?? faint attempt to start writing the paper ...

toric quasicoherent sheaves

in this paper we develop a concept of "toric quasicoherent sheaf over a toric variety" analogous to that of "quasicoherent sheaf over an ordinary variety".

[of course, this is probably already a well-known concept under some name somewhere, so even if there's still enough additional material to make the paper worthwhile i'd have to re-write the above to take the well-known-ness into account...]

just as the quasicoherent sheaves over a (decent?) variety form a cocomplete abelian category equipped with a nice tensor product, the toric quasicoherent sheaves over a toric variety form a grothendieck topos equipped with a nice tensor product.

the conceptual motivation behind this development is mostly not described here, though some of it may be relatively obvious. (variety : toric variety :: vector space : set.)

1. the cocone category of a fan

a toric variety can be considered as a so-called "fan", made out of "cones" which are conical subsets of a lattice (in the sense of fg free abelian group) which have "faces" which are other cones of lower dimension. more convenient for present purposes is the dual "co-fan" made out of the "co-cones" dual to the cones in the fan, bearing a "co-face" relationship where the dual cones bear a face relationship. (the co-fan may at first however seem less intuitive than the fan because the concept of "co-face" may be less familiar than that of "face"; a co-face is bigger than the original where a face is smaller than the original, and the way that co-faces "lose" a dimension is by becoming translation-invariant in that dimension.)

the cocones of the co-fan of a toric variety v form a category v# where the morphisms are translation maps. the main purpose of v# here is that the category v% of "toric quasicoherent sheaves over v" is defined as a certain subcategory of the category of set-valued functors on v#. the name "toric quasicoherent sheaf" is justified in part by the fact that abelian groups in v% are essentially the same thing as ordinary quasicoherent sheaves over v.

the cocone category v# has some notable structure which is of use in constructing the toric quasicoherent sheaf category v% and in establishing corresponding structure on it: v# has filtered colimits (a co-face is the union of an increasing sequence of subobjects isomorphic to the co-cone of which it's a co-face), and a semi-monoidal structure compatible with the filtered colimits. there's generally no unit object for the semi-monoidal structure except in the case where v is affine, though.

(in fact, in the case where v is affine, v% is the topos of actions of the commutative monoid given as the "coordinate monoid" of v, while v# is the category of models of that topos. this may serve as a guide to the way in which v% and v# are, like v itself, "glued together from affine pieces" in the general non-affine case.)