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From Ohkawa to Strong Generation via Approximable Triangulated Categories—A Variation on the Theme of Amnon Neeman’s Nagoya Lecture Series

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Bousfield Classes and Ohkawa's Theorem (BouCla 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 309))

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Abstract

This survey stems from Amnon Neeman’s lecture series at Ohakawa’s memorial workshop. Starting with Ohakawa’s theorem, this survey intends to supply enough motivation, background and technical details to read Neeman’s recent papers on his “approximable triangulated categories” and his \({{\,\mathrm{\mathbf {D}}}}^{b}_{\mathrm {coh}}(X)\) strong generation sufficient criterion via de Jong’s regular alteration, even for non-experts.

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Notes

  1. 1.

    The author also would like to thank Professors Mitsunori Imaoka, Takao Matumoto, Takeo Ohsawa, Katsumi Shimomura, and Masayuki Yamasaki, for coorganizing the workshop.

  2. 2.

    Actually, the author thought even such a short list is exciting.

  3. 3.

    Concerning this sentence, Professor Morava communicated the following thoughts to the author: “When I read it I was reminded of a quotation from the English writer Sir Thomas Browne (from ‘Urn Burial’, in 1658):

    What song the Sirens sang, or what name Achilles assumed when he hid himself among women, though puzzling questions, are not beyond all conjecture...

    I believe understanding the structure of Ohkawa’s set (perhaps by defining something like a topology on it) is very important, not just for homotopy theory but for mathematics in general. An analogy occurs to me, to other very complicated objects (like the Stone-Čech compactification of the rationals or the reals, or maybe the Mandelbrot set) which are very mysterious but can approached as limits of more comprehensible objects. Indeed I wonder if this is what Neeman’s theory of approximable triangulated categories points toward.” 

  4. 4.

    Let us briefly recall the localization in the abelian category setting: [41, III, 1] [43, p. 122, Exer. 9]. Just as we may start with thick triangulated categories for Verdier quotients, which we will see in Remark 2.3 (iii), to localize an abelian category \(\mathcal {A}\) by its full subcategory \(\mathcal {B},\) we start with assuming \(\mathcal {B}\) is a \(\underline{Serre\,\,subcategory}\), i.e.

    $$\begin{aligned} \text {for any exact sequene}\ 0 \rightarrow B' \rightarrow B \rightarrow B'' \rightarrow 0 \ \text {in}\, \mathcal {A},\qquad \left( B\in \mathcal {B}\iff ( B' \in \mathcal {B}\, \text {and}\, B'' \in \mathcal {B}) \right) \end{aligned}$$

    Then the \(\underline{ quotient~ category~ \mathcal {A}/\mathcal {B}, ~in~ the~ sense~ of~ Gabriel, ~Grothendieck, and ~ Serre}\), is of the following form:

    $$\begin{aligned} \begin{aligned} {\text {Ob}} \mathcal {A}/\mathcal {B}:= {\text {Ob}} \mathcal {A}; \qquad {\text {``}\!{{\,\mathrm{Hom}\,}}\!\text {''}}_{\mathcal {A}/\mathcal {B}} ( A, A' ) := \varinjlim _{\underline{A},\underline{A'} \ \text {s.t.}\ A/\underline{A} \in \mathcal {B}, \underline{A'} \in \mathcal {B}} \ {{{\,\mathrm{Hom}}}_{\mathcal {A}}} \left( \underline{A}, A'/\underline{A'} \right) \end{aligned} \end{aligned}$$

    Thus, an element of \({{\,\mathrm{Hom}}}_{\mathcal {A}/\mathcal {B}}( A, A' )\) is of the following form:

    However, if we consider a similar diagram in the setting of derived categories, we may take the homotopy pullback \(\widetilde{\underline{A}}\) as in the following diagram:

    Here, arrows with \(\bigstar \) are local maps, and so, this gives a pair of maps \(( A \xleftarrow {\bigstar } \widetilde{ \underline{A} } \rightarrow A' ),\) which is a typical element in the “\({{\,\mathrm{Hom}\,}}\)” class in the Verdier quotient.

