Related papers: On compact generation of deformed schemes

Let $R$ be an associative ring with unit and denote by $K({\rm R \mbox{-}Proj})$ the homotopy category of complexes of projective left $R$-modules. Neeman proved the theorem that $K({\rm R \mbox{-}Proj})$ is $\aleph_1$-compactly generated,…

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting…

We relate the variance theory for Cousin complexes -^# developed by Lipman, Nayak and the author to Grothendieck duality for Cousin complexes. Specifically for a Cousin complex F on (Y, \Delta)--with \Delta a codimension function on a…

Let $f \colon X \to Y$ be a morphism of concentrated schemes. We characterize $f$-perfect complexes $\mathcal{E}$ as those such that the functor $\mathcal{E} \otimes^{\mathbf{L}}_X \mathbf{L} f^*-$ preserves bounded complexes. We prove, as…

We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy…

We introduce a tensor compatibility condition for t-structures. For any Noetherian scheme $X$, we prove that there is a one-to-one correspondence between the set of filtrations of Thomason subsets and the set of aisles of compactly…

Grothendieck's formal functions theorem states that the coherent cohomology of a Noetherian scheme can be recovered from that of a blowup and the infinitesimal thickenings of the center and of the exceptional divisor of the blowup. In this…

We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using…

We study a form of d\'{e}vissage for generation in derived categories of Noetherian schemes. First, we extend a result of Takahashi from the affine context to the global setting, showing that the bounded derived category is classically…

This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. Additionally, we establish that the Krull dimension of a variety…

We show that a well-known exact sequence in K-theory for quotients of triangulated categories descends to numerical K-groups provided that the category, the quotient and the category we take the quotient with has a numerical K-group, and if…

We develop the deformation-obstruction calculus for morphisms of complexes with a fixed lift of the codomain, to derived categories of flat nilpotent deformations of abelian categories. As an application, we give an alternative proof that…

We introduce a notion of derived Azumaya's algebras over rings and schemes. We prove that any such algebra $B$ on a scheme $X$ provides a class $\phi(B)$ in $H^{1}_{et}(X,\mathbb{Z})\times H^{2}_{et}(X,\mathbb{G}_{m})$. We prove that for…

Weakly approximable triangulated categories, introduced by Neeman, provide a powerful framework for studying localization phenomena in triangulated categories. In this paper, we establish new localization theorems showing that, under mild…

If an element of the Grothendieck group of the derived category of a scheme is locally represented by perfect complexes, then can the original element be represented by a perfect complex? We provide a counterexample on a projective variety…

We prove new Brown representability theorems for triangulated categories using metric techniques as introduced in the work of Neeman. In the setting of algebraic geometry, this gives us new representability theorems for homological and…

We obtain some fundamental results, as Bokstedt-Neeman Theorem and Grothendieck duality, about the derived category of modules on a finite ringed space. Then we see how these results are transfered to schemes in a simple way and generalized…

Using the localization property, we construct a triangulated category of motives over quasi-projective T-schemes for any coefficient where T is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also…

Given a suitable Noetherian scheme, we classify tensor $t$-structures on the bounded derived category of coherent sheaves and its variants with prescribed support. Furthermore, we show that the existence of such $t$-structures restricting…

In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of…