3,233 research outputs found
Groups elementarily equivalent to a free nilpotent group of finite rank
In this paper we give a complete algebraic description of groups elementarily
equivalent to a given free nilpotent group of finite rank
Deformed preprojective algebras of type L: Kuelshammer spaces and derived equivalences
Bialkowski, Erdmann and Skowronski classified those indecomposable
self-injective algebras for which the Nakayama shift of every (non-projective)
simple module is isomorphic to its third syzygy. It turned out that these are
precisely the deformations, in a suitable sense, of preprojective algebras
associated to the simply laced ADE Dynkin diagrams and of another graph L_n,
which also occurs in the Happel-Preiser-Ringel classification of subadditive
but not additive functions. In this paper we study these deformed preprojective
algebras of type L via their Kuelshammer spaces, for which we give precise
formulae for their dimensions. These are known to be invariants of the derived
module category, and even invariants under stable equivalences of Morita type.
As main application of our study of Kuelshammer spaces we can distinguish many
(but not all) deformations of the preprojective algebra of type L up to stable
equivalence of Morita type, and hence also up to derived equivalence.Comment: 24 page
Decomposition spaces in combinatorics
A decomposition space (also called unital 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses (up to homotopy) composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In the present contribution, after establishing a formula for the section coefficients, we survey a large supply of examples, emphasising the notion's firm roots in classical combinatorics. The first batch of examples, similar to binomial posets, serves to illustrate two key points: (1) the incidence algebra in question is realised directly from a decomposition space, without a reduction step, and reductions are often given by CULF functors; (2) at the objective level, the convolution algebra is a monoidal structure of species. Specifically, we encounter the usual Cauchy product of species, the shuffle product of L-species, the Dirichlet product of arithmetic species, the Joyal-Street external product of q-species and the Morrison `Cauchy' product of q-species, and in each case a power series representation results from taking cardinality. The external product of q-species exemplifies the fact that Waldhausen's S-construction on an abelian category is a decomposition space, yielding Hall algebras. The next class of examples includes Schmitt's chromatic Hopf algebra, the Fa\`a di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and several variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. We finish by computing the M\Preprin
Relative Serre functor for comodule algebras
Let be a finite tensor category, and let be an
exact left -module category. A relative Serre functor of
, introduced by Fuchs, Schaumann and Schweigert, is an endofunctor
on together with a natural isomorphism
for , where is
the internal Hom functor of . In this paper, we discuss the case
where and
for a finite-dimensional Hopf algebra and a finite-dimensional exact left
-comodule algebra . We give an explicit description of a relative Serre
functor of and its twisted module structure in terms of
integrals of and the Frobenius structure of . We also study pivotal
structures on and give some explicit examples.Comment: 48 page
Toric Geometry and the Semple-Nash modification
This paper proposes some material towards a theory of general toric varieties
without the assumption of normality. Their combinatorial description involves a
fan to which is attached a set of semigroups subjected to gluing-up conditions.
In particular it contains a combinatorial construction of the blowing up of a
sheaf of monomial ideals on a toric variety. In the second part it is shown
that over an algebraically closed base field of zero characteristic the
Semple-Nash modification of a general toric variety is isomorphic to the
blowing up of the sheaf of logarithmic jacobian ideals and that in any
characteristic this blowing-up is an isomorphism if and only if the toric
variety is non singular. In the second part we prove that orders on the lattice
of monomials (toric valuations) of maximal rank are uniformized by iterated
Sempla-Nash modifications.Comment: New version. Appeared in "Revista de la Real Academia de Ciencias
Exactas, Fisicas y Naturales, Serie A Matematicas", October 2012 (Electronic
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