School of Mathematics
University of Manchester

Conference in memory of Gena Puninski, 6-9th April, 2018.
Mike Prest
Publications and preprints

## Books

Model Theory and Modules, London Mathematical Society Lecture Notes Series Vol. 130, Cambridge University Press, Cambridge, 1988 (400pp).

(with J. Humphreys) Numbers, Groups and Codes, Cambridge University Press, 1989 (288pp). A second, considerably revised edition was published in 2004 An extensive list of corrections is on this webpage.

Purity, Spectra and Localisation, Encyclopedia of Mathematics and its Applications, Vol. 121, Cambridge University Press, 2009 (xxviii+769pp).

## Papers: published and in press

Links are to drafts of papers which, in some cases, may differ somewhat from the published version.

1. Some model-theoretic aspects of torsion theories, J. Pure Applied Algebra, 12(3) (1978), 295-310.

2. Torsion and universal Horn classes of modules, J. London Math. Soc., 19(3) (1979), 411-416.

3. Model-completions of some theories of modules, J. London Math. Soc., 20(3) (1979), 369-372.

4. Elementary torsion theories and locally finitely presented Abelian categories, J. Pure Applied Algebra, 18(2) (1980), 205-212.

5. Quantifier elimination for modules, pp. 109-129 in "Proceedings of the Model Theory Meeting at Brussels and Mons 1980" in Bull. Soc. Math. Belgique, 33 (1981).

6. Elementary equivalence of sigma-injective modules, Proc. London Math. Soc., 45(1) (1982), 71-88.

7. Existentially complete prime rings, J. London Math. Soc., 28(2) (1983), 238-246.

8. (with A. Pillay) Forking and pushouts for modules, Proc. London Math. Soc., 46(2) (1983), 365-384.

9. Rings of finite representation type and modules of finite Morley rank, J. Algebra, 88(2) (1984), 502-533.

10. The generalised RK-order, orthogonality and regular types for modules, J. Symbolic Logic, 50(1) (1985), 202-219.

11. (with A. Pillay) Modules and stability theory, Trans. Amer. Math. Soc., 300(2) (1987), 641-662.

12. Model theory and representation type of algebras, pp. 219-260 in Logic Colloquium '86 (Proceedings of the European meeting of the Association for Symbolic Logic, Hull, 1986), North-Holland, Amsterdam, 1988.

13. (with F. Point) Decidability for theories of modules, J. London Math. Soc. 38(2) (1988), 193-206.

14. Duality and pure-semisimple rings, J. London Math. Soc. 38 (1988), 403-409.

15. (with A. Marcja and C. Toffalori) Classification theory for abelian groups with an endomorphism, Arch. Math. Logic, 31(2) (1991), 95-104.

16. Wild representation type and undecidability, Communications in Algebra, 19(3) (1991), 919-929.

17. (with T. Kucera) Imaginary modules, J. Symbolic Logic, 57(2) (1992), 698-723.

18. (with T. Kucera) Four concepts from geometrical'' stability theory in modules, J. Symbolic Logic, 57(2) (1992), 723-740.

19. (with A. Marcja and C. Toffalori) The stability classification for abelian-by-finite groups and modules over group rings, J. London Math. Soc., 47(2) (1993), 212-226.

20. (with A. Marcja and C. Toffalori) On wildness of abelian-by-finite groups, Ann. Pure Applied Logic, 62(2) (1993), 167-173.

21. Remarks on elementary duality, Ann. Pure Applied Logic, 62(2) (1993), 183-205.

22. (with Ph. Rothmaler and M. Ziegler) Extensions of elementary duality, J. Pure Applied Algebra, 93(1) (1994), 33-56.

23. (with Ph. Rothmaler and M. Ziegler) Absolutely pure modules and indiscrete rings, J. Algebra, 174(2) (1995), 349-372.

24. Epimorphisms of rings, interpretations of modules and strictly wild algebras, Comm. Algebra, 24(2) (1996), 517-531.

25. Representation embeddings and the Ziegler spectrum, J. Pure Applied Algebra, 113(3) (1996), 315-323.

26. (with S. Baratella) Pure-injective modules over the dihedral algebras, Comm. Algebra, 25(1) (1997), 11-32.

27. Interpreting modules in modules, Ann. Pure Applied Logic, 88(2-3) (1997), 193-215.

28. (with K. Burke) Rings of definable scalars and biendomorphism rings, pp.188-201 in Model Theory of Groups and Automorphisms, London Math. Soc. Lecture Note Ser. Vol. 244, 1997.