  5. 5.

    WARNING!: In this article, we follow the convention of [111, Def. 1.5.1] [78, 4.5] for a \(\underline{ triangulated~ subcategory}\), which is automatically full by this convention. On the other hand, it is not so in the convention of [131, p. 3,1.1].

  6. 6.

    Verdier quotient does not necessarily have small \({{\,\mathrm{Hom}\,}}\)-sets.

  7. 7.

    If \(\mathcal {T}\) is essentially small, this result also follows immediately from a general result reviewed later in Proposition 4.3.

  8. 8.

    We dot not require \(\mathcal {T}\) to have small coproducts in this definition.

  9. 9.

    This is an adjoint pair of functors between ordinary categories, and we are not considering any triangulated structure.

  10. 10.

    Goes back at least to Verdier.

  11. 11.

    Let us recall the following precursor of this result in the setting of abelian categories, which goes back at least to Gabriel (see also [126, Lem. 3.2]): If an exact functor \(F : \mathcal {A}\rightarrow \mathcal {B}\) between abelian categories has a fully faithful right adjoint G (i.e. the adjunction \(F\circ G \rightarrow {\text {Id}}_{\mathcal {B}}\) is an isomorphism, then \({{\,\mathrm{Ker}\,}}F\) is Serre subcategory of \(\mathcal {A},\) and F induces the following equivalence of abelian categories: \(\mathcal {A}/ {{\,\mathrm{Ker}\,}}F \ \xrightarrow {\cong } \ \mathcal {B},\) where the left hand side is the abelian quotient category in the sense of Gabriel, Grothendieck, and Serre.

  12. 12.

    We do not require \(\mathcal {T}\) to have small coproducts.

  13. 13.

    An arrow above is left adjoint to the arrow below.

  14. 14.

    A Bousfield colocalization functor means its opposite functor is a Bousfield localization functor [54, Def. 3.1.1] [78, 2.8]. WARNING: This terminology is not consistent with that of Bousfield [21] (see [54, Rem. 3.1.4]).

  15. 15.

    Strictly speaking, the definition here is slightly differently from Krause’s, but essentially the same.

  16. 16.

    This claim itself is a special case of Corollary 2.17.

  17. 17.

    This “perfectly generated” condition is used to apply Brown representability (Theorem 2.15) to construct two right adjoints in the recollement.

  18. 18.

    For the precise definition of recollement, consult [15, 1.4].

  19. 19.

    Strictly speaking, this is the telescope conjecture without smash (tensor) product, but coincides with the original Ravenel’s telescope conjecture for \(\mathcal {T}=\mathcal {S}\mathcal {H},\) and more generally for rigidly compactly generated tensor triangulated categories [54, Def. 3.3.2, Def. 3.3.8] (see also Proposition 2.28).

  20. 20.

    For a serious treatment of the definition of “tensor triangulated category,” consult [92].

  21. 21.

    For a concise summary of the academic life of Professor Tetsusuke Ohkawa, see [91] in this proceedings.

  22. 22.

    For short reviews of the Morel-Voevodsky stable homotopy category, cosult [72, 94] for instance.

  23. 23.

    If \(x\in \mathcal {T}\) is strongly dualizable, i.e. rigid, the natural map \(x \rightarrow D^2x\) is an isomorphism [81, Chapter III] [54, Th. A.2.5.(b)].

  24. 24.

    In a rigidly compactly generated tensor triangulated category, any compact object is rigid, for, by Proposition 2.23, any compact object is seen to be isomorphic to a direct summand of a finite extensions of finite coproducts of rigid elements. In particular, in a rigidly compactly generated tensor triangulated category, \(\mathbbm {1}\) is both rigid and compact.

