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Grothendieck Rings of Theories of Modules
The model-theoretic Grothendieck ring of a first order structure, as defined
by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the
definable subsets of finite powers of the structure. In this paper we compute
the Grothendieck ring, , of a right -module , where
is any unital ring. As a corollary we prove a conjecture of Prest
that is non-trivial, whenever is non-zero. The main proof uses
various techniques from the homology theory of simplicial complexes.Comment: 42 Page
Grothendieck rings of theories of modules
We consider right modules over a ring, as models of a first order theory. We explorethe definable sets and the definable bijections between them. We employ the notionsof Euler characteristic and Grothendieck ring for a first order structure, introduced byJ. Krajicek and T. Scanlon in [24]. The Grothendieck ring is an algebraic structurethat captures certain properties of a model and its category of definable sets.If M is a module over a product of rings A and B, then M has a decomposition into a direct sum of an A-module and a B-module. Theorem 3.5.1 states that then the Grothendieck ring of M is the tensor product of the Grothendieck rings of the summands.Theorem 4.3.1 states that the Grothendieck ring of every infinite module over afield or skew field is isomorphic to Z[X].Proposition 5.2.4 states that for an elementary extension of models of anytheory, the elementary embedding induces an embedding of the corresponding Grothendieck rings. Theorem 5.3.1 is that for an elementary embedding of modules, we have the stronger result that the embedding induces an isomorphism of Grothendieck rings.We define a model-theoretic Grothendieck ring of the category Mod-R and explorethe relationship between this ring and the Grothendieck rings of general right R-modules. The category of pp-imaginaries, shown by K. Burke in [7] to be equivalentto the subcategory of finitely presented functors in (mod-R; Ab), provides a functorial approach to studying the generators of theGrothendieck rings of R-modules. It is shown in Theorem 6.3.5 that whenever R andS are Morita equivalent rings, the rings Grothendieck rings of the module categories Mod-R and Mod-S are isomorphic.Combining results from previous chapters, we derive Theorem 7.2.1 saying that theGrothendieck ring of any module over a semisimple ring is isomorphic to a polynomialring Z[X1,...,Xn] for some n.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras
We study the representation theory of three towers of algebras which are
related to the symmetric groups and their Hecke algebras. The first one is
constructed as the algebras generated simultaneously by the elementary
transpositions and the elementary sorting operators acting on permutations. The
two others are the monoid algebras of nondecreasing functions and nondecreasing
parking functions. For these three towers, we describe the structure of simple
and indecomposable projective modules, together with the Cartan map. The
Grothendieck algebras and coalgebras given respectively by the induction
product and the restriction coproduct are also given explicitly. This yields
some new interpretations of the classical bases of quasi-symmetric and
noncommutative symmetric functions as well as some new bases.Comment: 12 pages. Presented at FPSAC'06 San-Diego, June 2006 (minor
explanation improvements w.r.t. the previous version
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