563 research outputs found
Valuative and geometric characterizations of Cox sheaves
We give an intrinsic characterization of Cox sheaves on Krull schemes in
terms of their valuative algebraic properties. We also provide a geometric
characterization of their graded relative spectra in terms of good quotients of
graded schemes, extending the work of Arzhantsev, Derenthal, Hausen and Laface
on relative spectra of Cox sheaves on normal varieties. Moreover, we obtain an
irredundant characterization of Cox rings which in turn produces a normality
criterion for certain graded rings.Comment: 25 page
Categorification of Hopf algebras of rooted trees
We exhibit a monoidal structure on the category of finite sets indexed by
P-trees for a finitary polynomial endofunctor P. This structure categorifies
the monoid scheme (over Spec N) whose semiring of functions is (a P-version of)
the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base
change to Z and collapsing H_0). The monoidal structure is itself given by a
polynomial functor, represented by three easily described set maps; we show
that these maps are the same as those occurring in the polynomial
representation of the free monad on P.Comment: 29 pages. Does not compile with pdflatex due to dependency on the
texdraw package. v2: expository improvements, following suggestions from the
referees; final version to appear in Centr. Eur. J. Mat
Representations of categories of G-maps
We study representations of wreath product analogues of categories of finite
sets. This includes the category of finite sets and injections (studied by
Church, Ellenberg, and Farb) and the opposite of the category of finite sets
and surjections (studied by the authors in previous work). We prove noetherian
properties for the injective version when the group in question is
polycyclic-by-finite and use it to deduce general twisted homological stability
results for such wreath products and indicate some applications to
representation stability. We introduce a new class of formal languages
(quasi-ordered languages) and use them to deduce strong rationality properties
of Hilbert series of representations for the surjective version when the group
is finite.Comment: 27 pages, split off from arXiv:1409.1670v1; v2: added Section 5.2 on
wreath product version of Murnaghan's stability theorem; v3: significant
rewrite; v4: corrected some proof
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