×

Endomorphism algebras of peak \(I\)-spaces over posets of infinite prinjective type. (English) Zbl 0884.16017

Let \(R\) be a commutative ring with an identity element and let \(I\) be a finite poset (i.e. partially ordered set) with the partial order \(\preceq\). Following the reviewer’s paper [J. Pure Appl. Algebra 90, No. 1, 77-103 (1993; Zbl 0815.16006)] the authors define the category \(P(I,R)\) of peak \(I\)-spaces over the ring \(R\) to be the category whose objects are the systems \({\mathbf M}=(M_i,\sigma_{ij})\) of \(R\)-modules \(M_i\), \(i\in I\), connected by \(R\)-homomorphisms \(\sigma_{ij}\colon M_i\to M_j\), \(i\preceq j\), satisfying the following conditions: (a) \(\sigma_{ii}\) is the identity map on \(M_i\) and \(\sigma_{js}\sigma_{ij}=\sigma_{is}\) for all \(i\preceq j\preceq s\), (b) \(\bigcap_{i\prec p\in\max I}\text{Ker}(\sigma_{ip})=0\) for any element \(i\in I\), where \(\max I\) is the set of maximal elements of \(I\).
The reviewer’s paper mentioned above contains a combinatorial characterization of posets \(I\) of infinite prinjective type, that is, those for which the category \(P(I,K)\) of peak \(I\)-spaces has infinitely many isomorphism classes of indecomposable objects (up to isomorphism) for any commutative field \(K\). It was shown there that a poset \(I\) is of infinite prinjective type if and only if \(I\) contains any of the critical posets \({\mathcal P}_1,\ldots,{\mathcal P}_{114}\) listed in the paper.
One of the main aims of the reviewed paper is to realize any \(R\)-algebra as an endomorphism algebra of an object in \(P(I,R)\) for any poset \(I\) of infinite prinjective type. The authors prove that if \(I\) is a poset of infinite prinjective type, \(\lambda\) is an infinite cardinal number and \(A\) is an \(R\)-algebra generated by at most \(\lambda\) elements, then there exists a \(\lambda\)-family of functors \(\text{Mod}(A)\to P(I,R)\), \(M\mapsto\{M_\beta\}_{\beta\subseteq\lambda}\), which is rigid for every \(A\)-module, that is, \(\text{Mor}(M_\alpha,N_\beta)\cong\operatorname{Hom}_A(M,N)\) if \(\alpha\subseteq\beta\), and \(\text{Mor}(M_\alpha,N_\beta)=0\) if \(\alpha\not\subseteq\beta\), for all \(A\)-modules \(M\) and \(N\). By applying this to \(M=N=A\) we get a ring isomorphism \(\text{End}(A_\beta)\cong A\). The problem is easily reduced to the case when \(I\) is any of the critical posets \({\mathcal P}_1,\ldots,{\mathcal P}_{114}\).
The proof is based on a general construction given in the paper and applies the so called “Shelah elevator” which allows us to derive the existence of large rigid families from the existence of small ones.
Reviewer: D.Simson (Toruń)

MSC:

16S50 Endomorphism rings; matrix rings
16G20 Representations of quivers and partially ordered sets
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
06A06 Partial orders, general

Citations:

