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Assume that $f:X\to Y$ is a morphism of complex varieties, and the homomorphisms $H^i_\text{sing}(f)$ are bijective for $0\le i\le 3$ (though possibly $3$ is too much here:)). Under which assumption …

Hartshorne's 1974 conjecture states that a smooth closed subvariety $X$ of $\mathbb{CP}^r$ of dimension $>2/3r$ is necessarily (globally) a complete intersection. Is anything interesting known about t …

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 …

If $X'$ is a minimal resolution of $X$ then $X''$ can be obtained from it by means of successive blow ups of smooth subvarieties, right? Then $M(X')$ is a direct summand of $M(X'')$; for example, see …

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 …

For Voevodsky-type motivic categories over various schemes and a morphism $f$ of base schemes there exist the following functors: $f_*$ exists unconditionally and $f_{\#}$ (which is closely related to …

Any characteristic zero field is an inductive limit of fields that can be embedded into complex numbers (i.e., those of characteristic 0 and of cardinality at most continuum). Hence the assumption t …

No, $\mathbb{Q}(X)$ is the presheaf "additively represented" by $X$; it is not constant. Now I will express my understanding of this matter; I did not check that it fits with Ayoub's definitions and n …

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 …

Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets …

Let $M$ be a $R$-linear Chow motif over a field $k$ that is perfect but not necessarily algebraically closed. Can one prove that $M$ is not a direct summand of itself (that is, $M\not\cong M\bigoplus …