Locally projective monoidal model structure for complexes of quasi-coherent sheaves on P1(k)

We will generalize the projective model structure in the categoryof unbounded complexes of modules over a commutative ring tothe category of unbounded complexes of quasi-coherent sheavesover the projective line. Concretely we will define a locallyprojective model structure in the category of complexes of quasi-coherentsheaves on the projective line. In this model structure thecofibrant objects are the dg-locally projective complexes. Wealso describe the fibrations of this model structure and showthat the model structure is monoidal. We point out that thismodel structure is necessarily different from other known modelstructures such as the injective model structure and the locallyfree model structure.