Zbl 0815.16006
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, Vol. 13. · Zbl 0301.16001
[2] David M. Arnold, Representations of partially ordered sets and abelian groups, Abelian group theory (Perth, 1987) Contemp. Math., vol. 87, Amer. Math. Soc., Providence, RI, 1989, pp. 91 – 109. · doi:10.1090/conm/087/995268
[3] -, Coxeter correspondences and invariants for representations of finite posets, Manuscript.
[4] David M. Arnold and Fred Richman, Field-independent representations of partially ordered sets, Forum Math. 4 (1992), no. 4, 349 – 357. · Zbl 0763.06001 · doi:10.1515/form.1992.4.349
[5] Walter Baur, On the elementary theory of quadruples of vector spaces, Ann. Math. Logic 19 (1980), no. 3, 243 – 262. · Zbl 0453.03010 · doi:10.1016/0003-4843(80)90011-X
[6] Claudia Böttinger and Rüdiger Göbel, Endomorphism algebras of modules with distinguished partially ordered submodules over commutative rings, J. Pure Appl. Algebra 76 (1991), no. 2, 121 – 141. · Zbl 0759.16006 · doi:10.1016/0022-4049(91)90055-7
[7] Claudia Böttinger and Rüdiger Göbel, Modules with two distinguished submodules, Abelian groups (Curaçao, 1991) Lecture Notes in Pure and Appl. Math., vol. 146, Dekker, New York, 1993, pp. 97 – 104. · Zbl 0808.16030
[8] Sheila Brenner and M. C. R. Butler, Endomorphism rings of vector spaces and torsion free abelian groups, J. London Math. Soc. 40 (1965), 183 – 187. · Zbl 0127.25802 · doi:10.1112/jlms/s1-40.1.183
[9] Sheila Brenner, Decomposition properties of some small diagrams of modules, Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 127 – 141.
[10] Sheila Brenner, On four subspaces of a vector space, J. Algebra 29 (1974), 587 – 599. · Zbl 0279.15002 · doi:10.1016/0021-8693(74)90092-1
[11] Sheila Brenner and M. C. R. Butler, Endomorphism rings of vector spaces and torsion free abelian groups, J. London Math. Soc. 40 (1965), 183 – 187. · Zbl 0127.25802 · doi:10.1112/jlms/s1-40.1.183
[12] Sheila Brenner and Claus Michael Ringel, Pathological modules over tame rings, J. London Math. Soc. (2) 14 (1976), no. 2, 207 – 215. · Zbl 0356.16010 · doi:10.1112/jlms/s2-14.2.207
[13] M. C. R. Butler, Some almost split sequences in torsionfree abelian group theory, Abelian group theory (Oberwolfach, 1985) Gordon and Breach, New York, 1987, pp. 291 – 301. · Zbl 0653.20058
[14] A. L. S. Corner, Endomorphism algebras of large modules with distinguished submodules., J. Algebra 11 (1969), 155 – 185. · Zbl 0214.05606 · doi:10.1016/0021-8693(69)90052-0
[15] A. L. S. Corner, Fully rigid systems of modules, Rend. Sem. Mat. Univ. Padova 82 (1989), 55 – 66 (1990). · Zbl 0712.16007
[16] Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57. · Zbl 0332.16015 · doi:10.1090/memo/0173
[17] Ju. A. Drozd, Coxeter transformations and representations of partially ordered sets, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 34 – 42 (Russian).
[18] Manfred Dugas and Bernhard Thomé, Countable Butler groups and vector spaces with four distinguished subspaces, J. Algebra 138 (1991), no. 1, 249 – 272. · Zbl 0724.20035 · doi:10.1016/0021-8693(91)90199-I
[19] M. Dugas, R. Göbel and W. May, Free modules with two distinguished submodules, Comm. Algebra, to appear. · Zbl 0887.16014
[20] Manfred Dugas and Rüdiger Göbel, Automorphism groups of fields, Manuscripta Math. 85 (1994), no. 3-4, 227 – 242. · Zbl 0824.12003 · doi:10.1007/BF02568195
[21] Christophe Reutenauer, Semisimplicity of the algebra associated to a biprefix code, Semigroup Forum 23 (1981), no. 4, 327 – 342. · Zbl 0481.68076 · doi:10.1007/BF02676657
[22] László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. László Fuchs, Infinite abelian groups. Vol. II, Academic Press, New York-London, 1973. Pure and Applied Mathematics. Vol. 36-II.
[23] L. Fuchs, Large indecomposable modules in torsion theories, Aequationes Math. 34 (1987), no. 1, 106 – 111. · Zbl 0631.13010 · doi:10.1007/BF01840130
[24] Pierre Gabriel, Représentations indécomposables des ensembles ordonnés, Séminaire P. Dubreil (26e année: 1972/73), Algèbre, Exp. No. 13, Secrétariat Mathématique, Paris, 1973, pp. 4 (French). D’après L. A. Nazarova et A. V. Roĭter (Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 28 (1972), 5 – 31).
[25] Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71 – 103; correction, ibid. 6 (1972), 309 (German, with English summary). · Zbl 0232.08001 · doi:10.1007/BF01298413
[26] Peter Gabriel, Indecomposable representations. II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971) Academic Press, London, 1973, pp. 81 – 104.
