Centre De Recerca Matematica Research Articles

Stratification and $$\pi $$ π -cosupport: finite groups

Acknowledgements Part of this article is based on work supported by the National Science Foundation under Grant No. 0932078000, while DB, SBI, and HK were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the 2012–2013 Special Year in Commutative Algebra. The authors thank the Centre de Recerca Matematica, Barcelona, for hospitality during a visit in April 2015 that turned out to be productive and pleasant. SBI and JP were partly supported by NSF Grants DMS-1503044 and DMS-0953011, respectively. We are grateful to Eric Friedlander for comments on an earlier version of this paper.

Arithmetic of Shimura curves

This is the note for a series of lectures that the author gave at the Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

Preface: Old and New Perspectives on the Tutte Polynomial

This paper introduces a special issue on the Tutte polynomial derived from the Sec- ond Workshop on Tutte Polynomials and Applications, 2005, held at the Centre de Recerca Matem atica, Bellaterra, Catalonia. We discuss the prehistory of Tutte polynomials and two cur- rent areas of research, to what extent a graph is determined by its chromatic or Tutte polynomial and generic versions of Tutte polynomials.

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups

Descending Chains of Modules and Jordan–Hölder Theorem

It is clear that there is a relation between the notion of unique factorization domain (a commutative domain in which every non-zero non-invertible element is a product of irreducible elements, and if a1a2 . . . an = a1a ′ 2 . . . a ′ m are any two such factorizations, then n = m and there exists a permutation σ such that ai and aσ(i) are associates for every i = 1, 2, . . . , n), the Krull-Schmidt Theorem (every module of finite length is a direct sum of indecomposable modules, and if A1⊕A2⊕· · ·⊕An = A1⊕A2⊕· · ·⊕Am are any two such decompositions, then n = m and there exists a permutation σ such that Ai ∼= Aσ(i) for every i = 1, 2, . . . , n), and the Jordan-Holder Theorem (every module A of finite length has a composition series, and if A = A0 ≥ A1 ≥ · · · ≥ An = 0 and A = A0 ≥ A1 ≥ · · · ≥ Am = 0 are any two composition series, then n = m and there exists a permutation σ such that Ai−1/Ai ∼= Aσ(i)−1/Aσ(i) for every i = 1, 2, . . . , n). The relation between these three contexts is that what we state is equivalent to saying that some commutative monoid is free in all these three cases. An integral domain R is a unique factorization domain if and only if the commutative monoid R∗/U(R) is free, where we have denoted by R∗ the multiplicative monoid R \ {0} and by U(R) the group of invertible elements of R. In this case, a free set of generators of R∗/U(R) is ∗Partially supported by Ministero dell’Universita e della Ricerca Scientifica e Tecnologica (Progetto di ricerca di rilevante interesse nazionale “Nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani”), Italy. This paper was written when the authors were visiting the Centre de Recerca Matematica, Barcelona. They wish to thank the Centre for the hospitality provided.

Baer duality for commutative rings

∗ The author is partially supported by and by the Hungarian National Foundation for Scientific Research, grant no. T029525. Part of this paper was written while the author was visiting the Centre de Recerca Matematica supported by a grant from the Ministerio de Educacion y Ciencia (Spain). † Part of this paper was written while the author was visiting the Universita di Ferrara with partial financial support from MURST (Italy), she wants to thank her host for its hospitality. She was also supported by the DGESIC (Spain) through the project PB980873, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. ‡ Part of this article was written while the first author was a member of G.N.S.A.G.A. of C.N.R. with partial financial support from M.U.R.S.T. and H.C.M. project n.CHRXCT93-0091.