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Dear Pieter,
Oct. 1st, 2014 04:08 amthe phenomenon is certainly related to (quasi)coherent extensions by zero in some way, as one can see from the fact that Deligne's proof (from his appendix to "Residues and Duality") uses the functor of extension by zero for pro-coherent sheaves (see section 5.16 in the contraherent manuscript, where the direct generalization of Deligne's arguments to coderived categories of unbounded complexes is worked out).
Moreover, the fact that the pro-coherent extension by zero is an exact functor is clearly closely related to the preservation of injectivity of quasi-coherent sheaves by restrictions to open subschemes of Noetherian schemes (both properties being deducible from the Artin-Rees lemma, after all; I recall this approach to proving Hartshorne's theorem about quasi-coherent injectives was used in Appendix A to my preprint arXiv:1102.0261).
Trying to speculate a bit further about this connection, one might say that the compact objects of the coderived category are bounded complexes of coherents, while those of the conventional derived category are the perfect complexes ( = bounded complexes of locally free sheaves of finite rank). And however difficult it might be to extend by zero a coherent sheaf, extending by zero a locally free sheaf (so as to remain approximately in the same class of objects) is harder still. I haven't thought much about this, though (or otherwise don't remember what I thought when I did).
Concerning the question about a specific counterexample, such is provided essentially in Example 6.5 to Neeman's paper in JAMS-1996. Let R be any "nontrivial enough" Noetherian commutative ring (such as the ring of integers Z or polynomials in one variable k[t], etc.) Let f be an element in R (Neeman denotes it by \gamma). Our scheme X will be the spectrum of the ring of dual numbers A = R[\eps]/(\eps^2) over R, and the open subscheme will be the principal affine one corresponing to f, that is U = Spec B and B = R[f^{-1}][\eps]/(\eps^2).
( Read more... )
I hope these comments will be of some help in clarifying the issues mentioned in your letter. Please do not hesitate to contact me if anything in the above exposition is unclear or raises doubts, or with any further questions or comments you might have. I am very happy to hear that somebody in the world is reading the contraherent manuscript, after all.
Best regards,
Leonid
Pieter Belmans @uantwerp 29 сен. в 12:13
> Dear Leonid,
> In your manuscript on contraherent cosheaves one reads on page 10 that:
> […] the restriction of a homotopy-injective complex of quasi-coherent sheaves to such a subschema may no longer be homotopy-injective.
> I’ve been puzzled about this for a while now, and my intuition tells me it’s related to the lack of a good “extension by zero” for quasicoherent sheaves, but I have not been able to come up with a concrete counterexample (nor any confirmation that my hunch is correct). Could you shed some light on this issue?
> With kind regards,
> Pieter
Moreover, the fact that the pro-coherent extension by zero is an exact functor is clearly closely related to the preservation of injectivity of quasi-coherent sheaves by restrictions to open subschemes of Noetherian schemes (both properties being deducible from the Artin-Rees lemma, after all; I recall this approach to proving Hartshorne's theorem about quasi-coherent injectives was used in Appendix A to my preprint arXiv:1102.0261).
Trying to speculate a bit further about this connection, one might say that the compact objects of the coderived category are bounded complexes of coherents, while those of the conventional derived category are the perfect complexes ( = bounded complexes of locally free sheaves of finite rank). And however difficult it might be to extend by zero a coherent sheaf, extending by zero a locally free sheaf (so as to remain approximately in the same class of objects) is harder still. I haven't thought much about this, though (or otherwise don't remember what I thought when I did).
Concerning the question about a specific counterexample, such is provided essentially in Example 6.5 to Neeman's paper in JAMS-1996. Let R be any "nontrivial enough" Noetherian commutative ring (such as the ring of integers Z or polynomials in one variable k[t], etc.) Let f be an element in R (Neeman denotes it by \gamma). Our scheme X will be the spectrum of the ring of dual numbers A = R[\eps]/(\eps^2) over R, and the open subscheme will be the principal affine one corresponing to f, that is U = Spec B and B = R[f^{-1}][\eps]/(\eps^2).
( Read more... )
I hope these comments will be of some help in clarifying the issues mentioned in your letter. Please do not hesitate to contact me if anything in the above exposition is unclear or raises doubts, or with any further questions or comments you might have. I am very happy to hear that somebody in the world is reading the contraherent manuscript, after all.
Best regards,
Leonid
Pieter Belmans @uantwerp 29 сен. в 12:13
> Dear Leonid,
> In your manuscript on contraherent cosheaves one reads on page 10 that:
> […] the restriction of a homotopy-injective complex of quasi-coherent sheaves to such a subschema may no longer be homotopy-injective.
> I’ve been puzzled about this for a while now, and my intuition tells me it’s related to the lack of a good “extension by zero” for quasicoherent sheaves, but I have not been able to come up with a concrete counterexample (nor any confirmation that my hunch is correct). Could you shed some light on this issue?
> With kind regards,
> Pieter