[27] I. M. Gel\(^{\prime}\)fand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970) North-Holland, Amsterdam, 1972, pp. 163 – 237. Colloq. Math. Soc. János Bolyai, 5.
[28] Rüdiger Göbel, Vector spaces with five distinguished subspaces, Results Math. 11 (1987), no. 3-4, 211 – 228. · Zbl 0637.15010 · doi:10.1007/BF03323270
[29] Rüdiger Göbel and Warren May, Four submodules suffice for realizing algebras over commutative rings, J. Pure Appl. Algebra 65 (1990), no. 1, 29 – 43. · Zbl 0716.16015 · doi:10.1016/0022-4049(90)90098-3
[30] Rüdiger Göbel , Abelian groups, Lecture Notes in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1993. · Zbl 0516.20032
[31] Rüdiger Göbel and Martin Ziegler, Very decomposable abelian groups, Math. Z. 200 (1989), no. 4, 485 – 496. · Zbl 0645.20034 · doi:10.1007/BF01160951
[32] H.J. v. Höhne, On weakly positive unit forms, Comment. Math. Helv. 63 (1988), 312-336. · Zbl 0662.15014
[33] S. Kasjan and D. Simson, A peak reduction functor for socle projective representations, J. Algebra 187 (1997), 49-70. CMP 97:06 · Zbl 0904.16008
[34] Otto Kerner, Partially ordered sets of finite representation type, Comm. Algebra 9 (1981), no. 8, 783 – 809. · Zbl 0465.16014 · doi:10.1080/00927878108822619
[35] M. M. Kleiner, Partially ordered sets of finite type, J. Soviet Math. 3 (1975), 607-615. · Zbl 0345.06001
[36] M. M. Kleiner, Faithful representations of partially ordered sets of finite type, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 28 (1972), 42 – 59 (Russian). Investigations on the theory of representations.
[37] Rüdiger Göbel , Abelian groups, Lecture Notes in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1993. · Zbl 0516.20032
[38] R. Mines, C. Vinsonhaler, and W. J. Wickless, Representations and duality, Abelian group theory and related topics (Oberwolfach, 1993) Contemp. Math., vol. 171, Amer. Math. Soc., Providence, RI, 1994, pp. 295 – 303. · Zbl 0821.16012 · doi:10.1090/conm/171/01780
[39] Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. · Zbl 0136.00604
[40] L. A. Nazarova, Partially ordered sets of infinite type, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 5, 963 – 991, 1219 (Russian).
[41] L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, J. Soviet. Math. 3 (1975), 585-607. · Zbl 0336.16031
[42] L. A. Nazarova and A. V. Roĭter, Representations of bipartite completed posets, Comment. Math. Helv. 63 (1988), no. 3, 498 – 526. · Zbl 0671.18002 · doi:10.1007/BF02566776
[43] Fred Richman and Elbert A. Walker, Ext in pre-Abelian categories, Pacific J. Math. 71 (1977), no. 2, 521 – 535. · Zbl 0354.18018
[44] Fred Richman and Elbert A. Walker, Valuated groups, J. Algebra 56 (1979), no. 1, 145 – 167. · Zbl 0401.20049 · doi:10.1016/0021-8693(79)90330-2
[45] C. M. Ringel, Tame Algebras (On algorithms for solving vector space problems II), pp. 137-287 in Representation Theory I, Lecture Notes in Math., Vol. 831, Springer, 1980.
[46] Claus Michael Ringel, Infinite-dimensional representations of finite-dimensional hereditary algebras, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977) Academic Press, London-New York, 1979, pp. 321 – 412.
[47] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. · Zbl 0546.16013
[48] C. M. Ringel and H. Tachikawa, QF-3 rings, J. Reine Angew. Math. 272 (1975), 49-72. · Zbl 0318.16006
[49] Saharon Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 18 (1974), 243 – 256. · Zbl 0318.02053 · doi:10.1007/BF02757281
[50] Saharon Shelah, On the possible number \?\?(\?)= the number of nonisomorphic models \?_{\infty ,\?}-equivalent to \? of power \?, for \? singular, Notre Dame J. Formal Logic 26 (1985), no. 1, 36 – 50. · Zbl 0567.03010 · doi:10.1305/ndjfl/1093870759
[51] D. Simson, Functor categories in which every flat object is projective, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 375 – 380 (English, with Russian summary). · Zbl 0328.18005
[52] Daniel Simson, On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math. 96 (1977), no. 2, 91 – 116. · Zbl 0361.18010
[53] Daniel Simson, Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Applications, vol. 4, Gordon and Breach Science Publishers, Montreux, 1992. · Zbl 0818.16009
[54] Daniel Simson, Posets of finite prinjective type and a class of orders, J. Pure Appl. Algebra 90 (1993), no. 1, 77 – 103. · Zbl 0815.16006 · doi:10.1016/0022-4049(93)90138-J
[55] A. G. Zavadskiĭ and L. A. Nazarova, Partially ordered sets of finite growth, Funktsional. Anal. i Prilozhen. 16 (1982), no. 2, 72 – 73 (Russian).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.