A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal $\mathfrak{p}$ in the cohomology ring of the group scheme.

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

We prove a Knörrer-periodicity-type equivalence between derived factorization categories of gauged Landau–Ginzburg models, which is an analogy of a theorem proved by Shipman and Isik independently. As an application, we obtain a gauged Landau–Ginzburg version of Orlov’s theorem describing a relationship between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces, by combining the above Knörrer periodicity type equivalence and the theory of variations of geometric invariant theory quotients due to Ballard, Favero and Katzarkov.

Let $R$ be a commutative noetherian local ring. As an analog of the notion of the dimension of a triangulated category defined by Rouquier, the notion of the dimension of a subcategory of finitely generated $R$-modules is introduced in this paper. We found evidence that certain categories over nice singularities have small dimensions. When $R$ is Cohen–Macaulay, under a mild assumption it is proved that finiteness of the dimension of the full subcategory consisting of maximal Cohen–Macaulay modules which are locally free on the punctured spectrum is equivalent to saying that $R$ is an isolated singularity. As an application, the celebrated theorem of Auslander, Huneke, Leuschke, and Wiegand is not only recovered but also improved. The dimensions of stable categories of maximal Cohen–Macaulay modules as triangulated categories are also investigated in the case where $R$ is Gorenstein, and special cases of the recent results of Aihara and Takahashi, and Oppermann and Št́ovíček are recovered and improved. Our key technique involves a careful study of annihilators and supports of $\mathsf{Tor}$, $\mathsf{Ext}$, and $\underline{\mathsf{Hom}}$ between two subcategories.

Bondarko defines and studies the notion of weight structure and he shows that there exists a weight structure over the category of Voevodsky motives with rational coefficients (over a field of characteristic 0). In this paper we extend this weight structure to the category of Beilinson motives (for any scheme of finite type over a base scheme which is excellent of dimension at most two) introduced and studied by Cisinsky and Déglise. We also check the weight exactness of the Grothendieck operations.

Let $f:X\to Y$ be a Cohen–Macaulay map of finite type between Noetherian schemes, and $g:Y'\to Y$ a map, with Y′ Noetherian. Let $f':X'\to Y'$ be the base change of f under g and $g':X'\to X$ the base change of g under f. We show that there is a canonical isomorphism $\theta_g^f: {g'}^*\omega_f \simeq \omega_{f'}$, where $\omega_f$ and $\omega_{f'}$ are the relative dualizing sheaves. The map $\theta_g^f$ is easily described when f is proper, and has a more subtle description when f is not. If f is smooth we show that $\theta_g^f$ corresponds to the canonical identification $g'^*\Omega_f^r= \Omega_{f'}^r$ of differential forms, where r is the relative dimension of f. This work is closely related to B. Conrad's work on base change. However, our approach to the problems and our viewpoint are very different from Conrad's: dualizing complexes and their Cousin versions, residual complexes, do not appear in this paper.