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International Conference on Bousfield Classes form a set: in Memory of Testusuke Ohkawa

BouCla 2015: Bousfield Classes and Ohkawa's Theorem pp 221–243Cite as

Some Observations About Motivic Tensor Triangulated Geometry over a Finite Field

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

Abstract

We give a brief introduction to tensor triangulated geometry, a brief introduction to various motivic categories, and then make some observations about the conjectural structure of the tensor triangulated spectrum of the Morel–Voevodsky stable homotopy category over a finite field.

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Notes

  1. 1.

    An object a in a \(\otimes \)-category is called strongly dualisable if there exists an object Da such that \(a \otimes -\) is left adjoint to \((Da) \otimes -\). A \(\otimes \)-category is called rigid if every object is strongly dualisable.

  2. 2.

    Recall that \(\text {Proj}\) of a non-negatively graded ring is the set of those proper homogeneous prime ideals which don’t contain all elements of positive degree. There is no such exclusion in the definition of \(\text {Spec}^h\).

  3. 3.

    This assumption does not appear in [51, Def. 4.2], but it seems it should be there.

  4. 4.

    In [51, Def. 4.2], the definition uses coherent abelian subcategories, which, as Oliver Braunling pointed out to me, are just thick subcategories containing a zero object.

  5. 5.

    An object a in an abelian category is called simple if every monomorphism \(b \rightarrow a\) (in the categorical sense) is either zero or an isomorphism. An object is semisimple if it is a sum of simple objects. An abelian category is semisimple if all of its objects are semisimple.

  6. 6.

    An adequate equivalence relation is a family of equivalence relations \(\sim _X\) on the \(\mathscr {Z}^*(X)\) which satisfy three properties, which essentially require that composition as defined above is well-defined, [53]. In short, pullback, pushforward, and intersection are well-defined.

  7. 7.

    Two cycles \(\alpha , \alpha ' \in \mathscr {Z}^i(X)\) are rationally equivalent if there is a cycle \(\beta \in \mathscr {Z}^i(\mathbb {P}^1_X)\) such that \(\beta \cdot [\{0\} \times X] = \alpha \) and \(\beta \cdot [\{\infty \} \times X] = \alpha '\). Rational equivalence is the coarsest equivalence relation.

  8. 8.

    A cycle \(\alpha \) is homologically equivalent to zero if its image under the cycle class map \(\mathscr {Z}^i(X) \rightarrow H^{2i}(X)\) is zero, for some prechosen Weil cohomology theory, such as étale cohomology \(H^i(X) = H_{et}^{2i}(X, \mathbb {Q}_l(i))\) for some \(l \not \mid \mathsf {char}\ p\). In other words, \(A_{\hom }^i(X)\) is the image of the cycle class map \(A_{\hom }^i(X) = \text {im}(\mathscr {Z}^i(X) \rightarrow H^{2i}(X))\).

  9. 9.

    Numerical equivalence is the coarsest equivalence relation which makes the intersection product \(A_{\text {num}}^i(X) \otimes A_{\text {num}}^{d - i}(X) \rightarrow A_{\text {num}}^d(X)\) nondegenerate, where \(d = \dim X\). That is, \(\alpha \in A^i_{\text {rat}}(X)\) is numerically equivalent to zero if and only if \(\alpha \cdot \beta = 0\) for all \(\beta \in A^{d - i}_{\text {rat}}(X)\).

  10. 10.

    The category \(\text {SmCor}(S)\) can be defined for any noetherian separated scheme S, but we have not mentioned this construction because branch points make composition more subtle. For a more general S, the group \(\hom _{\text {SmCor}(S)}(X, Y)\) is only a proper subgroup of the free abelian group we describe, since branches of X introduce an ambiguity in the composition of some cycles. In fact, it can be defined as the largest subgroup for which composition is well-defined, as soon as the notion of composition has been formalised appropriately, [33, Chap. 2].

  11. 11.

