Quasi locally free sheaves on primitive double schemes

Let X₂ be a complex primitive double scheme, X=(X₂)red. The ideal sheaf of X is a line bundle on X. Let E be a coherent sheaf on X₂. Then there exists a nonempty open subset U of X₂ and integers m₁, m₂, uniquely determined by E, such that E|U is locally isomorphic to (OX⊗ Cm₁)⊕(OX₂⊗ Cm₂). If we can take U=X₂, we say the E is quasi locally free. If m₁=1 we say that E is rigid. In this case, we show that deformations of E are of the same type. So locally free sheaves and rigid quasi locally free sheaves on X₂ play the same role on X₂ as locally free sheaves do on smooth schemes. In this paper we study the existence and description of quasi locally free sheaves, and flat families of such sheaves.