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Graded -modules and Local Cohomology
Let be a polynomial ring over a field of
characteristic let \m=(x_1,..., x_n) be the maximal ideal generated by
the variables, let be the naturally graded injective hull of R/\m and
let be degree shifted downward by We introduce the notion
of graded -modules (as a refinement of the notion of -modules) and show
that if a graded -module \M has zero-dimensional support, then \M, as a
graded -module, is isomorphic to a direct sum of a (possibly infinite)
number of copies of
As a consequence, we show that if the functors and are
defined by and where
are homogeneous ideals of then as a naturally graded
-module, the local cohomology module H^{i_0}_{\m}(T(R)) is isomorphic to
where is a finite number. If this question is
open even for Comment: Revised result in section
The Slow-Roll and Rapid-Roll Conditions in the Space-like Vector Field Scenario
In this note we derive the slow-roll and rapid-roll conditions for the
minimally and non-minimally coupled space-like vector fields. The function
represents the non-minimal coupling effect between vector fields and
gravity, the case is the minimal coupling case. For a clear comparison
with scalar field, we define a new function where
, is the "comoving" vector field. With
reference to the slow-roll and rapid-roll conditions, we find the small-field
model is more suitable than the large-field model in the minimally coupled
vector field case. And as a non-minimal coupling example, the F=0 case just has
the same slow-roll conditions as the scalar fields.Comment: no figur
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