  25. 25.

    Recall in this case \(\mathcal {T}\) becomes distributive, because for any objects \(x_{\lambda }\ (\lambda \in \Lambda ), y, z\) in \(\mathcal {T},\) \( {{\,\mathrm{Hom}\,}}\left( ( \oplus _{\lambda } x_{\lambda } ) \otimes y, z \right) \cong {{\,\mathrm{Hom}\,}}\left( \oplus _{\lambda } x_{\lambda }, \underline{{{\,\mathrm{Hom}\,}}}(y,z) \right) \cong \prod _{\lambda } {{\,\mathrm{Hom}\,}}\left( x_{\lambda }, \underline{{{\,\mathrm{Hom}\,}}}(y,z) \right) \cong \prod _{\lambda } {{\,\mathrm{Hom}\,}}( x_{\lambda }\otimes y, z ) \cong {{\,\mathrm{Hom}\,}}\left( \oplus _{\lambda } x_{\lambda }\otimes y, z \right) .\)

  26. 26.

    Conjecture 2.32 should be taken more seriously. In fact, Professor Peter May is very glad to see Conjecture 2.32 is advertised here.

  27. 27.

    Our presentation of “supports” in this definition and next proposition is somewhat different from those given in [2, Proof of Th. 4.12] [61, App. A], but the author hopes this would be more transparent to the reader.

  28. 28.

    So, should had been known to Verdier.

  29. 29.

    Let us recall the following precursor of this result in the setting of abelian category of quasi-coherent sheaves, which should go back at least to Gabriel (see e.g. [126, In the proof of Prop. 3.1]): \( {{\,\mathrm{QCoh}\,}}(X) \big / {{\,\mathrm{QCoh}}}_Z (X) \ \xrightarrow [\cong ]{\overline{j^*}} \ {{\,\mathrm{QCoh}\,}}(U), \) where the left hand side is the abelian quotient category in the sense of Gabriel, Grothendieck, and Serre.

  30. 30.

    Unlike Theorem 2.37 stated under the noetherian assumption, (14) is stated under more general quasicompact, quasiseparated assumption. Therefore, in this equality \(\left( {{{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}} \right) _Z (X) := \left\{ Y \in {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X) \ \mid \ {{\,\mathrm{Supp}\,}}Y \subseteq Z \right\} = \mathrm {Ker}\ {\mathbf {L}}j^*,\) we may not replace \({{\,\mathrm{Supp}\,}}\) with \({{\,\mathrm{supp}\,}}.\) In fact, without the noetherian hypothesis, Theorem 2.37 becomes very bad as was shown in [107]. The author is grateful to Professor Neeman for this reference.

  31. 31.

    Such a property is not usually satisfied for general triangulated categories. So, most effort to generalize the Hopkins–Smith theorem for a general triangulated category \(\mathcal {T}\) aim at a classification of thick (tensor) ideals of \(\mathcal {T}^c.\)

  32. 32.

    This telescope conjecture is equivalent to the telescope conjecture without product Conjecture 2.22 via Propositions 2.28 and 2.29.

  33. 33.

    A trend here is to apply the higher algebra technique of Lurie [85, 86] to understand chromatic phenomena [13, 133], where the latter contains a concise review of higher algebra technology. Different kinds of applications of Lurie’s higher algebra technique can be seen in [89, 90].

  34. 34.

    This splitting conjecture implies, for any p-completed finite spectrum F and any infinite subset \(\{ n_i \}_{i=1}^{\infty } \subseteq \mathrm {N},\) the natural map \(F \rightarrow \prod _{i=1}^{\infty } L_{K(n_i)}F\) is split injective. For this and much more, consult [9, 51].

  35. 35.

    It appears that [97, p. 4, Corollary] should be modified as in (27).

  36. 36.

    Strickland’s theorem for \(G=\mathbb {Z}/2\) has recently been generalized to arbitrary finite group G by Balmer–Sanders [8].

