140 research outputs found
The Internal Operads of Combinatory Algebras
We argue that operads provide a general framework for dealing with
polynomials and combinatory completeness of combinatory algebras, including the
classical -algebras, linear -algebras, planar
-algebras as well as the braided -algebras. We show that every extensional combinatory algebra gives rise to
a canonical closed operad, which we shall call the internal operad of the
combinatory algebra. The internal operad construction gives a left adjoint to
the forgetful functor from closed operads to extensional combinatory algebras.
As a by-product, we derive extensionality axioms for the classes of combinatory
algebras mentioned above
Arithmetic of positive characteristic L-series values in Tate algebras
The second author has recently introduced a new class of L-series in the
arithmetic theory of function fields over finite fields. We show that the value
at one of these L-series encode arithmetic informations of certain Drinfeld
modules defined over Tate algebras. This enables us to generalize Anderson's
log-algebraicity Theorem and Taelman's Herbrand-Ribet Theorem.Comment: final versio
Complete intersections and mod p cochains
We give homotopy invariant definitions corresponding to three well known
properties of complete intersections, for the ring, the module theory and the
endomorphisms of the residue field, and we investigate them for the mod p
cochains on a space, showing that suitable versions of the second and third are
equivalent and that the first is stronger. We are particularly interested in
classifying spaces of groups, and we give a number of examples.
This paper follows on from arXiv:0906.4025 which considered the classical
case of a commutative ring and arXiv:0906.3247 which considered the case of
rational homotopy theory.Comment: To appear in AG
Equations of tropical varieties
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as =(ℝ∪{−∞},max,+) by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of -points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of -schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting
Algebraic structures of tropical mathematics
Tropical mathematics often is defined over an ordered cancellative monoid
\tM, usually taken to be (\RR, +) or (\QQ, +). Although a rich theory has
arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted
algebraic structure theory, and also do not reflect certain valuation-theoretic
properties, thereby forcing researchers to rely often on combinatoric
techniques.
In this paper we describe an alternative structure, more compatible with
valuation theory, studied by the authors over the past few years, that permits
fuller use of algebraic theory especially in understanding the underlying
tropical geometry. The idempotent max-plus algebra of an ordered monoid
\tM is replaced by R: = L\times \tM, where is a given indexing semiring
(not necessarily with 0). In this case we say layered by . When is
trivial, i.e, , is the usual bipotent max-plus algebra. When
we recover the "standard" supertropical structure with its
"ghost" layer. When L = \NN we can describe multiple roots of polynomials
via a "layering function" . Likewise, one can define the layering
componentwise; vectors are called
tropically dependent if each component of some nontrivial linear combination
\sum \a_i v_i is a ghost, for "tangible" \a_i \in R. Then an
matrix has tropically dependent rows iff its permanent is a ghost.
We explain how supertropical algebras, and more generally layered algebras,
provide a robust algebraic foundation for tropical linear algebra, in which
many classical tools are available. In the process, we provide some new results
concerning the rank of d-independent sets (such as the fact that they are
semi-additive),put them in the context of supertropical bilinear forms, and lay
the matrix theory in the framework of identities of semirings.Comment: 19 page
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