Hopf Algebra of Building Sets

The combinatorial Hopf algebra on building sets $BSet$ extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized Dehn-Sommerville relations for the Hopf algebra $BSet$, we define a class of building sets called eulerian and show that eulerian building sets satisfy Bayer-Billera relations. We show the existence of the $\mathbf{c}\mathbf{d}-$index, the polynomial in two noncommutative variables associated to an eulerian building set. The complete characterization of eulerian building sets is given in terms of combinatorics of intersection posets of antichains of finite sets.

MONOMORPHISMS AND KERNELS IN THE CATEGORY OF FIRM MODULES

In this paper we consider for a non-unital ring R, the category of firm R-modules for a non-unital ring R, i.e. the modules M such that the canonical morphism μM : R ⊗RM → M given by r ⊗ m ↦ rm is an isomorphism. This category is a natural generalization of the usual category of unitary modules for a ring with identity and shares many properties with it. The only difference is that monomorphisms are not always kernels. It has been proved recently that this category is not Abelian in general by providing an example of a monomorphism that is not a kernel in a particular case. In this paper we study the lattices of monomorphisms and kernels, proving that the lattice of monomorphisms is a modular lattice and that the category of firm modules is Abelian if and only if the composition of two kernels is a kernel.

Log-isocristaux surconvergents et holonomie

Let 𝒱 be a complete discrete valuation ring of unequal characteristic with perfect residue field. Let $\X $ be a separated smooth formal 𝒱-scheme, 𝒵 be a normal crossing divisor of $\X $, $\X ^\#:= (\X , \ZZ )$ be the induced formal log-scheme over 𝒱 and $u: \X ^\# \rightarrow \X $ be the canonical morphism. Let X and Z be the special fibers of $\X $ and 𝒵, T be a divisor of X and ℰ be a log-isocrystal on $\X ^\#$ overconvergent along T, that is, a coherent left $\D ^\dag _{\X ^\#} (\hdag T) _{\Q }$-module, locally projective of finite type over $ \O _{\X } (\hdag T) _{\Q }$. We check the relative duality isomorphism: $u_{T,+} (\E ) \riso u_{T,!} (\E (\ZZ ))$. We prove the isomorphism $u_{T,+} (\E ) \riso \D ^\dag _{\X } (\hdag T) _{\Q } \otimes _{\D ^\dag _{\X ^\#} (\hdag T) _{\Q }} \E (\ZZ )$, which implies their holonomicity as $\D ^\dag _{\X } (\hdag T) _{\Q }$-modules. We obtain the canonical morphism ρℰ : uT,+(ℰ)→ℰ(†Z). When ℰ is moreover an isocrystal on $\X $ overconvergent along T, we prove that ρℰ is an isomorphism.

On the Classification of Lie Pseudo-Algebras

For every ( differentiable) bundle E over a manifold M, Jk(E) denotes the set of all k-jets of local (differentiable) sections of the bundle E. Jk(E) is a bundle over M such that if X is a section of E, thenis a (differentiable) section of Jk(E). If E is a vector bundle, Jk(E) is a vector bundle and we have the canonical exact sequence of vector bundleswhere Sk(T*) is the symmetric Whitney tensor product of the cotangent vector bundle T* of M. and π is the canonical morphism which associates to each k-jet of section its jet of inferior order.