of quasi-coherent \(\mathcal{O}_X\)-modules on a noetherian, separated scheme \(X\). The classification in stated in terms of ``Thomason filtration'', that is filtrations of subsets of \(X\) where each subset is Thomason. At the end the authors consider the bounded derived category of coherent sheaves over \(\mathbb{P}^1_k\), \(\mathbf{D}^b(\operatorname{Coh} \mathbb{P}^1_k)\), defined over a field \(k\), and classify the \(\otimes\)-preaisles, \(\otimes\)-aisles and \(\otimes\)-copreaisles for this category. Using this, they conclude that there are no non-trivial tensor weight structures on \(\mathbf{D}^b(\operatorname{Coh} \mathbb{P}^1_k)\).