29. Morphisms between finitely presented modules and infinite-dimensional representations, Canad. Math. Soc. Conf. Proc. Ser., Vol. 24 (1998), 447-455.

30. The representation theories of elementarily equivalent rings, J. Symbolic Logic, 63(2) (1998), 439-450.

31. Ziegler spectra of tame hereditary algebras, J. Algebra, 207(1) (1998), 146-164.

32. The sheaf of locally definable scalars over a ring, pp. 339-351 in Models and Computability, London Math. Soc. Lecture Note Ser., Vol. 259, Cambridge University Press, 1999.

33. (with G. Puninski and Ph. Rothmaler) Rings described by various purities, Comm. Algebra, 27(5) (1999), 2127-2162.

34. Tensor product and theories of modules, J. Symbolic Logic, 64(2) (1999), 617-628.

35. (with V. Puninskaya) Vaught's conjecture for modules over a commutative Prüfer ring, Algebra i Logika, 38(4) (1999), 419-435.

36. (with V. Puninskaya) Modules with few types over a commutative Prüfer ring, (in Russian), Uspekhi Mat. Nauk, 54(3) (1999), 149-150.

37. (with G. Puninski) \Sigma-pure-injective modules over a commutative Prüfer ring, Comm. Algebra, 27(2) (1999), 961-971.

38. (with G. Puninski) Some model theory over hereditary noetherian domains, J. Algebra, 211(1) (1999), 268-297.

39. Topological and geometric aspects of the Ziegler spectrum, in Proceedings of Conference on Infinite Length Modules, Bielefeld, 1998, Birkhäuser, 2000.

40. (with K. Burke) The Ziegler and Zariski spectra of some domestic string algebras, Algebras and Representation Theory, 5(3) (2002), 211-234. .pdf

41. (with V. Puninskaya) Modules with few types over some finite-dimensional algebras, J. Symbolic Logic, 67(2) (2002), 841-858.

42. (with J. Schröer) Serial functors, Jacobson radical and representation type, J. Pure Applied Algebra, 170(2-3) (2002), 295-307/

43. (with A. Marcja and C. Toffalori) The torsionfree part of the Ziegler spectrum of $RG$ where $R$ is a Dedekind domain and $G$ is a finite group, J. Symbolic Logic, 67(3) (2002), 1126-1140.

44. (with G. Puninski) Pure injective envelopes of finite length modules over a Generalised Weyl Algebra, J. Algebra, 251(1) (2002), 150-177. .pdf

45. Model theory for algebra, pp. 199-226 in M. Hazewinkel (ed.), Handbook of Algebra, Vol. 3, Elsevier, 2003. .pdf

46. Model theory and modules, pp. 227-253 in M. Hazewinkel (ed.), Handbook of Algebra, Vol. 3, Elsevier, 2003. .pdf

47. (with S. Crivei and G. Reynders) Model theory of comodules, J. Symbolic Logic, 69(1) (2004), 137-142. .pdf

48. (with R. Wisbauer) Finite presentation and purity in categories sigma[M], Colloq. Math., 99(2) (2004), 189-202. .pdf

49. (with V. Puninskaya and A. Ralph) Model theory of sheaves of modules, J. Symbolic Logic, 69(4) (2004), 1187-1200. .pdf

50. (with G. Garkusha) Injective objects in triangulated categories, J. Algebra Applications, 3(4) (2004), 367-389. .pdf

51. (with G. Puninski) One-directed indecomposable pure-injective modules over string algebras, Colloq. Math., 101(1) (2004), 89-112. .ps

52. Ideals of mod-R and the omega-radical, J. London Math. Soc., 71(2) (2005), 321-334. .pdf

53. (with G. Garkusha) Triangulated categories and the Ziegler spectrum, Algebras and Representation Theory, 8(4) (2005), 499-523. .pdf

54. (with G. Puninski) Krull-Gabriel dimension of 1-domestic string algebras, Algebras and Representation Theory, 9(4) (2006), 337-358. .pdf

55. (with S. L'Innocente) Rings of definable scalars of Verma modules, J. Algebra Apps., 6(5) (2007), 779-787 .pdf

56. (with G. Garkusha) Classifying Serre subcategories of finitely presented modules, Proc. Amer. Math. Soc., 136(3) (2008), 761-770 .pdf

57. (with G. Garkusha) Reconstructing projective schemes from Serre subcategories, J. Algebra, 319(2) (2008), 1132-1153 .pdf

58. (with G. Garkusha) Torsion classes of finite type and spectra, pp. 393-412 in K-theory and Noncommutative Geometry, European Math. Soc., 2008. .pdf

59. (with R. Rajani) Model-theoretic imaginaries and coherent sheaves, Applied Categorical Structures, 17(6) (2009), 517-559. .pdf.

