Dear Pieter,
Oct. 1st, 2014 04:08 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
posicthe 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).
The quasi-coherent sheaves over X and U are simply modules over A and B, so in the sequel I will be talking about complexes of modules over these rings. The complex we want to consider is the unbounded complex of A-modules C_A with all the terms being copies of the free A-module A and all the differentials vanishing. This is not a homotopy injective complex, of course, so let us construct a homotopy injective resolution for it. For this purpose, it suffices to pick an injective right resolution I_A of the A-module A and consider the direct product of the complexes of A-modules I_A[n] over all the integers n\in Z. Denote this complex by J_A; then there is a natural quasi-isomorphism of complexes of A-modules C_A \to J_A.
Note that the direct sum of the complexes I_A[n] over all n\in Z would be also a complex of injective A-modules quasi-isomorphic to C_A. However, unlike J_A, this direct sum complex is not homotopy injective (generally speaking).
Restricting the complex J_A, viewed as a complex of quasi-coherent sheaves on X, to the open subscheme U, one obtains the complex of B-modules J_A[f^{-1}]. On the other hand, applying the above construction to the ring B instead of A we get a complex of B-modules J_B equal to the direct product of the complexes I_B[n] over all n\in Z. We can use any injective right resolution of the free B-module B in the role of I_B in this construction, so let us take I_B = I_A[f^{-1}} (which is a complex of injective B-modules by Hartshorne's theorem).
Now there is a natural morphism of complexes of injective B-modules J_A[f^{-1}] \to J_B (which is not an isomorphism, because the localization does not preserve infinite products). This morphism is a quasi-isomorphism, though (both complexes being quasi-isomorphic to C_B, as both the localization and the infinite product of modules are exact functors). The complex J_B is clearly a homotopy injective complex of B-modules. It is claimed that the complex J_A[f^{-1}] is not.
To prove as much, it suffices to pick some left exact functor F on the category of B-modules, with its right derived functor RF defined on the unbounded derived category D(B-mod) using homotopy injective resolutions, and show that applying the functor F termwise to the complexes J_A[f^{-1}] and J_B produces complexes with different cohomology. Recall that the affine scheme X = Spec A was constructed as an infinitesimal thickening of the scheme Y = Spec R, and similarly U = Spec B is an infinitesimal thickening of V = Spec R[f^{-1}]. Let i_B: V \to U denote the related closed embedding of schemes; the left exact functor that we will apply is the restriction with scheme-theoretic supports i_B^!.
In terms of modules, the functor F_B assigns to a B-module N its maximal submodule annihilated by the action of \eps (which is a module over R[f^{-1}]). The functor F_A: A-mod \to R-mod is defined similarly. Note that the functors F preserve the infinite products of modules (and the infinite direct sums, too). They also commute with the localizations.
It should not be difficult to compute that the complex F_A(I_A) has cohomology modules isomorphic to R at all the nonnegative cohomological degrees m. Consequently, the cohomology of the complex F_A(J_A) is the direct product of copies of R indexed by the nonnegative integers m, sitting at every cohomological degree n\in Z. Accordingly, the cohomology of F_B(J_A[f^{-1}]) = F_A(J_A)[f^{-1}] is the localization of the infinite product, (\prod_m R)[f^{-1}], sitting at every cohomological degree n. On the other hand, the cohomology of F_B(J_B) is the infinite product of localizations, \prod_m R[f^{-1}], sitting at every cohomological degree.
Since the natural map (\prod_m R)[f^{-1}] \to \prod_m R[f^{-1}] is not, generally speaking, an isomorphism (e.g., for R = Z and f any noninvertible integer), the cohomology of the two complexes differ. So the map J_A[f^{-1}] \to J_B cannot be a homotopy equivalence of complexes of A-modules. This map being their quasi-isomorphism, they cannot be both homotopy injective complexes.
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).
The quasi-coherent sheaves over X and U are simply modules over A and B, so in the sequel I will be talking about complexes of modules over these rings. The complex we want to consider is the unbounded complex of A-modules C_A with all the terms being copies of the free A-module A and all the differentials vanishing. This is not a homotopy injective complex, of course, so let us construct a homotopy injective resolution for it. For this purpose, it suffices to pick an injective right resolution I_A of the A-module A and consider the direct product of the complexes of A-modules I_A[n] over all the integers n\in Z. Denote this complex by J_A; then there is a natural quasi-isomorphism of complexes of A-modules C_A \to J_A.
Note that the direct sum of the complexes I_A[n] over all n\in Z would be also a complex of injective A-modules quasi-isomorphic to C_A. However, unlike J_A, this direct sum complex is not homotopy injective (generally speaking).
Restricting the complex J_A, viewed as a complex of quasi-coherent sheaves on X, to the open subscheme U, one obtains the complex of B-modules J_A[f^{-1}]. On the other hand, applying the above construction to the ring B instead of A we get a complex of B-modules J_B equal to the direct product of the complexes I_B[n] over all n\in Z. We can use any injective right resolution of the free B-module B in the role of I_B in this construction, so let us take I_B = I_A[f^{-1}} (which is a complex of injective B-modules by Hartshorne's theorem).
Now there is a natural morphism of complexes of injective B-modules J_A[f^{-1}] \to J_B (which is not an isomorphism, because the localization does not preserve infinite products). This morphism is a quasi-isomorphism, though (both complexes being quasi-isomorphic to C_B, as both the localization and the infinite product of modules are exact functors). The complex J_B is clearly a homotopy injective complex of B-modules. It is claimed that the complex J_A[f^{-1}] is not.
To prove as much, it suffices to pick some left exact functor F on the category of B-modules, with its right derived functor RF defined on the unbounded derived category D(B-mod) using homotopy injective resolutions, and show that applying the functor F termwise to the complexes J_A[f^{-1}] and J_B produces complexes with different cohomology. Recall that the affine scheme X = Spec A was constructed as an infinitesimal thickening of the scheme Y = Spec R, and similarly U = Spec B is an infinitesimal thickening of V = Spec R[f^{-1}]. Let i_B: V \to U denote the related closed embedding of schemes; the left exact functor that we will apply is the restriction with scheme-theoretic supports i_B^!.
In terms of modules, the functor F_B assigns to a B-module N its maximal submodule annihilated by the action of \eps (which is a module over R[f^{-1}]). The functor F_A: A-mod \to R-mod is defined similarly. Note that the functors F preserve the infinite products of modules (and the infinite direct sums, too). They also commute with the localizations.
It should not be difficult to compute that the complex F_A(I_A) has cohomology modules isomorphic to R at all the nonnegative cohomological degrees m. Consequently, the cohomology of the complex F_A(J_A) is the direct product of copies of R indexed by the nonnegative integers m, sitting at every cohomological degree n\in Z. Accordingly, the cohomology of F_B(J_A[f^{-1}]) = F_A(J_A)[f^{-1}] is the localization of the infinite product, (\prod_m R)[f^{-1}], sitting at every cohomological degree n. On the other hand, the cohomology of F_B(J_B) is the infinite product of localizations, \prod_m R[f^{-1}], sitting at every cohomological degree.
Since the natural map (\prod_m R)[f^{-1}] \to \prod_m R[f^{-1}] is not, generally speaking, an isomorphism (e.g., for R = Z and f any noninvertible integer), the cohomology of the two complexes differ. So the map J_A[f^{-1}] \to J_B cannot be a homotopy equivalence of complexes of A-modules. This map being their quasi-isomorphism, they cannot be both homotopy injective complexes.
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
