Torsion on non-integral schemes and singular sheaves of dimension one

In Part I we analyze the torsion of a module over a non-integral ring and the torsion of a sheaf on a non-integral scheme. We give an explicit definition of the torsion subsheaf of a quasi-coherent sheaf and prove a condition under which it is also quasi-coherent. Using the associated primes of a module and the primary decomposition of ideals, we review the main criteria for torsion-freeness and purity of a sheaf. We also discuss properties of the restriction of a coherent sheaf to its annihilator and its Fitting support and prove that sheaves of pure dimension are torsion-free on their support. Part II deals with the problem of determining "how many" sheaves in the fine Simpson moduli spaces M=M_{dm-1}(P2) of stable sheaves on the projective plane with linear Hilbert polynomial dm-1 for d>3 are not locally free on their support. Such sheaves are called singular and form a closed subvariety M' in M. We describe sheaves in an open subvariety of M as twisted ideal sheaves of curves of degree d. In order to determine the singular ones, we characterize free ideals in terms of the absence of two coeffcients in the defining polynomial to conclude that M' is singular of codimension 2.