1,589 research outputs found
An algebraic study of exactness in partial contexts
DMF@?s are the natural algebraic tool for modelling reasoning with Korner@?s partial predicates. We provide two representation theorems for DMF@?s which give rise to two adjunctions, the first between DMF and the category of sets and the second between DMF and the category of distributive lattices with minimum. Then we propose a logic L"{"1"} for dealing with exactness in partial contexts, which belongs neither to the Leibniz, nor to the Frege hierarchies, and carry on its study with techniques of abstract algebraic logic. Finally a fully adequate and algebraizable Gentzen system for L"{"1"} is given
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Abstract representation theory of Dynkin quivers of type A
We study the representation theory of Dynkin quivers of type A in abstract
stable homotopy theories, including those associated to fields, rings, schemes,
differential-graded algebras, and ring spectra. Reflection functors, (partial)
Coxeter functors, and Serre functors are defined in this generality and these
equivalences are shown to be induced by universal tilting modules, certain
explicitly constructed spectral bimodules. In fact, these universal tilting
modules are spectral refinements of classical tilting complexes. As a
consequence we obtain split epimorphisms from the spectral Picard groupoid to
derived Picard groupoids over arbitrary fields.
These results are consequences of a more general calculus of spectral
bimodules and admissible morphisms of stable derivators. As further
applications of this calculus we obtain examples of universal tilting modules
which are new even in the context of representations over a field. This
includes Yoneda bimodules on mesh categories which encode all the other
universal tilting modules and which lead to a spectral Serre duality result.
Finally, using abstract representation theory of linearly oriented
-quivers, we construct canonical higher triangulations in stable
derivators and hence, a posteriori, in stable model categories and stable
-categories
Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes
We study the exactness of certain combinatorially defined complexes which
generalize the Orlik-Solomon algebra of a geometric lattice. The main results
pertain to complex reflection arrangements and their restrictions. In
particular, we consider the corresponding relation complexes and give a simple
proof of the -formality of these hyperplane arrangements. As an application,
we are able to bound the Castelnouvo-Mumford regularity of certain modules over
polynomial rings associated to Coxeter arrangements (real reflection
arrangements) and their restrictions. The modules in question are defined using
the relation complex of the Coxeter arrangement and fiber polytopes of the dual
Coxeter zonotope. They generalize the algebra of piecewise polynomial functions
on the original arrangement
Tame Class Field Theory for Global Function Fields
We give a function field specific, algebraic proof of the main results of
class field theory for abelian extensions of degree coprime to the
characteristic. By adapting some methods known for number fields and combining
them in a new way, we obtain a different and much simplified proof, which
builds directly on a standard basic knowledge of the theory of function fields.
Our methods are explicit and constructive and thus relevant for algorithmic
applications. We use generalized forms of the Tate-Lichtenbaum and Ate
pairings, which are well-known in cryptography, as an important tool.Comment: 25 pages, to appear in Journal of Number Theor
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