    A Nisnevich distinguished square is a cartesian square \(\underset{U}{{\mathop {\downarrow }\limits ^{U{\times }_XV}}}\underset{\rightarrow }{{\mathop {\phantom {\downarrow }}\limits ^{\rightarrow }}}\underset{X}{{\mathop {\downarrow }\limits ^{V}}}\) such that \(U \rightarrow X\) is an open immersion, \(V \rightarrow X\) is étale, and \((X {-} U)_{red}{\times }_X V \rightarrow (X {-} U)_{red}\) is an isomorphism.

  12. 12.

    In [58] Voevodsky only uses Zariski distinguished squares, i.e., those squares for which j is also an open immersion. However, [58, Thm. 3.1.12] implies that, at least when the base is a perfect field, the Zariski and Nisnevich versions produce the same category. On the other hand, it is the Nisnevich descent property which is often used in most of the proofs in [58]. Nisnevich locally, closed immersions of smooth schemes look like zero sections of trivial affine bundles, cf. [47, Proof of Lemma 2.28].

  13. 13.

    In his MathSciNet review, Röndigs attributes this quote to Suslin.

  14. 14.

    The projective model structure is the model structure for which a morphism is a fibration (resp. weak equivalence) if and only if it is a fibration (resp. weak equivalence) of symmetric \(S^1\)-spectra after evaluation on every \(X \in \text {Sm}(S)\).

  15. 15.

    The Yoneda functor produces a presheaf of pointed sets \(\hom _{\text {Sm}(S)}(-, \mathbb {P}^1_S)\), and then working schemewise, we associated to every pointed set its induced pointed simplicial set, and from there its associated symmetric \(S^1\)-spectrum.

  16. 16.

    The projective model structure has the nice property that representable presheaves are cofibrant, however, by representable we mean the image of a scheme with a disjoint base point, cf. Footnote 17. Presheaves which are the image of pointed schemes whose basepoint is not disjoint are not in general cofibrant. Hence, we need to take some cofibrant model. For example, the pushout of \(S_+ \wedge \Delta ^1_+ {\mathop {\leftarrow }\limits ^{0}} S_+ {\mathop {\rightarrow }\limits ^{\infty }} \mathbb {P}^1_+\) is a cofibrant model for \((\mathbb {P}^1_S, \infty )\), where \(\Delta ^1_+\) is the constant presheaf corresponding to the simplicial interval. This is exactly the analogue of the complex \(([S] {\mathop {\rightarrow }\limits ^{\infty }} [\mathbb {P}^1_S])\) which we could have used to define \(\mathbb {Z}(1)\) in \(DM_{gm}^{eff}(S)\).

  17. 17.

    We first equip any S-scheme X with a base point by replacing it with \(X_+ = X \sqcup S\). Then using the procedure described in Footnote 15 we get a functor \(\text {Sm}(S) \rightarrow Sp_{S^1}(S)\), which we compose with the canonical functor \(Sp_{S^1}(S) \rightarrow Sp_TSp_{S^1}(S)\).

  18. 18.

    This is actually the composition of the constant presheaf functor \(Sp_{S^1} \rightarrow Sp_{S^1}(S)\) from symmetric \(S^1\)-spectra to presheaves of symmetric \(S^1\)-spectra, and the canonical functor \(Sp_{S^1}(S) \rightarrow Sp_TSp_{S^1}(S)\).

  19. 19.

    There are some weird sign and bracket conventions in [23]. In [23], \(\hat{\eta }\) and \(\{a\}\) are used to denote what [46] writes as \(\eta \) and [a]. On the other hand, [23] use \(\eta \) and [a] for the elements in \(K^{W} {\mathop {=}\limits ^{def}} K^{MW} / h\) corresponding to our \(\eta \) and \(-[a]\).

  20. 20.

    In fact, the Conservativity Conjecture described in [5] is one of the major problems in the area. It is a kind of triangulated analogue of the Hodge conjecture.

  21. 21.

