On cohomological dimension and height relative to bigraded modules

Let [Formula: see text] be a field, and let [Formula: see text] denote a standard bigraded polynomial ring over [Formula: see text]. Let [Formula: see text] be a finitely generated bigraded [Formula: see text]-module, and set [Formula: see text]. This paper investigates various algebraic invariants of [Formula: see text] with respect to [Formula: see text], including the bigraded versions of mcd, height, and cohomological dimension, as well as the behavior of its graded components [Formula: see text], viewed as finitely generated graded [Formula: see text]-modules. Several applications and examples are presented.