In this chapter, we explore two fundamental procedures. The first focuses on understanding the behavior of an operator space tensor norm in finite dimensions, while the second deals with quotients by subspaces of finite codimension. In particular, we introduce the concepts of finitely and cofinitely-generated o.s. tensor norms, which parallel the corresponding notions in the Banach space setting. We establish key properties, provide illustrative examples, and demonstrate that the property of being finitely-generated is preserved under the intersection and sum procedures.