Pre-lie algebras and incidence categories of colored rooted trees

The incidence category \C_{\F} of a family \F of colored posets closed under disjoint unions and the operation of taking convex sub-posets was introduced by the author in \cite{Sz}, where the Ringel-Hall algebra \H_{\F} of \C_{\F} was also defined. We show that if the Hasse diagrams underlying \F are rooted trees, then the subspace \n_{\F} of primitive elements of \H_{\F} carries a pre-Lie structure, defined over ℤ, and with positive structure constants. We give several examples of \n_{\F}, including the nilpotent subalgebras of n, Ln, and several others

Twisted modules and co-invariants for commutative vertex algebras of jet schemes

Let Z⊂k be an affine scheme over \C and \J Z its jet scheme. It is well-known that \mathbb{C}[\J Z], the coordinate ring of \J Z, has the structure of a commutative vertex algebra. This paper develops the orbifold theory for \mathbb{C}[\J Z]. A finite-order linear automorphism g of Z acts by vertex algebra automorphisms on \mathbb{C}[\J Z]. We show that \mathbb{C}[\J^g Z], where \J^g Z is the scheme of g--twisted jets has the structure of a g-twisted \mathbb{C}[\J Z] module. We consider spaces of orbifold coinvariants valued in the modules \mathbb{C}[\J^g Z] on orbicurves [Y/G], with Y a smooth projective curve and G a finite group, and show that these are isomorphic to ℂ[ZG]

Quasicoherent sheaves on projective schemes over F_1

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves on MProj(A), and we prove several basic results regarding these. We show that: 1. every quasicoherent sheaf F on MProj(A) can be constructed from a graded A-set in analogy with the construction of quasicoherent sheaves on from graded R-modules 2. if F is coherent on MProj(A), then F(n) is globally generated for large enough n, and consequently, that F is a quotient of a finite direct sum of invertible sheaves 3. if F is coherent on MProj(A), then gamma(MProj(A)) is finitely generated over A0 (and hence a finite set if A0 = {0, 1}). The last part of the paper is devoted to classifying coherent sheaves on P_1 in terms of certain directed graphs and gluing data. The classification of these over F_1 is shown to be much richer and combinatorially interesting than in the case of ordinary P_1, and several new phenomena emerge.55203915 - Simons Foundatio

Quasicoherent sheaves on projective schemes over F_1

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves (of pointed sets) on MProj(A), and we prove several basic results regarding these. We show that: 1.) Every quasicoherent sheaf F on MProj(A) can be constructed from a graded A--set in analogy with the construction of quasicoherent sheaves on Proj(R) from graded R--modules. 2.) High enough twists of coherent sheaves are generated by finitely many global sections, hence that every coherent sheaf is a quotient of a locally free sheaf. 3.) Coherent sheaves have finite spaces of global sections. The last part of the paper is devoted to classifying coherent sheaves on P^1 in terms of certain directed graphs and gluing data. The classification of these over F_1 is shown to be much richer and combinatorially interesting than in the case of ordinary P^1, and several new phenomena emerge.Comment: arXiv admin note: text overlap with arXiv:1009.357