60. (with R. Rajani) Pure-injectivity and model theory for G-sets, J. Symbolic Logic, 74(2) (2009), 474-488 .pdf

61. Definable additive categories, in S. Breaz, S. Crivei and A. Marcus (Eds.), Proceedings of the International Conference on Modules and Representation Theory (Cluj-Napoca, July 7-12, 2008), Cluj University Press, Cluj-Napoca, 2009 .pdf

62. Pure-injective modules, AJSE-Mathematics, 1(1) (2009), 175-191 .pdf

63. (with S. Crivei and B. Torrecillas) Covers in finitely accessible categories, Proc. Amer. Math. Soc., 138 (2010), 1213-1221 .pdf

64. (with R. Rajani) Structure sheaves of definable additive categories, J. Pure Applied Algebra, 214 (2010), 1370-1383 .pdf

65. Definable additive categories: purity and model theory, Mem. Amer. Math. Soc., Vol. 210, 2011 .pdf

66. Model theory in additive categories, in "Models, Logics and Higher-Dimensional Categories: a Tribute to the Work of Mihaly Makkai", CRM Proceedings and Lecture Notes, Vol. 53, Amer. Math. Soc., 2011.

67. (with S. Estrada, P. A. Guil Asensio and J. Trlifaj) Model category structures arising from Drinfeld vector bundles, Adv. in Math., 231(3-4) (2012), 1417-1438. .pdf

68. (with H. Krause) The Gabriel-Roiter filtration of the Ziegler spectrum, Quart. J. Math., 64(3) (2013), 891-901 .pdf

69. Superdecomposable pure-injective modules, pp. 263-296 in Advances in Representation Theory of Algebras (ICRA Bielefeld 2012), European Math. Soc., 2014 .pdf

70. (with A. R. Mehdi) Almost dual pairs and definable classes of modules, Comm. Algebra, 43(4) (2015), 1387-1397. arXiv

71. (with R. Harland) Modules with irrational slope over tubular algebras, Proc. London Math. Soc., 110(3) (2015), 695-720. on arXiv

72. (with G. Puninski) Ringel's conjecture for domestic string algebras, Math. Zeit., 282(1) (2016), 61-77, DOI:10.1007/s00209-015-1532-6. on arXiv

73. (with L. Gregory) Representation embeddings, interpretation functors and controlled wild algebras, J. London Math. Soc., 94(3) (2016), 747-766, DOI:10.1112/jlms/jdw055. on arXiv

74. (with K. K. Arnesen, R. Laking and D. Pauksztello) The Ziegler spectrum for derived-discrete algebras, Adv. in Math., 319 (2017), 653-698. DOI:10.1016/j.aim.2017.07.016 on arXiv

75. (with L. Barbieri-Viale) Definable categories and T-motives, Rendiconti Sem. Mat. Univ. Padova, 139 (2018), 205-224. DOI:10.4171/RSMUP/139-8 on arXiv

76. (with R. Laking and G. Puninski) Krull-Gabriel dimension of domestic string algebras, Trans. Amer. Math. Soc., to appear. on arXiv

77. Modules as exact functors, Proceedings of 2016 Auslander Distinguished Lectures and Conference, Contemporary Mathematics, Vol. 716, Amer. Math. Soc., to appear. on arXiv

## Papers: submitted and preprints

The Zariski spectrum of the category of finitely presented modules, preprint, 1998, revised 2004, 2006 and 2008. .pdf

(with A. Ralph) Locally finitely presented categories of sheaves of modules. .ps, now split into a shortened paper of the same title .pdf and On sheafification of modules, .pdf, an exposition of some background.

