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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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