Strong generators in $\mathbf {D}^{\mathrm {perf}}(X)$ and $\mathbf {D}^b_{\mathrm {coh}}(X)$

Abstract

We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category $\mathbf D^{\rm perf}( X)$ is strongly generated whenever $X$ is a quasicompact, separated scheme, admitting a cover by open affine subsets $\mathrm {Spec}({R_i})$ with each $R_i$ of finite global dimension. We also prove that, for a noetherian scheme $X$ of finite type over an excellent scheme of dimension $\leq 2$, the derived category $\mathbf {D}^b_{\mathrm {coh}}(X)$ is strongly generated. The known results in this direction all assumed equal characteristic; we have no such restriction.

The method is interesting in other contexts: our key lemmas turn out to give a simple proof that, if $f\colon X\rightarrow Y$ is a separated morphism of quasicompact, quasiseparated schemes such that $\mathbf{R} f_*\colon \mathbf{D}_{\mathrm{\mathbf{qc}}}(X) \rightarrow \mathbf{D}_{\mathrm{\mathbf{qc}}}(Y)$ takes perfect complexes to complexes of bounded-below Tor-amplitude, then $f$ must be of finite Tor-dimension.

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    @ARTICLE{Lipman-Neeman07,
      author = {Lipman, Joseph and Neeman, Amnon},
      title = {Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor},
      journal = {Illinois J. Math.},
      fjournal = {Illinois Journal of Mathematics},
      volume = {51},
      year = {2007},
      number = {1},
      pages = {209--236},
      issn = {0019-2082},
      mrclass = {14A15},
      mrnumber = {2346195},
      mrreviewer = {Stefan Schröer},
      doi = {10.1215/ijm/1258735333},
      url = {https://doi.org/10.1215/ijm/1258735333},
      zblnumber = {1124.14003},
      }
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      pages = {2977--3003},
      issn = {0021-8693},
      mrclass = {18E30 (14E15 14F05)},
      mrnumber = {2609187},
      mrreviewer = {Jon Eivind Vatne},
      doi = {10.1016/j.jalgebra.2009.12.023},
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      url = {https://doi.org/10.1016/j.aim.2009.05.002},
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      author = {Neeman, Amnon},
      title = {Approximable triangulated categories},
      note = {to appear in \emph{Contemp. Math.}},
      year = {2018},
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      }
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    @MISC{Neeman17A,
      author = {Neeman, Amnon},
      title = {Triangulated categories with a single compact generator and a {Brown} representability theorem},
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    @ARTICLE{Neeman92A,
      author = {Neeman, Amnon},
      title = {The connection between the {$K$}-theory localization theorem of {T}homason, {T}robaugh and {Y}ao and the smashing subcategories of {B}ousfield and {R}avenel},
      journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
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      volume = {25},
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      mrreviewer = {Steven E. Landsburg},
      doi = {10.24033/asens.1659},
      url = {https://doi.org/10.24033/asens.1659},
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      }
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    @ARTICLE{Neeman96,
      author = {Neeman, Amnon},
      title = {The {G}rothendieck duality theorem via {B}ousfield's techniques and {B}rown representability},
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      doi = {10.1090/S0894-0347-96-00174-9},
      url = {https://doi.org/10.1090/S0894-0347-96-00174-9},
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      }
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      author = {Neeman, Amnon},
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      series = {Ann. of Math. Stud.},
      volume = {148},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {2001},
      pages = {viii+449},
      isbn = {0-691-08685-0; 0-691-08686-9},
      mrclass = {18E30 (55-02 55N20 55U35)},
      mrnumber = {1812507},
      mrreviewer = {Stanis\l aw Betley},
      doi = {10.1515/9781400837212},
      url = {https://doi.org/10.1515/9781400837212},
      zblnumber = {0974.18008},
      }
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    @MISC{Neeman18A,
      author = {Neeman, Amnon},
      title = {The categories $\ct^c$ and $\ct^b_c$ determine each other},
      year = {2018},
      arxiv = {1806.06471},
      zblnumber = {},
      }
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      author = {Neeman, Amnon},
      title = {The category $\big[\mathcal{T}^c\big]^{\hbox{\rm\tiny op}}$ as functors on $\mathcal{T}^b_c$},
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      zblnumber = {},
      }
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      author = {Neeman, Amnon},
      title = {Strong generation of some derived categories of schemes},
      booktitle = {Research Perspectives CRM Barcelona},
      series = {Trends in Math.},
      volume = {5},
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      }
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      mrclass = {14F05 (14A22 18E30)},
      mrnumber = {1998775},
      mrreviewer = {Bal\'{a}zs Szendrői},
      doi = {10.1070/RM2003v058n03ABEH000629},
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      mrnumber = {3545926},
      mrreviewer = {Shintarou Yanagida},
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      }

Authors

Amnon Neeman

Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra, Australia