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Let $f:X\to S$ be a morphism of Noetherian schemes; in the case I am interested in $S=\operatorname{Spec}R$ is affine and $f$ is proper. For a complex $C$ a complex of quasi-coherent sheaves on $X$ I …

In the Stacks project and in a book of Brian Conrad the "main" derived category of a scheme is the one of $\mathcal{O}_X$-modules. I would like to understand whether $D(\mathcal{O}_X)$ is better than …

Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful? If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full …

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 …

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 …

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 …

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 …

For which smooth projective $P$ over a field there exists a bounded $t$-structure $t$ on the bounded derived category of coherent sheaves $D^b(P)$ such the heart $Ht$ of $t$ has enough injectives? Not …

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 …

Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex $D$ of coherent sheaves. Then (if I understand these matters and the answer to the question Characterization of …

Let me try to sketch an argument (though I am not quite sure in details). A theorem of Neeman implies that a compact object $M$ in a compactly generated triangulated category $T$ is a generator if an …

For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the "sta …

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. …

Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ instea …

I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are si …