Cohomology of presheaves with oriented weak transfers
Author:
Joseph Ross
Journal:
J. Algebraic Geom. 26 (2017), 557-602
DOI:
https://doi.org/10.1090/jag/684
Published electronically:
November 23, 2016
MathSciNet review:
3647793
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Abstract |
References |
Additional Information
Abstract: Over a field of characteristic zero, we establish the homotopy invariance of the Nisnevich cohomology of homotopy invariant presheaves with oriented weak transfers, and the agreement of Zariski and Nisnevich cohomology for such presheaves. This generalizes a foundational result in Voevodsky’s theory of motives. The main idea is to find explicit smooth representatives of the correspondences which provide the input for Voevodsky’s cohomological architecture.
References
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- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- J. Heller, M. Voineagu, and P. A. Østvær, Equivariant cycles and cancellation for motivic cohomology, Doc. Math. 20 (2015), 269–332. MR 3398714
- Jens Hornbostel and Serge Yagunov, Rigidity for Henselian local rings and $\Bbb A^1$-representable theories, Math. Z. 255 (2007), no. 2, 437–449. MR 2262740, DOI https://doi.org/10.1007/s00209-006-0049-4
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- Bruno Kahn, Foncteurs de Mackey à réciprocité, http://arxiv.org/abs/1210.7577 (2012).
- Bruno Kahn, Shuji Saito, and Takao Yamazaki, Reciprocity sheaves, I, with two appendices by Kay Rülling, http://arxiv.org/abs/1402.4201 (2014).
- Shane Kelly, Triangulated categories of motives in positive characteristic, http://arxiv.org/abs/1305.5349v1 (2013).
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- Marc Levine, Slices and transfers, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 393–443. MR 2804260
- Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284
- Fabien Morel, $\Bbb A^1$-algebraic topology over a field, Lecture Notes in Mathematics, vol. 2052, Springer, Heidelberg, 2012. MR 2934577
- Fabien Morel and Vladimir Voevodsky, ${\bf A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224
- Manuel Ojanguren and Ivan Panin, Rationally trivial Hermitian spaces are locally trivial, Math. Z. 237 (2001), no. 1, 181–198. MR 1836777, DOI https://doi.org/10.1007/PL00004859
- I. Panin, Oriented cohomology theories of algebraic varieties, $K$-Theory 30 (2003), no. 3, 265–314. Special issue in honor of Hyman Bass on his seventieth birthday. Part III. MR 2064242, DOI https://doi.org/10.1023/B%3AKTHE.0000019788.33790.cb
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- Ivan Panin and Serge Yagunov, Rigidity for orientable functors, J. Pure Appl. Algebra 172 (2002), no. 1, 49–77. MR 1904229, DOI https://doi.org/10.1016/S0022-4049%2801%2900134-7
- Ivan Panin and Kirill Zainoulline, Variations on the Bloch-Ogus theorem, Doc. Math. 8 (2003), 51–67. MR 2029160
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR 0338129
- Joseph Ross, Cohomology theories with supports, Int. Math. Res. Not. IMRN 8 (2015), 2273–2309. MR 3344668, DOI https://doi.org/10.1093/imrn/rnt352
- Markus Spitzweck, Relations between slices and quotients of the algebraic cobordism spectrum, Homology Homotopy Appl. 12 (2010), no. 2, 335–351. MR 2771593
- Anastasia Stavrova, A note on transfers for non-stable ${K}_1$-functors of classical type, http://arxiv.org/abs/1408.5398v1 (2014).
- Andrei Suslin and Vladimir Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), no. 1, 61–94. MR 1376246, DOI https://doi.org/10.1007/BF01232367
- Vladimir Voevodsky, Cohomological theory of presheaves with transfers, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 87–137. MR 1764200
- Vladimir Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238. MR 1764202
- Mark Edward Walker, Motivic complexes and the K-theory of automorphisms, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 2694778
- Serge Yagunov, Rigidity. II. Non-orientable case, Doc. Math. 9 (2004), 29–40. MR 2054978
- Oscar Zariski, The theorem of Bertini on the variable singular points of a linear system of varieties, Trans. Amer. Math. Soc. 56 (1944), 130–140. MR 11572, DOI https://doi.org/10.1090/S0002-9947-1944-0011572-3
- Oscar Zariski, On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 791–796. MR 95846, DOI https://doi.org/10.1073/pnas.44.8.791
References
- Frédéric Déglise, Orientable homotopy modules, Amer. J. Math. 135 (2013), no. 2, 519–560. MR 3038720, DOI https://doi.org/10.1353/ajm.2013.0019
- Steven Diaz and David Harbater, Strong Bertini theorems, Trans. Amer. Math. Soc. 324 (1991), no. 1, 73–86. MR 986689 (91f:14003), DOI https://doi.org/10.2307/2001496
- Eric M. Friedlander and Andrei Suslin, The spectral sequence relating algebraic $K$-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 6, 773–875 (English, with English and French summaries). MR 1949356 (2004b:19006), DOI https://doi.org/10.1016/S0012-9593%2802%2901109-6
- Eric M. Friedlander and Vladimir Voevodsky, Bivariant cycle cohomology, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 138–187. MR 1764201
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181 (33 \#7330)
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 \#3116)
- J. Heller, M. Voineagu, and P. A. Østvær, Equivariant cycles and cancellation for motivic cohomology, Doc. Math. 20 (2015), 269–332. MR 3398714
- Jens Hornbostel and Serge Yagunov, Rigidity for Henselian local rings and $\mathbb {A}^1$-representable theories, Math. Z. 255 (2007), no. 2, 437–449. MR 2262740 (2008b:14027), DOI https://doi.org/10.1007/s00209-006-0049-4
- Florian Ivorra and Kay Rülling, K-groups of reciprocity functors, http://arxiv. org/abs/1209.1217v3 (2014).
