Timeline for Pseudo-coherent complexes over sheaves of non-commutative rings
Current License: CC BY-SA 4.0
9 events
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| Feb 16, 2022 at 22:17 | history | edited | Flavius Aetius | CC BY-SA 4.0 |
added 8 characters in body
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| Feb 8, 2022 at 9:56 | comment | added | Flavius Aetius | @Z.M. Also, I do not quite see how the fact that quasi-coherent $\mathcal{O}_X$-modules do not form an abelian category is a problem for the category $\mathcal{R}_X-Mod$ of all modules over $\mathcal{R}_X$ to be abelian? I think that $\mathcal{R}_X-Mod$ is abelian. And that is all we need to define its derived category $D(\mathcal{R}_X)$. | |
| Feb 8, 2022 at 9:43 | comment | added | Flavius Aetius | @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? | |
| Feb 6, 2022 at 9:22 | comment | added | Z. M | 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 | |
| Feb 6, 2022 at 0:26 | comment | added | Flavius Aetius | @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. | |
| Feb 6, 2022 at 0:14 | comment | added | Flavius Aetius | @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? | |
| Feb 5, 2022 at 5:18 | comment | added | Z. M | 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. | |
| Feb 4, 2022 at 17:52 | history | edited | Flavius Aetius | CC BY-SA 4.0 |
two typos were fixed;
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| Feb 4, 2022 at 15:24 | history | asked | Flavius Aetius | CC BY-SA 4.0 |