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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 30, Issue 5

# Contractibility of the stability manifold for silting-discrete algebras

David Pauksztello
• Corresponding author
• Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, United Kingdom
• Email
• Other articles by this author:
/ Manuel Saorín
/ Alexandra Zvonareva
Published Online: 2018-04-21 | DOI: https://doi.org/10.1515/forum-2017-0120

## Abstract

We show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for a silting-discrete algebra is contractible.

MSC 2010: 18E30; 16G10

## References

• [1]

T. Adachi, The classification of two-term tiling complexes for Brauer graph algebras, Proceedings of the 48th Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory Organ. Comm., Yamanashi (2016), 1–5. Google Scholar

• [2]

T. Adachi, O. Iyama and I. Reiten, τ-tilting theory, Compos. Math. 150 (2014), no. 3, 415–452.

• [3]

T. Adachi, Y. Mizuno and D. Yang, Silting-discreteness of triangulated categories and contractibility of stability spaces, preprint (2017), http://arxiv.org/abs/1708.08168.

• [4]

T. Aihara, Tilting-connected symmetric algebras, Algebr. Represent. Theory 16 (2013), no. 3, 873–894.

• [5]

T. Aihara and O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668.

• [6]

T. Aihara and Y. Mizuno, Tilting complexes over preprojective algebras of Dynkin type, Proceedings of the 47th Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory Organ. Comm., Okayama (2015), 14–19. Google Scholar

• [7]

L. Angeleri Hügel, F. Marks and J. Vitória, Silting modules, Int. Math. Res. Not. IMRN (2016), no. 4, 1251–1284. Google Scholar

• [8]

M. Auslander, Representation theory of Artin algebras. I, Comm. Algebra 1 (1974), 177–268.

• [9]

M. Auslander, Representation theory of Artin algebras. II, Comm. Algebra 1 (1974), 269–310.

• [10]

A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and Topology on Singular Spaces. I (Luminy 1981), Astérisque 100, Société Mathématique de France, Paris (1982), 5–171. Google Scholar

• [11]

A. Beligiannis and I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, 1–207. Google Scholar

• [12]

T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345.

• [13]

N. Broomhead, D. Pauksztello and D. Ploog, Averaging t-structures and extension closure of aisles, J. Algebra 394 (2013), 51–78.

• [14]

N. Broomhead, D. Pauksztello and D. Ploog, Discrete derived categories II: The silting pairs CW complex and the stability manifold, J. Lond. Math. Soc. (2) 93 (2016), no. 2, 273–300.

• [15]

S. E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235.

• [16]

L. Demonet, O. Iyama and G. Jasso, τ-tilting finite algebras, g-vectors and brick-τ-rigid correspondence, preprint (2015), http://arxiv.org/abs/1503.00285.

• [17]

G. Dimitrov, F. Haiden, L. Katzarkov and M. Kontsevich, Dynamical systems and categories, The Influence of Solomon Lefschetz in Geometry and Topology, Contemp. Math. 621, American Mathematical Society, Providence (2014), 133–170.Google Scholar

• [18]

G. Dimitrov and L. Katzarkov, Bridgeland stability conditions on the acyclic triangular quiver, Adv. Math. 288 (2016), 825–886.

• [19]

L. Fiorot, F. Mattiello and A. Tonolo, A classification theorem for t-structures, J. Algebra 465 (2016), 214–258.

• [20]

F. Haiden, L. Katzarkov and M. Kontsevich, Flat surfaces and stability structures, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 247–318.

• [21]

D. Happel, I. Reiten and S. O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, 1–88. Google Scholar

• [22]

O. Iyama, P. Jø rgensen and D. Yang, Intermediate co-t-structures, two-term silting objects, τ-tilting modules, and torsion classes, Algebra Number Theory 8 (2014), no. 10, 2413–2431.

• [23]

B. Keller and D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 239–253. Google Scholar

• [24]

A. King and Y. Qiu, Exchange graphs and Ext quivers, Adv. Math. 285 (2015), 1106–1154.

• [25]

S. Koenig and D. Yang, Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438. Google Scholar

• [26]

E. Macrì, Stability conditions on curves, Math. Res. Lett. 14 (2007), no. 4, 657–672.

• [27]

P. Nicolás, M. Saorín and A. Zvonareva, Silting theory in triangulated categories with coproducts, preprint (2015), http://arxiv.org/abs/1512.04700.

• [28]

S. Okada, Stability manifold of ${ℙ}^{1}$, J. Algebraic Geom. 15 (2006), no. 3, 487–505. Google Scholar

• [29]

A. Polishchuk, Constant families of t-structures on derived categories of coherent sheaves, Mosc. Math. J. 7 (2007), no. 1, 109–134, 167. Google Scholar

• [30]

C. Psaroudakis and J. Vitória, Realisation functors in tilting theory, Math. Z. 288 (2018), no. 3–4, 965–1028.

• [31]

Y. Qiu, Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math. 269 (2015), 220–264.

• [32]

Y. Qiu and J. Woolf, Contractible stability spaces and faithful braid group actions, preprint (2014), http://arxiv.org/abs/1407.5986.

• [33]

M. Saorín and J. Šťovíček, On exact categories and applications to triangulated adjoints and model structures, Adv. Math. 228 (2011), no. 2, 968–1007.

• [34]

D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. Vol. 2, London Math. Soc. Stud. Texts 71, Cambridge University Press, Cambridge, 2007. Google Scholar

• [35]

J. Woolf, Stability conditions, torsion theories and tilting, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 663–682.

Revised: 2018-02-27

Published Online: 2018-04-21

Published in Print: 2018-09-01

Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 16-31-60089

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2016-77445

Alexandra Zvonareva is supported by the RFBR Grant 16-31-60089. Manuel Saorín is supported by research projects from the Ministerio de Economía y Competitividad of Spain (MTM2016-77445P) and from the Fundación “Séneca” of Murcia (19880/GERM/15), both with a part of FEDER funds.

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1255–1263, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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