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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 would like to define the "relative" support of $C$ as the set of those $s\in S$ such that $C\otimes R_s\neq 0$.

My question is: did any consider this condition/definition in the literature; do any nice reformulations for it exist? Note that I don't want to assume $f$ to be finite. Moreover, if $f$ were affine then this support would probably coincide with that of $f_*C$; yet to deal with the proper case one should look at an affine cover of $X/S$.

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  • $\begingroup$ Sheaves on X form a (quasicoherent) sheaf of categories over S (in your case of S affine this just means R-linear category), and you're asking for the support of the section of this sheaf, which I believe is defined in a standard way, but I don't know a reference. My go-to for all things "QC sheaves of categories" though is Gaitsgory's "A notion of 1-affineness".. $\endgroup$
    – David Ben-Zvi
    Commented Sep 2, 2023 at 20:31
  • $\begingroup$ What do you mean by $C\otimes R_s\neq 0$? $\endgroup$
    – Jason Starr
    Commented Sep 5, 2023 at 12:23
  • $\begingroup$ I mean the tensor product over $R$. In my case the category $D(R)$ is just $R$-linear; so it is easy to define $C\otimes_R R_s$. $\endgroup$
    – Mikhail Bondarko
    Commented Sep 5, 2023 at 14:48

1 Answer 1

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Perhaps what you need is an action of of a tt-category (in your case $D(R)$) on the category $D_{qc}(X)$ which gives a support in $S$ to quasi-coherent complexes over $X$. This is discussed with greater generality in the paper:

Stevenson, Greg: Support theory via actions of tensor triangulated categories. J. Reine Angew. Math. 681 (2013), 219–254.

(also available in arXiv).

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  • $\begingroup$ Thank you very much! I will have a look at it. However the paper numdam.org/item/ASENS_2008_4_41_4_575_0 cited by Stevenson is better for my purposes. $\endgroup$
    – Mikhail Bondarko
    Commented Sep 5, 2023 at 16:55
  • $\begingroup$ @MikhailBondarko Glad it is useful! Stevenson's paper is more general, so perhaps you find some uses for it later. $\endgroup$
    – Leo Alonso
    Commented Sep 5, 2023 at 16:58
  • $\begingroup$ Yes; these papers are really useful for me; thank you! I wonder whether anybody has studied their relation to algebraic geometry. $\endgroup$
    – Mikhail Bondarko
    Commented Sep 6, 2023 at 5:46

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