7
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Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned by objects of a given type (subcategories means here that they are closed under finite (co-)limits aka shits and cones).

  1. Perfect - Complexes quasi-isomorphic to some bounded complex of flat finite modules.
  2. Coherent - Complexes with non-zero coherent cohomology in finitely many degrees.
  3. Finite tor dimension - Complexes quasi-isomorphic to some bounded complex of flat modules
  4. Finite injective dimension - Complexes quasi-isomorphic to some bounded complex of injective modules
  5. Pseudo-coherent - Complexes with coherent cohomology concentrated in a bounded above range.

There is a famous characterization of perfect complexes as the compact objects in $QCoh(X)$.

Question 1: Is there a similarly slick (stable $\infty$-)categorical characterization for 2, 3, 4, 5 above? Suggestions:

  • Perhaps $\mathcal{M}$ is of finite tor dimension iff $\mathcal{M} \otimes_{\mathcal{O}_X} (-)$ commutes with arbitrary products
  • Perhaps $\mathcal{M}$ is of finite injective dimension iff $\mathcal{Hom}_{\mathcal{O}_X}(-, \mathcal{M})$ sends products to coproducts
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    $\begingroup$ Although obvious from the statement, it is amazing that the characterization of (1) does not invoke neither the monoidal structure or the t-structure. $\endgroup$ – Yosemite Sam Jan 24 '17 at 20:05
  • $\begingroup$ @YosemiteSam I agree. This is a very strange kind of miracle. $\endgroup$ – Saal Hardali Jan 24 '17 at 20:06
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    $\begingroup$ You might want to look at § 7.2 of Lurie's Higher Algebra. For example, coherent = almost perfect + eventually coconnective ($\pi_i=0$ for $i \gg 0$). Your suggestion about finite tor-dimension is incorrect: the condition that $M \otimes_{O_X} -$ commutes with products is the same (assuming you mean derived tensor product) as dualizability (which is the same as perfectness). $\endgroup$ – AAK Nov 13 '17 at 16:05
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    $\begingroup$ Maybe coherent is the smallest subcategory containing compact objects and closed under truncation? For this you need to have the $t$-structure, though. $\endgroup$ – Sasha Nov 13 '17 at 22:54

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