  37. 37.

    See Defjnition 4.22 for this concept.

  38. 38.

    Or, researchers might prefer “\(\heartsuit \)-felt” \({{\,\mathrm{\mathbf {D}}}}^{b}_{\mathrm {coh}}(X) \cong \mathcal {D}^b( {{\,\mathrm{Coh}\,}}(X) )\) (although separated, not mere quasi-separated, assumption is needed for this equivalence) over simply formal \({{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}(X) \cong {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X)^c\)...

  39. 39.

    Having arbitrary small coproducts was an indispensable assumption for Brown representability and Bousfield localization (Theorem 2.15, Corollary 2.16).

  40. 40.

    If we apply (29) in order to obtain the isomorphism (32), we must require the extra “separated” assumption, for then we should also use the isomorphism:

    $$\begin{aligned} {{\,\mathrm{\mathbf {D}}}}^{b}_{\mathrm {coh}}(X) = \mathcal {D}^b( {{\,\mathrm{Coh}\,}}(X) ), \end{aligned}$$
    (32)

    which requires the “separated” assumption of X. This fact, and the above approach to use (30) was communicated to the author by Professor Neeman.

  41. 41.

    Professor Takeo Ohsawa is the AMS Stefan Bergman Prize 2014 recipient. His survey paper [114] in this proceedings is a concise summary of his work for which this prize was awarded. It was his Bergman Prize money which enabled us to invite distinguished lecturers to Ohkawa’s memorial conference at Nagoya University in the summer of 2015. Takeo Ohsawa was also Tetsusuke Ohkawa’s highschool classmate at Kanazawa University High School in Kanazawa, Japan.

  42. 42.

    X being proper over \({{\,\mathrm{Spec}\,}}(\mathbb {C})\) implies (as part of the definition of properness) that it is separated, hence \({{\,\mathrm{\mathbf {D}}}}^b ({{\,\mathrm{Coh}\,}}(X)) = {{\,\mathrm{\mathbf {D}}}}^{b}_{\mathrm {coh}}(X).\) Hence, these two isomorphisms are trivial consequences of the isomorphism \(\phi ^* : {{\,\mathrm{Coh}\,}}(X) \xrightarrow {\cong } {{\,\mathrm{Coh}\,}}(X^{\text {an}}).\) These two isomorphism are supplied just for reader’s information.

  43. 43.

    Of course, there are many other mathematical approaches to physics. For instance, some of Costello’s approach to quantum field theory via Lurie’s higher algebra [85, 86] point of view are touched upon in Matsuoka’s surveys [89, 90] in this proceedings.

  44. 44.

    Theorem 4.5 is reduced to Theorem 2.33 for \({{\,\mathrm{QCoh}\,}}(X) \cong {\text {Ind}} {{\,\mathrm{Coh}\,}}(X)\) under the Noetherian hypothesis [84, Lem. 3.9]. See also [25, p. 2] [121].

  45. 45.

    As we shall briefly review later, Bridgeland’s space of stability conditions is a kind of moduli space of “enriched hearts” of a triangulated category.

  46. 46.

    WARNING! We had already introduced the same notation \({{\,\mathrm{supp}\,}}\) back in Definition 2.35. However, from Proposition 2.36, Theorem 4.11, these two usages of \({{\,\mathrm{supp}\,}}\) coincide for the most fundamental example of \(\mathcal {K}= {{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}(X).\)

  47. 47.

    Let us recall the following related result in the setting of abelian category of quasi-coherent sheaves, which should go back at least to Gabiriel (see e.g. [126, Prop. 3.1]): \( {{\,\mathrm{Coh}\,}}(X) \big / {{\,\mathrm{Coh}}}_Z (X) \ \xrightarrow [\cong ]{\overline{j^*}} \ {{\,\mathrm{Coh}\,}}(U), \) where the left hand side is the abelian quotient category in the sense of Gabriel, Grothendieck, and Serre.