    They are in bijection with the homogeneous prime ideals of \(K^{MW}(\mathbb {F}_q)_{red} \cong \mathbb {Z}[\eta ] / (2 \cdot \eta )\). Then we have \(\text {Spec}(\mathbb {Z}[\eta ] / (2 \cdot \eta )) = \text {Spec}(\mathbb {Z}) \cup \text {Spec}(\mathbb {Z}/ 2[\eta ])\). Clearly, apart from \(\text {Spec}(\mathbb {Z}/ 2) = \text {Spec}(\mathbb {Z}) \cap \text {Spec}(\mathbb {Z}/ 2[\eta ])\), the graded ring \(\mathbb {Z}/ 2[\eta ]\) has exactly one other homogeneous prime: \((\eta )\).

References

  1. Artin, M., Mazur, B.: Etale homotopy. Lecture Notes in Mathematics, vol. 100. Springer, Berlin (1986). Reprint of the 1969 original

    Google Scholar 

  2. Aoki, K.: The weight complex functor is symmetric monoidal (2019). Arxiv preprint arXiv:1904.01384

  3. Ayoub, J.: Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II. Astérisque 315(2008), vi+364 (2007)

    Google Scholar 

  4. Ayoub, J.: Note sur les opérations de Grothendieck et la réalisation de Betti. J. Inst. Math. Jussieu 9(2), 225–263 (2010)

    Article  MathSciNet  Google Scholar 

  5. Ayoub, J.: Motive and algebraic cycles: a selection of conjectures and open questions (2015)

    Google Scholar 

  6. Bachmann, T.: On the invertibility of motives of affine quadrics. Doc. Math. 22, 363–395 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Balmer, P.: The spectrum of prime ideals in tensor triangulated categories. J. für Die Reine Und Angew. Math. 2005(588), 149–168 (2005)

    Article  MathSciNet  Google Scholar 

  8. Balmer, P.: Supports and filtrations in algebraic geometry and modular representation theory. Am. J. Math. 1227–1250 (2007)

    Google Scholar 

  9. Balmer, P.: Spectra, spectra, spectra-tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol. 10(3), 1521–1563 (2010)

    Article  MathSciNet  Google Scholar 

  10. Balmer, P.: Tensor triangular geometry. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 85–112. Hindustan Book Agency, New Delhi (2010)

    Google Scholar 

  11. Bloch, S.: Algebraic cycles and higher \(K\)-theory. Adv. Math. 61(3), 267–304 (1986)

    Article  MathSciNet  Google Scholar 

  12. Bondarko, M.V.: Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; voevodsky versus hanamura. J. Inst. Math. Jussieu 8(01), 39–97 (2009)

    Google Scholar 

  13. Bondarko, M.V.: \(\mathbb{Z}[1/p]\)-motivic resolution of singularities. Compos. Math. 147(5), 1434–1446 (2011)

    Article  MathSciNet  Google Scholar 

  14. Balmer, P., Sanders, B.: The spectrum of the equivariant stable homotopy category of a finite group (2015)

    Google Scholar 

  15. Cisinski, D.C., Déglise, F.: Triangulated categories of mixed motives (2012). Arxiv preprint arXiv:0912.2110

  16. Cisinski, D.C.: Descente par éclatements en \(K\)-théorie invariante par homotopie. Ann. Math. (2), 177(2), 425–448 (2013)

    Google Scholar 

  17. Cisinski, D.C.: Conversation (2016)

    Google Scholar 

  18. Dugger, D., Isaksen, D.C.: The motivic Adams spectral sequence. Geom. Topol. 14(2), 967–1014 (2010)

    Article  MathSciNet  Google Scholar 

  19. Dugger, D., Isaksen, D.C.: Motivic Hopf elements and relations. New York J. Math. 19, 823–871 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Deninger, C., Murre, J.: Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422, 201–219 (1991)

    Google Scholar 

  21. Friedlander, E.M.: Fibrations in etale homotopy theory. Inst. Hautes Études Sci. Publ. Math. 42, 5–46 (1973)

    Article  MathSciNet  Google Scholar 

  22. Friedlander, E.M., Voevodsky, V.: Bivariant cycle cohomology. In: Cycles, Transfers, and Motivic uomology Theories. Annals of Mathematics Studies, vol. 143, pp. 138–187. Princeton University Press, Princeton, NJ (2000)

