Proprietà di fasci algebrici coerenti e lisci su varietà algebriche affini normali (Q2536761)

In this paper all the results obtained in the preceding one [Zbl 0186.26001] for varieties with unique factorization algebra, are extended to the normal affine varieties. This is possible because the singular locus \(H\) of a normal variety \((V,\mathcal A_V)\) has \(\text{cdm} >1\) in \(V\), and the affine open sets of \(V\) which do not intersect \(H\), are algebraic affine varieties with locally unique factorization algebra. Because of this, given a torsion-free coherent algebraic sheaf of rank \(r>1\) it becomes possible to determine a subsheaf \(\overline{\mathcal M}\) of \((\mathcal A_V^r/(V-H)\) isomorphic to \((\mathcal M^{(r)}/(V-H)\), which admits some sections (on \(V-H)\) whose set of zeros is free from components of \(\text{cdm}\,1\) in \(V\). One succeeds then in constructing a subsheaf of \(\mathcal A_V^r\) isomorphic to \(\mathcal M^{(r)}\) the restriction of which to \(V-H\) coincides with \(\overline{\mathcal M}\) and which has some sections with set of zeros of \(\text{cdm} >1\) in \(V\). The existence of the short exact sequences: \[ 0\longrightarrow \mathcal A_V^h\longrightarrow \mathcal M^{(r)} \longrightarrow \mathcal M^{(r-h)} \longrightarrow 0 \qquad (1\le h\le r-1) \] is deduced as in the paper reviewed above (loc. cit.).