### Presheaves of chain complexes, by J.F. Jardine

This paper gives the basic constructions for homology theory in the
category of modules over a presheaf of commutative rings with
unit. The category of simplicial modules inherits a proper closed
simplicial model structure from the category of simplicial
presheaves. The corresponding stable category is described by several
models, including infinitely graded chain complexes, spectrum objects
in simplicial modules, and symmetric spectrum objects in simplicial
modules. The tensor product of simplicial modules induces a symmetric
monoidal tensor product on the category of symmetric spectrum objects,
by analogy with the construction of the smash product for symmetric
spectra.

This paper is in preliminary form only, and is expected
to pass through several revisions. Proofs of the displayed results are
in place, but it is expected that more material on Tor functors and
the relation with motivic homotopy theory will be added later.

The paper is available in dvi, ps and pdf formats at
Jardine's home page:

J.F. Jardine <jardine@uwo.ca>