## Krull–Gabriel dimension of domestic string algebras

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- by Rosanna Laking, Mike Prest and Gena Puninski PDF
- Trans. Amer. Math. Soc.
**370**(2018), 4813-4840 Request permission

## Abstract:

We calculate the Krull–Gabriel dimension of the category of modules over any domestic string algebra, in particular showing that it is finite, thus confirming a conjecture of Schröer. We also compute the Cantor–Bendixson rank of each point of its Ziegler spectrum and determine the topology on this space.## References

- K. Burke,
*Some model-theoretic properties of functor categories for modules*, Doctoral Thesis, University of Manchester, 1994. - Kevin Burke and Mike Prest,
*The Ziegler and Zariski spectra of some domestic string algebras*, Algebr. Represent. Theory**5**(2002), no. 3, 211–234. MR**1921759**, DOI 10.1023/A:1016581927685 - M. C. R. Butler and Claus Michael Ringel,
*Auslander-Reiten sequences with few middle terms and applications to string algebras*, Comm. Algebra**15**(1987), no. 1-2, 145–179. MR**876976**, DOI 10.1080/00927878708823416 - W. W. Crawley-Boevey,
*Maps between representations of zero-relation algebras*, J. Algebra**126**(1989), no. 2, 259–263. MR**1024991**, DOI 10.1016/0021-8693(89)90304-9 - William Crawley-Boevey,
*Infinite-dimensional modules in the representation theory of finite-dimensional algebras*, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 29–54. MR**1648602** - S. Garavaglia,
*Dimension and rank in the model theory of modules*, preprint, University of Michigan, 1979, revised 1980. - Werner Geigle,
*The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences*, Manuscripta Math.**54**(1985), no. 1-2, 83–106. MR**808682**, DOI 10.1007/BF01171701 - Werner Geigle,
*Krull dimension and Artin algebras*, Representation theory, I (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1177, Springer, Berlin, 1986, pp. 135–155. MR**842464**, DOI 10.1007/BFb0075263 - R. Harland,
*Pure-injective modules over tubular algebras and string algebras*, Doctoral Thesis, University of Manchester, 2011, available at www.maths.manchester.ac.uk/~mprest/publications.html - Richard Harland and Mike Prest,
*Modules with irrational slope over tubular algebras*, Proc. Lond. Math. Soc. (3)**110**(2015), no. 3, 695–720. MR**3342102**, DOI 10.1112/plms/pdu068 - Ivo Herzog,
*The endomorphism ring of a localized coherent functor*, J. Algebra**191**(1997), no. 1, 416–426. MR**1444506**, DOI 10.1006/jabr.1997.6920 - Henning Krause,
*Maps between tree and band modules*, J. Algebra**137**(1991), no. 1, 186–194. MR**1090218**, DOI 10.1016/0021-8693(91)90088-P - Henning Krause,
*Generic modules over Artin algebras*, Proc. London Math. Soc. (3)**76**(1998), no. 2, 276–306. MR**1490239**, DOI 10.1112/S0024611598000094 - Henning Krause,
*The spectrum of a module category*, Mem. Amer. Math. Soc.**149**(2001), no. 707, x+125. MR**1803703**, DOI 10.1090/memo/0707 - Mike Prest,
*Model theory and modules*, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR**933092**, DOI 10.1017/CBO9780511600562 - Mike Prest,
*Representation embeddings and the Ziegler spectrum*, J. Pure Appl. Algebra**113**(1996), no. 3, 315–323. MR**1417396**, DOI 10.1016/0022-4049(95)00155-7 - Mike Prest,
*Morphisms between finitely presented modules and infinite-dimensional representations*, Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 447–455. MR**1648645**, DOI 10.2307/2586842 - Mike Prest,
*Ziegler spectra of tame hereditary algebras*, J. Algebra**207**(1998), no. 1, 146–164. MR**1643078**, DOI 10.1006/jabr.1998.7472 - Mike Prest,
*Purity, spectra and localisation*, Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009. MR**2530988**, DOI 10.1017/CBO9781139644242 - Mike Prest and Gena Puninski,
*One-directed indecomposable pure injective modules over string algebras*, Colloq. Math.**101**(2004), no. 1, 89–112. MR**2106184**, DOI 10.4064/cm101-1-6 - Gena Puninski,
*Band combinatorics of domestic string algebras*, Colloq. Math.**108**(2007), no. 2, 285–296. MR**2291639**, DOI 10.4064/cm108-2-10 - Gena Puninski,
*Krull-Gabriel dimension and Cantor-Bendixson rank of 1-domestic string algebras*, Colloq. Math.**127**(2012), no. 2, 185–211. MR**2968961**, DOI 10.4064/cm127-2-4 - Gena Puninski and Mike Prest,
*Ringel’s conjecture for domestic string algebras*, Math. Z.**282**(2016), no. 1-2, 61–77. MR**3448374**, DOI 10.1007/s00209-015-1532-6 - Claus Michael Ringel,
*Some algebraically compact modules. I*, Abelian groups and modules (Padova, 1994) Math. Appl., vol. 343, Kluwer Acad. Publ., Dordrecht, 1995, pp. 419–439. MR**1378216** - Claus Michael Ringel,
*The Ziegler spectrum of a tame hereditary algebra*, Colloq. Math.**76**(1998), no. 1, 105–115. MR**1611289**, DOI 10.4064/cm-76-1-105-115 - J. Schröer,
*Hammocks for string algebras*, PhD Thesis, Sonderforschungsbereich 343, Ergänzungreihe 97–010, Universität Bielefeld, 1997. - Jan Schröer,
*On the Krull-Gabriel dimension of an algebra*, Math. Z.**233**(2000), no. 2, 287–303. MR**1743438**, DOI 10.1007/PL00004799 - Jan Schröer,
*On the infinite radical of a module category*, Proc. London Math. Soc. (3)**81**(2000), no. 3, 651–674. MR**1781151**, DOI 10.1112/S0024611500012600 - Jan Schröer,
*The Krull-Gabriel dimension of an algebra—open problems and conjectures*, Infinite length modules (Bielefeld, 1998) Trends Math., Birkhäuser, Basel, 2000, pp. 419–424. MR**1789229** - Martin Ziegler,
*Model theory of modules*, Ann. Pure Appl. Logic**26**(1984), no. 2, 149–213. MR**739577**, DOI 10.1016/0168-0072(84)90014-9

## Additional Information

**Rosanna Laking**- Affiliation: School of Mathematics, Alan Turing Building, University of Manchester, Manchester M13 9PL, United Kingdom
- Address at time of publication: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- Email: rlaking@mpim-bonn.mpg.de
**Mike Prest**- Affiliation: School of Mathematics, Alan Turing Building, University of Manchester, Manchester M13 9PL, United Kingdom
- MR Author ID: 141975
- Email: mprest@manchester.ac.uk
**Gena Puninski**- Affiliation: Department of Mechanics and Mathematics, Belarusian State University, Praspekt Nezalezhnosti 4, Minsk 220030, Belarus
- Received by editor(s): May 19, 2016
- Received by editor(s) in revised form: September 27, 2016
- Published electronically: December 26, 2017
- Additional Notes: This paper was started during a visit of the third author to the University of Manchester, supported by EPSRC grant EP/K022490/1, and was completed during a visit of the second author to the Belarusian State University. The authors thank both universities and EPSRC for their support
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 4813-4840 - MSC (2010): Primary 16G20, 16G60, 03C60
- DOI: https://doi.org/10.1090/tran/7093
- MathSciNet review: 3812097

Dedicated: Gena Puninski died on April 29, 2017. The other two authors dedicate this paper to his memory