Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the lattice $L$ of subspaces of a Hilbert space $H$ and homological properties (flatness and injectivity) of $H$ regarded as a natural module over the reflexive algebra $\operatorname{Alg}L$ that consists of all operators leaving invariant each element of the lattice $L$. It follows from these results that the cohomology groups with coefficients in $\mathscr B(H)$ are trivial for a broad class of reflexive algebras.