Finitistic Weak Dimension of Commutative Arithmetical Rings
  It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $\leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases. 
Francois Couchot (LMNO)
math.AC