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For triangulated categories $T,T'$ I would like to define "weakly exact" functors as those that respect cones, that is, $F(Cone f)\cong Cone(F(f))$ for any $T$-morphism $f$, and I do not require any f …

Assume that $T$ is a symmetric monoidal triangulated category, and $X$ is an object in it. Then the functors $X\otimes -$ and $-\otimes X: T\to T$ are not necessarily exact since they send distinguish …

Let $X$ be a noetherian scheme (actually, I need the case where $X$ is proper over an affine scheme), $C$ is an object of the derived category $D_{coh}(X)$ of coherent sheaves on $X$. Under which ass …

Let us call a commutative square $$ \require{AMScd} \begin{CD} A @>{g'}>> B \\ @V{f'}VV @VV{f}V \\ C @>>{g}> D \end{CD} $$ in a triangulated category split homotopy cartesian if the ("s …

A somewhat vague question: for which rings there exist "interesting" exact abelian subcategories of $R-\operatorname{Mod}$ that are closed with respect to products? Actually, I would like the correspo …

In the case where C is bounded above this statement was established in Neeman's https://arxiv.org/abs/1804.02240v4. Now I will try to extend his "approximation" statements to the case where $C$ is an …

Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R …

Let $A$ be a semi-simple abelian subcategory of a triangulated category $C$ that "generates" $A$ (that is, $C$ equals its own smallest triangulated subcategory that is closed under direct summands and …

Assume $R$ is a commutative Noetherian ring of finite Krull dimension; $R'$ is a not commutative ring that contains $R$ in its center and also finitely generated as an $R$-module. If the (left) global …

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely generate …

Let $T$ be a compactly generated triangulated category, that is, $T$ is closed with respect to small coproducts and equals its own smallest triangulated subcategory closed with respect to coproducts a …

As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-commutativ …

The "motivic motivation" is that by idempotent completing correspondences over a finite field one obtains a category of homological motives where Kunneth decompositions of diagonals are available. Mo …

Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the homo …

I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of complexe …