  We study Gorenstein categories. We show that such a category has Tate
cohomological functors and Avramov-Martsinkovsky exact sequences connecting the
Gorenstein relative, the absolute and the Tate cohomological functors. We show
that such a category has what Hovey calls an injective model structure and also
a projective model structure in case the category has enough projectives. As
examples we show that if X is a locally Gorenstein projective scheme then the
category Qco(X) of quasi-coherent sheaves on $X$ is such a category and so has
these features.