    Google Scholar 

  23. Gille, S., Scully, S., Zhong, C.: Milnor-Witt \(K\)-groups of local rings. Adv. Math. 286, 729–753 (2016)

    Google Scholar 

  24. Hu, P., Kriz, I., Ormsby, K.: Remarks on motivic homotopy theory over algebraically closed fields. J. K-Theory 7(1), 55–89 (2011)

    Google Scholar 

  25. Heller, J., Ormsby, K.: Primes and fields in stable motivic homotopy theory (2016)

    Google Scholar 

  26. Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory II. Ann. Math. 148(1), 1–49 (1998)

    Google Scholar 

  27. Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press, Demand (2006)

    Google Scholar 

  28. Ivorra, F.: Réalisation \(l\)-adique des motifs mixtes. Comptes Rendus Math. Acad. Sci. Paris 342(7), 505–510 (2006)

    Article  MathSciNet  Google Scholar 

  29. Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107(3), 447–452 (1992)

    Article  MathSciNet  Google Scholar 

  30. Jardine, J.F.: Motivic symmetric spectra. Doc. Math. 5, 445–553 (electronic) (2000)

    Google Scholar 

  31. Joachimi, R.: Thick ideals in equivariant and motivic stable homotopy categories. To appear in this proceedings

    Google Scholar 

  32. Kahn, B.: Algebraic \(K\)-theory, algebraic cycles and arithmetic geometry. Handbook of K-Theory, pp. 351–428. Springer, Berlin (2005)

    Google Scholar 

  33. Kelly, S.: Triangulated categories of motives in positive characteristic (2013)

    Google Scholar 

  34. Kleiman, S.: The standard conjectures. Motives (Seattle, WA, 1991). Proceedings of Symposia in Pure Mathematics Part 1, vol. 55, pp. 3–20. A. M. S., Providence, RI

    Google Scholar 

  35. Kock, J., Pitsch, W.: Hochster duality in derived categories and point-free reconstruction of schemes (2013)

    Google Scholar 

  36. Kashiwara, M., Schapira, P.: Categories and Sheaves. vol. 332. Springer Science & Business Media (2005)

    Google Scholar 

  37. Kelly, S., Saito, S.: Weight homology of motives. Int. Math. Res. Not. 13, 3938–3984 (2017)

    Google Scholar 

  38. Lam, T.Y.: Introduction to quadratic forms over fields. Graduate Studies in Mathematics, vol. 67. American Mathematical Society, Providence, RI (2005)

    Google Scholar 

  39. Levine, M.: A comparison of motivic and classical stable homotopy theories. J. Topol. 7(2), 327–362 (2014)

    Article  MathSciNet  Google Scholar 

  40. Levine, M., Yang, Y., Gufang, Z.: Algebraic elliptic cohomology theory and flops, I (2013)

    Google Scholar 

  41. Milnor, J.: Algebraic \(K\)-theory and quadratic forms. Invent. Math. 9, 318–344 (1969/1970)

    Google Scholar 

  42. Morel, F.: An introduction to \({\bf A}^1\)-homotopy theory. In: Contemporary Developments in Algebraic \(K\)-Theory. ICTP Lecture Notes, vol. XV, pp. 357–441 (electronic). Abdus Salam International Centre for Theoretical Physics, Trieste (2004)

    Google Scholar 

  43. Morel, F.: On the motivic \(\pi _0\) of the sphere spectrum. In: Axiomatic, Enriched and Motivic Homotopy Theory. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 131, pp. 219–260. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  44. Morel, F.: Sur les puissances de l’idéal fondamental de l’anneau de Witt. Comment. Math. Helv. 79(4), 689–703 (2004)