(with R. Rajani) Positive imaginaries and coherent affine functors .pdf

Abelian categories and definable additive categories on arXiv

Categories of imaginaries for definable additive categories on arXiv

Spectra of small abelian categories on arXiv

(with L. Barbieri-Viale and A. Huber) Tensor structure for Nori motives on arXiv

Multisorted modules and their model theory on arXiv

## Notes of Seminars, Lecture Series etc.

Notes on Model Theory; notes from lectures on intermediate model theory from the MODNET Summer School held at Leeds, December 2005.

Sheaves of Rings of Definable Scalars ; Conference in honour of Helmut Lenzing, Paderborn, 2006

Model Theory in Additive Categories; Model Theory conference, Seoul, 2010

The Functor Category Notes for lectures given at the conference in Bristol on Categorical Methods in Representation Theory, September 2012.

Summer School and Workshop on Infinite-dimensional Representations of Finite-dimensional Algebras, Manchester, September 2015

Model-theoretic Imaginaries and Localisation for Additive Categories; Topos Theory conference, IHES, Paris, 2015

The Compactness Theorem, Notes and Beamer slides, London Mathematical Society Undergraduate Summer School, Manchester 2017.

## Recent Postdoctoral Researchers

Zoe Gosling (zoe.gosling@postgrad.manchester.ac.uk) (co-supervisor, with Prof. Daniela Caselli)
Rose Wagstaffe
Alex Slavik (visiting student)

## PhD Students' Theses

Sarah Mattingly: On the Structure of Modules of Finite Rank (1994)

Kevin Burke: Some Model-theoretic Properties of Functor Categories for Modules (1994)

Robert Combley: Ordering the Formulae of a Stable Theory by Representability (1995)

Shahabaddin Ebrahimi Atani On Indecomposable Pure-Injective Modules over Pullback Rings (1996)

Geert Reynders: Ziegler Spectra over Serial Rings and Coalgebras (1998) .ps

Nicholas Granger: Stability, Simplicity and the Model Theory of Bilinear Forms (1999) .ps

Alexandra Ralph: An Approach to the Model Theory of Sheaves of Modules (2001)

Ravi Rajani: Model-Theoretic Imaginaries and Ziegler Spectra in General Categories (2007) .ps

Richard Harland: Pure-injective Modules over Tubular Algebras and String Algebras (2011).pdf

Lorna Gregory: Ziegler Spectra of Valuation Rings (2011).pdf

Simon Perera: Grothendieck Rings of Theories of Modules (2011).pdf

Philip Bridge: Essentially Algebraic Theories and Localizations in Toposes and Abelian Categories (2012) .pdf

Akeel Ramadan Mehdi: Purity Relative to Classes of Finitely Presented Modules (2013) .pdf

Amit Kuber: K-Theory of Theories of Modules and Algebraic Varieties (2014) .pdf

Rosanna Laking: String Algebras in Representation Theory (2016) .pdf

Samuel Dean: Dualities and Finitely Presented Functors (2017) .pdf

## MSc and MPhil Students' Dissertations

Nicholas Mosley: The Problem of Quantifier-Elimination for the Real Number Field with Exponential Function (1989)

John Smith: On Connections between Category Theory and Logic (1994)

Tristram de Piro: The Model Theory of Lie Algebras (1996)

Alexandra Ralph: An Approach to the Model Theory of Sheaves (1996)

Cecily Crampin: Models as Functors for T^{eq} (1997)

Timothy Mellor: Elimination of Quantifiers for Modules over the Path Algebra \tilde{A_1}(k) (1997)

Julie Fowler: Induction, Restriction and Elementary equivalence (1998)

James Gray: On Definable Sets over Finite Fields (1998)

Owain Derrick: Model Theory of Pseudofinite Fields (1999)

Ralph Sherrington: Quantum Computation (2001)

Martin Carpenter: Model Theory and Grothendieck Topologies (2001)

Owen Jones: Elimination of Quantifiers for Modules (2001)

Mahmood Sohrabi: Localization in Abelian and Accessible Categories (2002)

Jonathan Roberts: Model Theory of Fields (2005)

Ray-Ming Chen: O-Minimality (2006)

John Bourke: Interpretations and Automorphism Groups (2006)

David Crowther: Grothendieck Rings of Theories (2006)

Victoria Fellows: Set Theory (2006)

Andrew Bannatyne: The Whitehead Problem (2006)

François Wirion: Presenting Theories in Category Theory (2008)

Vladimir Lukiyanov: Well Generated Triangulated Categories (2010)

Vladimir Lukiyanov: Endofinite Modules (MPhil) (2011)

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