I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra over a simplicial ring.
- Let $A$ be a simplicial commutative ring. Is the category of simplicial modules over $A$ abelian?
- If so, does its derived category exist?
- Does any simplicial module have a projective resolution? a flat resolution? an injective resolution?
Consider the associated DG-algebra $N(A)$. Is there any relation between the derived category of DG-modules and the derived category I was hoping exists in 3 above?
Any references that discuss these basic issues?