  48. 48.

    The following interesting historical account on the difficulty of generalizing statements in \({{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}\) (14) (15):

    $$\begin{aligned} {\left\{ \begin{array}{ll} {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X) \big / \left( {{{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}} \right) _Z (X) \ \xrightarrow [\cong ]{\overline{{\mathbf {L}}j^*}} \ {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(U) \\ L = {\mathbf {R}}j_* {\mathbf {L}}j^* = \left( {\mathbf {R}}j_*\mathcal {O}_U \right) \otimes _{\mathcal {O}_X}^{{\mathbf {L}}} - : \ {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X) \ \rightarrow \ {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X) \big / \left( {{{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}} \right) _Z (X) \ \xrightarrow [\cong ]{\overline{{\mathbf {L}}j^*}} \ {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(U) \xrightarrow {{\mathbf {R}}j_*} {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X) \end{array}\right. } \end{aligned}$$

    and the precursor in the setting of abelian categories reviewed in footnote 27:

    $$\begin{aligned} {{\,\mathrm{QCoh}\,}}(X) \big / {{\,\mathrm{QCoh}}}_Z (X) \ \xrightarrow [\cong ]{\overline{j^*}} \ {{\,\mathrm{QCoh}\,}}(U) \end{aligned}$$

    to the setting of \({{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}},\) has been communicated to the author by Professor Neeman:

    ... But the right adjoints \(j_* : {{\,\mathrm{QCoh}\,}}(U) \rightarrow {{\,\mathrm{QCoh}\,}}(X)\) and \({\mathbf {R}}j_* : {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(U) \rightarrow {{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X)\) fail to preserve the finite subcategories \({{\,\mathrm{Coh}\,}}(-)\) and \({{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}(- ).\) For these categories some work is needed. Especially in the case of \({{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}( - )\); for a long time all that was known was that \({\mathbf {L}}j^* : {{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}(X) \rightarrow {{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}(U)\) isn’t surjective on objects, hence the natural map

    $$\begin{aligned} \frac{ {{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}(X) }{ {{\,\mathrm{Ker}\,}}({\mathbf {L}}j^*) } \ \longrightarrow {{\,\mathrm{\mathbf {D}}}}^{\mathrm {perf}}(U) \ \end{aligned}$$

    couldn’t be an equivalence. So the assumption was that this map had to be worthless.

    Thomason’s ingenious insight was that the old counterexamples were a red herring. Up to idempotent completion this map is an equivalence, and in particular induces an isomorphism in higher K-theory. This of course required proof. Thomason gave a rather involved proof, following SGA6, and I noticed that the proof simplifies and generalizes when one uses the methods of homotopy theory.

    It was an amusing role reversal: Thomason, the homotopy theorist, had the brilliant idea but gave a clumsy proof using the techniques of algebraic geometry, while I, the algebraic geometer, simplified the argument with the techniques of homotopy theory.

  49. 49.

    This is the involved part of this proof, for the existence of recollement there requires Brown representability.

  50. 50.

    For the fact that the idempotent completion of a triangulated category has a natural structure of a triangulated category, there is a proof in Balmer–Schlichting [6].

  51. 51.

    The weaker reconstruction just as a topological space was already shown by Thomason (see Theorem 4.11) in the course of his establishing a \({{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X)\) analogue of the Hopkins–Smith theorem (see Theorems 4.12 and 4.10).

  52. 52.

    It was Neeman’s insight to notice surprising usefullness of introducing related categories \(\mathopen {\langle }\mathcal {A}\mathclose {\rangle }_{l}^{[m,n]}\) and \(\overline{\mathopen {\langle }\mathcal {A}\mathclose {\rangle }}_{l}^{[m,n]}\) as well.

  53. 53.

    The author is grateful to Professor Neeman for this reference.

  54. 54.