    Article  MathSciNet  Google Scholar 

  45. Morel, F.: Rationalized motivic sphere spectrum (2006)

    Google Scholar 

  46. Fabien, M.: \(\mathbb{A}^1\)- algebraic topology over a field. Lecture Notes in Mathematics, vol. 2052. Springer, Heidelberg (2012)

    Google Scholar 

  47. Morel, F., Voevodsky, V.: \({\bf A}^1\)-homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math. 90(2001), 45–143 (1999)

    Google Scholar 

  48. Mazza, C., Voevodsky, V., Weibel, C.: Lecture notes on motivic cohomology. Clay Mathematics Monographs, vol. 2 American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2006)

    Google Scholar 

  49. Ormsby, K.M., Østvær, P.A.: Motivic Brown-Peterson invariants of the rationals. Geom. Topol. 17(3), 1671–1706 (2013)

    Google Scholar 

  50. Ormsby, K.M., Østvær, P.A.: Stable motivic \(\pi _1\) of low-dimensional fields. Adv. Math. 265, 97–131 (2014)

    Google Scholar 

  51. Peter, T.J.: Prime ideals of mixed Artin-Tate motives. J. K-Theory 11(2), 331–349 (2013)

    Article  MathSciNet  Google Scholar 

  52. Robalo, M.: \(K\)-theory and the bridge from motives to noncommutative motives. Adv. Math. 269, 399–550 (2015)

    Article  MathSciNet  Google Scholar 

  53. Samuel, P.: Relations d’équivalence en géométrie algébrique. In: Proceedings of the International Congress of Mathematicians 1958, pp. 470–487. Cambridge University Press, New York (1960)

    Google Scholar 

  54. Scholl, A.J.: Classical motives. In: Motives (Seattle, WA, 1991). Proceedings of Symposia in Pure Mathematics, vol. 55, pp. 163–187. American Mathematical Society, Providence, RI (1994)

    Google Scholar 

  55. Suslin, A.A.: Higher Chow groups and etale cohomology. In: Cycles, Transfers, and Motivic Homology Theories. Annals of Mathematics Studies, vol. 143, pp. 239–254. Princeton University Press, Princeton, NJ (2000)

    Google Scholar 

  56. Thornton, R.: The homogeneous spectrum of Milnor-Witt \(K\)-theory (2015)

    Google Scholar 

  57. Voevodsky, V.: A\(^1\)-homotopy theory. In: Proceedings of the International Congress of Mathematicians, vol. I. number Extra Vol. I, pp. 579–604 (electronic) Berlin (1998)

    Google Scholar 

  58. Voevodsky, V.: Triangulated categories of motives over a field. In: Cycles, Transfers, and Motivic Homology Theories. Annals of Mathematics Studies, vol. 143, pp. 188–238. Princeton University Press, Princeton, NJ (2000)

    Google Scholar 

  59. Voevodsky, V.: Cancellation theorem. Doc. Math. (Extra volume: Andrei A. Suslin sixtieth birthday):671–685, 2010

    Google Scholar 

  60. Wilson, G.M., Østvær, P.A.: Two-complete stable motivic stems over finite fields (2016)

    Google Scholar 

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Acknowledgements

I thank the organisers of the conference “Bousfield classes form a set: a workshop in memory of Tetsusuke Ohkawa” for the invitation to speak which led me to think about these things, and also for having organised such an interesting conference. I also thank Paul Balmer, Jens Hornbostel, and Denis-Charles Cisinski for interesting discussions about potential future work, Marc Hoyois for discussions about the étale homotopy type, and Jeremiah Heller and Kyle Ormsby for pointing out that an “elementary fact” I was using in the proof of Proposition 4.5 is actually a combination of theorems of Ayoub, Balmer, Gabber, and Riou.

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  1. Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

    Shane Kelly

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  1. Graduate School of Mathematics, Nagoya University, Nagoya, Aichi, Japan

    Takeo Ohsawa

  2. Department of Computer Science, Nagoya Institute of Technology, Nagoya, Japan

    Norihiko Minami

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Kelly, S. (2020). Some Observations About Motivic Tensor Triangulated Geometry over a Finite Field. 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_7

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