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I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.

Assume that $\mathcal{R}_X$ is a sheaf of (not necessarily commutative) rings on a topological space $X$. We also do not assume that $\mathcal{R}_X$ is noetherian or coherent. Let $D^b(\mathcal{R}_X)$ be the bounded derived category of $\mathcal{R}_X-Mod$. My question is: Is the subcategory $D^b_{coh}(\mathcal{R}_X)$ of pseudo-coherent complexes in $D^b(\mathcal{R}_X)$ defined? And if yes, does the definition of pseudo-coherent complexes of modules over commutative rings apply to the context of modules over sheaves of noncommutative rings?

In other words, if the notion $D^b_{coh}(\mathcal{R}_X)$ exists in this setting, can we view it as the (triangulated) subcategory of $D^b(\mathcal{R}_X)$ consisting of complexes which are locally quasi-isomorphic to bounded above complexes of locally free $\mathcal{R}_X$-modules of finite rank.

I append Joseph Lipman's notes "NOTES ON DERIVED FUNCTORS AND GROTHENDIECK DUALITY" (https://www.math.purdue.edu/~jlipman/Duality.pdf) where in the second paragraph of Section 4.3, he defines $D_{coh}^b(O_X)$ for the commutative structure sheaf $\mathcal{O}_X$ of a scheme $X$. Basically, I try to determine if his definition is general enough so that it goes true when one replaces $\mathcal{O}_X$ with a sheaf of noncommutative rings $\mathcal{R}_X$.

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  • $\begingroup$ Let me first point out that pseudocoherent complexes do not live in the bounded derived category but (cohomologically) bounded-above derived category. Next, there is still a step from the version that you cited to a sheaf of noncommutative rings: the reference only considers the case that $\mathcal O_X$ is the structure sheaf of a scheme, not for arbitrary ringed spaces, and in general, quasi-coherent sheaves do not form an abelian category (stacks.math.columbia.edu/tag/01BD), although the category of coherent sheaves do form an abelian category for all ringed spaces. $\endgroup$
    – Z. M
    Commented Feb 5, 2022 at 5:18
  • $\begingroup$ @Z.M. Why don't they live in the bounded derived category? For instance, in Hotta's book personal.math.ubc.ca/~cautis/dmodules/hottaetal.pdf, the category of pseudo-coherent complexes of $\mathcal{D}_X$-modules is defined as a subcategory of the bounded derived category $D^b(\mathcal{D}_X)$, no? Look Lemma $2.6.13$ on page $74$ in the book. Or is this some special case? $\endgroup$
    – Flavius Aetius
    Commented Feb 6, 2022 at 0:14
  • $\begingroup$ @Z.M. Wait! I thought that given any sheaf of rings $\mathcal{R}_X$ on a topological space $X$, the category $\mathcal{R}_X-Mod$ is automatically an abelian categeory. See, for example, page $87$ in Kashiwara-Schapira's book "Sheaves on Manifolds". In particular, this should still hold true for sheaves of rings $\mathcal{R}_X$ with a morphism $\mathcal{O}_X\to\mathcal{R}_X$ which are quasi-coherent as left $\mathcal{O}_X$-modules. $\endgroup$
    – Flavius Aetius
    Commented Feb 6, 2022 at 0:26
  • $\begingroup$ Pseudocherence is usually only assumed to be locally bounded above, say, in SGA6. Another point that I forgot to mention: pseudocoherent complexes over the structure sheaf are only assumed to be locally quasi-isomorphic to a bounded above complex of finite free modules, not necessarily globally quasi-isomorphic to a complex of vector bundles (in general, it is difficult to produce or classify vector bundles on a scheme). See stacks.math.columbia.edu/tag/08CA $\endgroup$
    – Z. M
    Commented Feb 6, 2022 at 9:22
  • $\begingroup$ @Z.M. What does then the notation $D_c^b(\mathcal{D}_X)$ in Lemma $2.6.13$ in Hotta's book personal.math.ubc.ca/~cautis/dmodules/hottaetal.pdf mean? Isn't that the "bounded" derived category? $\endgroup$
    – Flavius Aetius
    Commented Feb 8, 2022 at 9:43

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