    Unlike (14) and Theorem 4.17, the general case (where X is quasicompact and quasiseparated) is still open—see [104, Just above Lem. 3.5].

  55. 55.

    There is some subtlety here. See e.g. [108, footnote 4 in Proof of Lem. 5; Sketch 7.19.(i)].

  56. 56.

    In fact, when X is affine, strong generation of \({{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X)\) has been proved by Iyengar and Takahashi [60] under different hypotheses, and using quite different techniques, from Neeman’s Theorem 5.16. And they give examples where strong generation fails; see [60] and references therein.

  57. 57.

    (Gabber’s strengthening [40] of) de Jong’s alteration is now widely used in the Morel–Voevodsky motivic stable homotopy theory. See e.g. [55, 71]. For an introductory review of de Jong’s alteration, consult Oort’s [117] for instance.

  58. 58.

    This proof does not directly use the approximability of \({{\,\mathrm{\mathbf {D}}}}_{\mathrm {qc}}(X),\) the approximability enters only indirectly, when we appeal to Theorem 5.10. What we want to highlight here, following a strong suggestion of Professor Neeman, is “the pivotal role that the homotopy-theoretical ideas of Bousfield, Ohkawa, Hopkins–Smith and many others play in the reduction.”

  59. 59.

    \(\underline{\mathrm{WARNING!}}\) In [109, Proof that Theorem 2.3 follows from Theorem 2.4], Neeman concluded the existence of an honest map \(H \rightarrow {\mathbf {R}}f_* \left( \mathcal {O}_Y \oplus \varSigma \mathcal {O}_Y\right) \) corresponding to (56). However, this is quite problematic, and usually, such an honest map \(H \rightarrow {\mathbf {R}}f_* \mathcal {O}_Y \oplus \varSigma {\mathbf {R}}f_* \mathcal {O}_Y\) does not exist. Thus, some sort of patch is needed. The “patch” presented above was communicated to the author by Professor Neeman, and the author replaced his own patch, which concentrates on \(\widetilde{R}\) (see (63)), with Professor Neeman’s “patch” , which concentrates on \(\widetilde{H}\) (see (63)), because Professor Neeman’s patch delivers a simple message how to read [109, Proof that Theorem 2.3 follows from Theorem 2.4]: just replace H with \(\widetilde{H}\) and pretend the map \(\widetilde{\psi }' : \widetilde{H} \rightarrow {\mathbf {R}}f_* \left( \mathcal {O}_Y \oplus \varSigma \mathcal {O}_Y \right) \) obtained in (63) as our “honest map” \(H \rightarrow {\mathbf {R}}f_* \left( \mathcal {O}_Y \oplus \varSigma \mathcal {O}_Y \right) ,\) and then, just proceed as is written in [109, Proof that Theorem 2.3 follows from Theorem 2.4].

    According to Professor Neeman, this leap and omission of justification is standard. So, the reader is required to come up with this kind of patch spelled out in terms of elementary Bousfield (or Miller’s finite) localization instantaneously at the top of his or her head. Thus, homotopy theoretical insight is prerequisite to read Professor Neeman’s papers!

  60. 60.

    In Neeman’s corresponding calculation [109, 1st paragraph in p. 24], the extension length of \({\text {Coprod}}\) was doubled to be \(2(M+N)\) rather than \(M+N\) given in (73). However, the author does not see such a need, and so, the author opted to present as in (73).

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    Norihiko Minami

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Minami, N. (2020). From Ohkawa to Strong Generation via Approximable Triangulated Categories—A Variation on the Theme of Amnon Neeman’s Nagoya Lecture Series. In: Ohsawa, T., Minami, N. (eds) Bousfield Classes and Ohkawa's Theorem. BouCla 2015. Springer Proceedings in Mathematics & Statistics, vol 309. Springer, Singapore. https://doi.org/10.1007/978-981-15-1588-0_3

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