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Properties of abelian categories via recollements. (English) Zbl 1428.18004
A recollement of an abelian category $$\mathcal{A}$$ by abelian categories $$\mathcal{X}$$ and $$\mathcal{Y}$$ is a diagram of additive functors
$\begin{tikzcd}[sep = huge] \mathcal{Y} \arrow[r, "i_{\ast}"] & \mathcal{A} \arrow[l, bend right, "i^{\ast}"'] \arrow[l, bend left, "i^{!}"'] \arrow[r, "j^{\ast}"] & \mathcal{X} \arrow[l, bend right, "j_{!}"'] \arrow[l, bend left, "j_{\ast}"'] \end{tikzcd}$ such that $$i_{\ast}, j_{!}, j_{\ast}$$ are fully faithful, $$(i^{\ast}, i_{\ast}, i^{!})$$ and $$(j_{!}, j^{\ast}, j_{\ast})$$ are adjoint triples, and Im($$i_{\ast}$$) = Ker ($$j^{\ast}$$).
Recollements allow the export of structural data from $$\mathcal{A}$$ to $$\mathcal{X}$$ and $$\mathcal{Y}$$ and also the glueing of structural information from $$\mathcal{X}$$ and $$\mathcal{Y}$$ to $$\mathcal{A}$$. In this paper, the authors study how basic properties of abelian categories behave in the presence of a recollement. For instance, $$\mathcal{A}$$ is well-powered if and only if $$\mathcal{X}$$ and $$\mathcal{Y}$$ are well-powered (see Proposition 3.2).
It is shown that (assuming $$\mathcal{A}$$ has Yoneda Ext sets) if $$\mathcal{A}$$ AB3 (respectively, AB3$$^*$$, AB4, AB4$$^*$$, AB5, AB5$$^*$$, Grothendieck) then so are $$\mathcal{X}$$ and $$\mathcal{Y}$$ (see Corollary 2.4, Proposition 3.5 and Lemma 4.1). The converses to the above statements are not true in general and the authors give an example demonstrating this for the Grothendieck property (see §5.1). The concept of a ‘directed’ recollement is introduced and the authors show that if $$\mathcal{X}$$ and $$\mathcal{Y}$$ are AB5 (respectively, AB5$$^*$$) then so is $$\mathcal{A}$$ when the recollement is directed (see Theorem 3.9). Finally, several criteria for $$\mathcal{A}$$ to be Grothendieck when $$\mathcal{X}$$ and $$\mathcal{Y}$$ are Grothendieck (see Corollaries 4.5 and 5.3, Propositions 4.7 and 4.9, and Theorem 5.9).

##### MSC:
 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 18E10 Abelian categories, Grothendieck categories 18G80 Derived categories, triangulated categories 18E35 Localization of categories, calculus of fractions 18E40 Torsion theories, radicals
##### Keywords:
recollement; Grothendieck category; t-structure
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##### References:
 [1] Alonso Tarrío, L.; López, A. Jeremías; Souto Salorio, M. J., Localization in categories of complexes and unbounded resolutions, Can. J. Math., 52, 225-247, (2000) · Zbl 0948.18008 [2] Angeleri Hügel, L.; Marks, F.; Vitória, J., Torsion pairs in silting theory, Pac. J. Math., 291, 2, 257-278, (2017) · Zbl 1391.18014 [3] Beilinson, A.; Bernstein, J.; Deligne, P., Faisceaux Pervers, (Analysis and Topology on Singular Spaces, I. Analysis and Topology on Singular Spaces, I, Luminy, 1981. Analysis and Topology on Singular Spaces, I. Analysis and Topology on Singular Spaces, I, Luminy, 1981, Astérisque, vol. 100, (1982), Soc. Math. France: Soc. Math. France Paris), 5-171, (in French) [4] Beligiannis, A., Relative homological algebra and purity in triangulated categories, J. Algebra, 227, 268-361, (2000) · Zbl 0964.18008 [5] Beligiannis, A.; Reiten, I., Homological and homotopical aspects of torsion theories, Mem. Am. Math. Soc., 188, 883, (2007), viii+207 pp · Zbl 1124.18005 [6] Bondarko, M., Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory, 6, 3, 387-504, (2010) · Zbl 1303.18019 [7] Breitsprecher, S., Lokal endlich präsentierbare Grothendieck-Kategorien, Mitt. Math. Sem. Giessen, 85, 1-25, (1970) · Zbl 0281.18010 [8] Cline, E.; Parshall, B.; Scott, L., Finite dimensional algebras and highest weight categories, J. Reine Angew. Math., 391, 85-99, (1988) · Zbl 0657.18005 [9] Crawley-Boevey, W., Locally finitely presented additive categories, Commun. Algebra, 22, 1641-1674, (1994) · Zbl 0798.18006 [10] Dickson, S. E., A torsion theory for abelian categories, Trans. Am. Math. Soc., 121, 223-235, (1966) · Zbl 