ag.algebraic geometry – Example of pseudocoherent advanced which isn’t domestically quasi-isomorphic to a strict pseudocoherent one Answer

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ag.algebraic geometry – Example of pseudocoherent advanced which isn’t domestically quasi-isomorphic to a strict pseudocoherent one

I question this query right here plane if I posted it too on math.stackexchange (recieving no respond up to now) as a result of I’ve learn some analogous query however for consummate complexes on this web site, plane although sadly I’ve not sufficient background to grasp the respond. Anyway, fastened a strategy $$X$$ and following the definitions of Thomason-Trobaugh, I convene a advanced $$E^{cdot}$$ of $$mathcal{O}_X$$-modules strict$$n$$pseudocoherent whether it is bounded above and $$E^ok$$ is domestically freed from finite rank in every diploma $$kgeq n$$. I convene it strict pseudocoherent if it is strict-$$n$$-pseudocoherent for any integer $$n$$. I convene it $$n$$pseudocoherent if for each $$xin X$$ there exists a neighborhood $$U$$, a strict $$n$$-pseudocoherent advanced $$F^{cdot}$$ of $$mathcal{O}_U$$-modules and a quasi-isomorphism $$F^{cdot}to E^{cdot}|_U$$ (birefly, $$E^{cdot}$$ is domestically quasi-isomorphic to a strict $$n$$-pseudocoherent advanced). Eventually, $$E^{cdot}$$ is claimed to breathe pseudocoherent whether it is $$n$$-pseudocoherent for each integer $$n$$.

Now, clearly for a pseudocoherent advanced and a set $$x$$, there’s for any $$n$$ an launch neighborhood of $$x$$, relying on $$n$$, over which it is quasi-isomorphic to a strict $$n$$-pseudocoherent advanced. It appears cheap that it couldn’t exists an launch neighborhood of $$x$$ such that restricted to it the advanced is domestically quasi-isomorphic to a strict $$n$$-pseudocoherent for each $$n$$ (principally, as a result of the intersection of the earlier ones might now not breathe launch). But learn how to present a concrete instance of this occurring?

Now, so far as I do know (from the paper of Thomason-Trobaugh, Proposition 2.3.1) we can’t anticipate to search out an instance amongst complexes of quasi-coherent $$mathcal{O}_X$$-modules. My try (which I notice to breathe fairly grievance) is right here: https://math.stackexchange.com/questions/3950352/example-of-pseudocoherent-complex-which-is-not-locally-quasi-isomorphic-to-a-str, the place I used to be searching for an instance over the affine line, contemplating a advanced of skyscraper sheaves. Don’t actually know whether or not it could labor with apt changes or whether or not the respond ought to breathe create in a distinct context.

Any ameliorate is appreciated, thanks in forward!

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