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Let $X$ be some smooth algebraic variety. I would like to understand the relation between the following two categories:

  1. $D^b_{cd,1}\text{Coh}(X) \subset D^b\text{Coh}(X)$: the full subcategory of the derived category of coherent sheaves with cohomologies being supported in codimension $\geq 1$.

  2. $D^b\text{Coh}_{cd,1}(X)$: The derived category of the abelian category of coherent sheaves supported in codimension $\geq 1$.

I was trying to show that for a bounded complex $F^\bullet$ of coherent sheaves with cohomologies supported in codimension $\geq 1$ there exists a quasi-isomorphic complex $E^\bullet$ with $E^i$ being supported in codimension $\geq 1$, which would show that they are equivalent, but I have ran into some problems like $i_*\circ i^*$ for $i$ a closed immersion not being exact (the idea was to take the union of the support of cohomologies and restrict the complex to this union). Is it still true that these categories are equivalent? If not, are they at least related in some nice way?

Edit: For now, I can only say that the natural t-structures on both of these triangulated categories give the same hearts, but I would need more than that to show equivalence.

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  • $\begingroup$ Isn't this just checkable at the generic points? Taking stalks is exact so I don't think there is a problem. $\endgroup$
    – GTA
    Commented Sep 4, 2019 at 16:41
  • $\begingroup$ I am not sure I understand what you mean. Are you saying that I can construct a quasi-isomorphic complex by taking stalks at the generic points? $\endgroup$
    – Arkadij
    Commented Sep 4, 2019 at 17:03
  • $\begingroup$ What I'm saying is that being supported in codimension >=1 for coherent sheaf is the same as having zero stalk for each generic point, and taking stalk at a generic point is exact so it is actually a functor of derived categories. Say there is one generic point $i:\eta\rightarrow X$ then $Li^{*}=i^{*}:D^{b}Coh(X)\rightarrow D^{b}Coh(\eta)$ and both conditions are equivalent to complexes mapped to zero complex via $i^{*}$ because a complex with zero cohomology is quasi-isomorphic to zero complex. $\endgroup$
    – GTA
    Commented Sep 4, 2019 at 17:49
  • $\begingroup$ I think one can construct the complex by taking a large enough subscheme of codimension $\ge 1$, for example the subscheme given by $\ker(\mathcal{O}_X\to \mathcal{End}(\mathcal{H}^*F))$. Then $\mathcal{H}^*F$ will be acyclic with respect to the pullback onto this subscheme $\endgroup$
    – Wille Liu
    Commented Sep 4, 2019 at 17:53
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    $\begingroup$ Maybe I am pointing out the obvious, but note that the coh. sheaves supported in codim $\geq1$ form a Serre subcategory (even better, torsion class) in the category of coh. sheaves. The question in this setting is somewhat considered in stacks project: stacks.math.columbia.edu/tag/06UP but the criterion for equivalence there does not seem to fit this setting. $\endgroup$
    – Pavel Čoupek
    Commented Sep 6, 2019 at 2:40

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