- Bruno Kahn, Foncteurs de Mackey à réciprocité, http://arxiv.org/abs/1210.7577 (2012).
- Bruno Kahn, Shuji Saito, and Takao Yamazaki, Reciprocity sheaves, I, with two appendices by Kay Rülling, http://arxiv.org/abs/1402.4201 (2014).
- Shane Kelly, Triangulated categories of motives in positive characteristic, http://arxiv.org/abs/1305.5349v1 (2013).
- Marc Levine, Chow’s moving lemma and the homotopy coniveau tower, $K$-Theory 37 (2006), no. 1-2, 129–209. MR 2274672 (2007m:19001), DOI https://doi.org/10.1007/s10977-006-0004-5
- Marc Levine, Slices and transfers, Doc. Math. 2010, Extra volume: Andrei A. Suslin sixtieth birthday, 393–443. MR 2804260 (2012d:14033)
- Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284 (2007e:14035)
- Fabien Morel, $\mathbb {A}^1$-algebraic topology over a field, Lecture Notes in Mathematics, vol. 2052, Springer, Heidelberg, 2012. MR 2934577
- Fabien Morel and Vladimir Voevodsky, $\textbf {A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224 (2002f:14029)
- Manuel Ojanguren and Ivan Panin, Rationally trivial Hermitian spaces are locally trivial, Math. Z. 237 (2001), no. 1, 181–198. MR 1836777 (2002e:13022), DOI https://doi.org/10.1007/PL00004859
- I. Panin, Oriented cohomology theories of algebraic varieties, Special issue in honor of Hyman Bass on his seventieth birthday. Part III, $K$-Theory 30 (2003), no. 3, 265–314. MR 2064242 (2005f:14043), DOI https://doi.org/10.1023/B%3AKTHE.0000019788.33790.cb
- I. Panin, Homotopy invariance of the sheaf $W_\textrm {Nis}$ and of its cohomology, Quadratic forms, linear algebraic groups, and cohomology, Dev. Math., vol. 18, Springer, New York, 2010, pp. 325–335. MR 2648736 (2011g:14056), DOI https://doi.org/10.1007/978-1-4419-6211-9_21
- Ivan Panin and Serge Yagunov, Rigidity for orientable functors, J. Pure Appl. Algebra 172 (2002), no. 1, 49–77. MR 1904229 (2003e:14014), DOI https://doi.org/10.1016/S0022-4049%2801%2900134-7
- Ivan Panin and Kirill Zainoulline, Variations on the Bloch-Ogus theorem, Doc. Math. 8 (2003), 51–67 (electronic). MR 2029160 (2005f:14037)
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129 (49 \#2895)
- Joseph Ross, Cohomology theories with supports, Int. Math. Res. Not. IMRN 8 (2015), 2273–2309. MR 3344668, DOI https://doi.org/10.1093/imrn/rnt352
- Markus Spitzweck, Relations between slices and quotients of the algebraic cobordism spectrum, Homology, Homotopy Appl. 12 (2010), no. 2, 335–351. MR 2771593 (2011m:14033)
- Anastasia Stavrova, A note on transfers for non-stable ${K}_1$-functors of classical type, http://arxiv.org/abs/1408.5398v1 (2014).
- Andrei Suslin and Vladimir Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), no. 1, 61–94. MR 1376246 (97e:14030), DOI https://doi.org/10.1007/BF01232367
- Vladimir Voevodsky, Cohomological theory of presheaves with transfers, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 87–137. MR 1764200
- Vladimir Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238. MR 1764202
- Mark Edward Walker, Motivic complexes and the K-theory of automorphisms, Thesis (Ph.D.)–University of Illinois at Urbana-Champaign, 1996, ProQuest LLC, Ann Arbor, MI. MR 2694778
- Serge Yagunov, Rigidity. II. Non-orientable case, Doc. Math. 9 (2004), 29–40. MR 2054978 (2005h:14055)
- Oscar Zariski, The theorem of Bertini on the variable singular points of a linear system of varieties, Trans. Amer. Math. Soc. 56 (1944), 130–140. MR 0011572 (6,186a)
- Oscar Zariski, On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 791–796. MR 0095846 (20 \#2344)
Additional Information
Joseph Ross
Affiliation:
Department of Mathematics, University of Southern California, 3620 South Vermont Avenue, Los Angeles, California 90089-0894
MR Author ID:
891309
Email:
joseph.ross@gmail.com
Received by editor(s):
February 26, 2015
Received by editor(s) in revised form:
November 8, 2015
Published electronically:
November 23, 2016
Article copyright:
© Copyright 2016
University Press, Inc.