Title of paper: Presheaves of chain complexes Author: J.F. Jardine AMS Classification numbers: 55P42 55U15 18G15 Address of Author: Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada Email: jardine@uwo.ca 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 different 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.