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On perfectly generated weight structures and adjacent t-structures

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Abstract

This paper is dedicated to the study of smashing weight structures (these are the weight structures "coherent with coproducts"), and the application of their properties to t-structures. In particular, we prove that the hearts of compactly generated t-structures are Grothendieck abelian categories; this statement strengthens earlier results of several other authors. The central theorem of the paper is as follows: any perfect (as defined by Neeman) set of objects of a triangulated category generates a weight structure; we say that weight structures obtained this way are perfectly generated. An important family of perfectly generated weight structures are (the opposites to) the ones right adjacent to compactly generated t-structures; they give injective cogenerators for the hearts of the latter. Moreover, we establish the following not so explicit result: any smashing weight structure on a well generated triangulated category (this is a generalization of the notion of a compactly generated category that was also defined by Neeman) is perfectly generated; actually, we prove more than that. Furthermore, we give a classification of compactly generated torsion theories (these generalize both weight structures and t-structures) that extends the corresponding result of D. Pospisil and J. Šťovíček to arbitrary smashing triangulated categories. This gives a generalization of a t-structure statement due to B. Keller and P. Nicolas.

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Notes

  1. Note also that Theorem B of ibid. says that \({{\underline{Ht}}}\) is an AB5 abelian category whenever \({\underline{C}}\) is a "strong stable derivator" triangulated category, whereas Theorem C of ibid. gives the existence of generators for a wide class of t-structures.

  2. Respectively, loc. cit. is just a little weaker than Theorem 0.2. Note also that [8] is much more self-contained than the current paper. Respectively, ibid. is quite long and and rather difficult to read. For this reason the author has decided to split it and publish the resulting texts separately (see Remark 0.5 of ibid.); note also that these newer texts contain some results not contained in ibid., and the exposition in them is more accurate.

  3. Note that in ibid. the term complete Hom-orthogonal pair was used. In some other papers torsion theories are called torsion pairs.

  4. Clearly, if C is triangulated or abelian, then X is a retract of Y if and only if X is its direct summand.

  5. Recall that different choices of cones are connected by non-unique isomorphisms.

  6. Recall that a class \({\mathcal {Q}}\subset {\text {Obj}}{{\underline{Ht}}}\) is said to generate \({{\underline{Ht}}}\) whenever for any non-zero \({{\underline{Ht}}}\)-morphism h there exists \(Q\in {\mathcal {Q}}\) such that the homomorphism \({{\underline{Ht}}}(Q,h)\) is non-zero as well. Since \({{\underline{Ht}}}\) is is closed with respect to small coproducts, if \({\mathcal {Q}}\) is essentially small then this condition is fulfilled if and only if any object of \({{\underline{Ht}}}\) is a quotient of a coproduct of elements of \({\mathcal {P}}\).

  7. The terminology we introduce is new; yet big hulls were essentially considered in (Theorem 3.7 of) [35].

  8. Note also that this convention is compatible with the one used for weights of mixed complexes of étale sheaves in (§5.1.5 of) [4]; see Proposition 3.17 of [7].

  9. An argument even more closely related to our one was used in the proof of [28, Lemma 2.2]; yet the assumptions of that lemma appear to require a correction.

  10. Note however that weak weight structures (one replaces the orthogonality axiom in Definition 2.2.1 by \({\underline{C}}_{w\le 0}\perp {\underline{C}}_{w\ge 2}\)) were essentially considered in [14] (cf. Remark 2.1.2 of ibid.), in Theorem 3.1.3(2,3) of [16], in §3.6 of [7], and in (Remark 6.3(4) of) [18].

  11. It appears that this statement originates from §2 of [26]; cf. Theorem B of loc. cit. for an important application of this argument.

  12. Actually, \({\underline{C}}^{{\aleph _0}}\) is essentially small itself in "reasonable" cases; in this case the essential smallness of classes of objects in it is automatic.

  13. This statement was previously proved in [35] and our argument is just slightly different from the one of Pospisil and Šťovíček; see Lemma 3.9 of ibid.

  14. Note that these objects will automatically be \(\beta \)-compact; see the previous part of this definition.

  15. Actually, the standard convention is to say that \({\underline{\coprod }}{\mathcal {P}}\) is contravariantly finite if this condition is fulfilled; yet our version of this term is somewhat more convenient for the purposes of this section.

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  1. St.Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

    Mikhail V. Bondarko

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§§1.1–2.2 and §3 were supported by the Russian Science Foundation grant no. 16-11-00073. The results of sections 2.3 and 2.4 were supported the Russian Science Foundation grant no. 20-41-04401.

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Bondarko, M.V. On perfectly generated weight structures and adjacent t-structures. Math. Z. 300, 1421–1454 (2022). https://doi.org/10.1007/s00209-021-02815-6

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