2,469 research outputs found
Dagger closure in regular rings containing a field
We prove that dagger closure is trivial in regular domains containing a field
and that graded dagger closure is trivial in polynomial rings over a field. We
also prove that Heitmann's full rank one closure coincides with tight closure
in positive characteristic under some mild finiteness conditions. Furthermore,
we prove that dagger closure is always contained in solid closure and that the
forcing algebra for an element contained in dagger closure is parasolid.Comment: 12 pages, v2: added one corollary and two references, v3: Major
simplification in proof of main thm due to the suggestion of a referee, minor
changes in expositio
Dagger closure and solid closure in graded dimension two
We introduce a graded version of dagger closure and prove that it coincides
with solid closure for homogeneous ideals in two dimensional
-graded domains of finite type over a field.Comment: 26 page
An embedding theorem for Hilbert categories
We axiomatically define (pre-)Hilbert categories. The axioms resemble those
for monoidal Abelian categories with the addition of an involutive functor. We
then prove embedding theorems: any locally small pre-Hilbert category whose
monoidal unit is a simple generator embeds (weakly) monoidally into the
category of pre-Hilbert spaces and adjointable maps, preserving adjoint
morphisms and all finite (co)limits. An intermediate result that is important
in its own right is that the scalars in such a category necessarily form an
involutive field. In case of a Hilbert category, the embedding extends to the
category of Hilbert spaces and continuous linear maps. The axioms for
(pre-)Hilbert categories are weaker than the axioms found in other approaches
to axiomatizing 2-Hilbert spaces. Neither enrichment nor a complex base field
is presupposed. A comparison to other approaches will be made in the
introduction.Comment: 24 page
An inclusion result for dagger closure in certain section rings of abelian varieties
We prove an inclusion result for graded dagger closure for primary ideals in
symmetric section rings of abelian varieties over an algebraically closed field
of arbitrary characteristic.Comment: 11 pages, v2: updated one reference, fixed 2 typos; final versio
Local acyclicity in -adic cohomology
We prove an analogue for -adic coefficients of the Deligne--Laumon theorem
on local acyclicity for curves. That is, for an overconvergent -isocrystal
on a relative curve admitting a good compactification,
we show that the cohomology sheaves of are overconvergent
isocrystals if and only if has constant Swan conductor at infinity.Comment: 45 pages, comments welcom
H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics
A certain class of Frobenius algebras has been used to characterize
orthonormal bases and observables on finite-dimensional Hilbert spaces. The
presence of units in these algebras means that they can only be realized
finite-dimensionally. We seek a suitable generalization, which will allow
arbitrary bases and observables to be described within categorical
axiomatizations of quantum mechanics. We develop a definition of H*-algebra
that can be interpreted in any symmetric monoidal dagger category, reduces to
the classical notion from functional analysis in the category of (possibly
infinite-dimensional) Hilbert spaces, and hence provides a categorical way to
speak about orthonormal bases and quantum observables in arbitrary dimension.
Moreover, these algebras reduce to the usual notion of Frobenius algebra in
compact categories. We then investigate the relations between nonunital
Frobenius algebras and H*-algebras. We give a number of equivalent conditions
to characterize when they coincide in the category of Hilbert spaces. We also
show that they always coincide in categories of generalized relations and
positive matrices.Comment: 29 pages. Final versio
- …