  For a noetherian scheme, we introduce its unbounded stable derived category.
This leads to a recollement which reflects the passage from the bounded derived
category of coherent sheaves to the quotient modulo the subcategory of perfect
complexes. Some applications are included, for instance an analogue of maximal
Cohen-Macaulay approximations, a construction of Tate cohomology, and an
extension of the classical Grothendieck duality. In addition, the relevance of
the stable derived category in modular representation theory is indicated.