0138.01801 [11] Feng, J.; Zhang, P., Types of Serre subcategories of Grothendieck categories, J. Algebra, 508, 16-34, (2018) · Zbl 1391.18019 [12] Franjou, V.; Pirashvili, T., Comparison of abelian categories recollements, Doc. Math., 9, 41-56, (2004) · Zbl 1060.18008 [13] Gao, N.; Psaroudakis, C., Ladders of compactly generated triangulated categories and preprojective algebras, Appl. Categ. Struct., 26, 4, 657-679, (2018) · Zbl 1428.18021 [14] Gentle, R., T.T.F. Theories in Abelian categories, Commun. Algebra, 16, 877-908, (1996) · Zbl 0652.18005 [15] Geigle, W.; Lenzing, H., Perpendicular categories with applications to representations and sheaves, J. Algebra, 144, 2, 273-343, (1991) · Zbl 0748.18007 [16] Grothendieck, A., Sur quelques points d’algèbre homologique, Tôhoku Math. J., 9, 119-221, (1957) · Zbl 0118.26104 [17] Happel, D.; Reiten, I.; Smalø, S., Tilting in abelian categories and quasitilted algebras, Mem. Am. Math. Soc., 120, 575, (1996), viii+ 88 · Zbl 0849.16011 [18] Joiţa, D.; Nǎstǎsescu, C.; Nǎstǎsescu, L., Recollement of Grothendieck categories. Applications to schemes, Bull. Math. Soc. Sci. Math. Roum. (N.S.), 56(104), 1, 109-116, (2013) · Zbl 1313.18014 [19] Keller, B., Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (4), 27, 1, 63-102, (1994) · Zbl 0799.18007 [20] Krause, H., On Neeman’s well generated triangulated categories, Doc. Math., 6, 121-126, (2001) · Zbl 0987.18012 [21] Krause, H., Coherent functors in stable homotopy theory, Fundam. Math., 173, 33-56, (2002) · Zbl 1001.55022 [22] Krause, H., Localization theory for triangulated categories, (Triangulated Categories. Triangulated Categories, Lond. Math. Soc. Lect. Note Ser., vol. 375, (2010), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 161-235 · Zbl 1232.18012 [23] Krause, H., Deriving Auslander’s formula, Doc. Math., 20, 669-688, (2015) · Zbl 1348.18018 [24] Kuhn, N. J., The generic representation theory of finite fields: a survey of basic structure, (Infinite Length Modules. Infinite Length Modules, Bielefeld, 1998. Infinite Length Modules. Infinite Length Modules, Bielefeld, 1998, Trends Math., (2000), Birkhäuser: Birkhäuser Basel), 193-212 · Zbl 0980.20033 [25] Liu, Q.; Vitória, J.; Yang, D., Glueing silting objects, Nagoya Math. J., 216, 117-151, (2014) · Zbl 1342.18022 [26] Modoi, G. C., The dual of Brown representability for some derived categories, Ark. Mat., 54, 2, 485-498, (2016) · Zbl 1373.18010 [27] Murfet, D., Derived Categories Part II, available online at [28] Neeman, A., Triangulated Categories, Ann. Math. Stud., vol. 148, (2001), Princeton University Press · Zbl 0974.18008 [29] Neeman, A., On the derived category of sheaves on a manifold, Doc. Math., 6, 483-488, (2001) · Zbl 1026.18006 [30] Neeman, A., Non-left-complete derived categories, Math. Res. Lett., 18, 5, 827-832, (2011) · Zbl 1244.18009 [31] Parra, C.; Saorin, M., Direct limits in the heart of a t-structure: the case of a torsion pair, J. Pure Appl. Algebra, 219, 9, 4117-4143, (2015) · Zbl 1333.18017 [32] Parshall, B.; Scott, L., Derived categories, quasi-hereditary algebras, and algebraic groups, Carlton Univ. Math. Notes, 3, 1-111, (1989) [33] Pauksztello, D., Compact corigid objects in triangulated categories and co-t-structures, Cent. Eur. J. Math., 6, 1, 25-42, (2008) · Zbl 1152.18009 [34] Psaroudakis, C., Homological theory of recollements of abelian categories, J. Algebra, 398, 63-110, (2014) · Zbl 1314.18016 [35] Psaroudakis, C., A representation-theoretic approach to recollements of abelian categories, (Surveys in Representation Theory of Algebras. Surveys in Representation Theory of Algebras, Contemp. Math., vol. 716, (2018), Amer. Math. Soc.), 67-154 [36] Psaroudakis, C.; Vitória, J., Recollements of module categories, Appl. Categ. Struct., 22, 4, 579-593, (2014) · Zbl 1305.18055 [37] Psaroudakis, C.; Vitória, J., Realisation functors in tilting theory, Math. Z., 288, 3, 965-1028, (2018) · Zbl 1407.18014 [38] Stenström, B., Rings of Quotients, (1975), Springer
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