104,910 research outputs found

### Graded $F$-modules and Local Cohomology

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of
characteristic $p>0,$ let \m=(x_1,..., x_n) be the maximal ideal generated by
the variables, let $^*E$ be the naturally graded injective hull of R/\m and
let $^*E(n)$ be $^*E$ degree shifted downward by $n.$ We introduce the notion
of graded $F$-modules (as a refinement of the notion of $F$-modules) and show
that if a graded $F$-module \M has zero-dimensional support, then \M, as a
graded $R$-module, is isomorphic to a direct sum of a (possibly infinite)
number of copies of $^*E(n).$
As a consequence, we show that if the functors $T_1,...,T_s$ and $T$ are
defined by $T_{j}=H^{i_j}_{I_j}(-)$ and $T=T_1\circ...\circ T_s,$ where
$I_1,..., I_s$ are homogeneous ideals of $R,$ then as a naturally graded
$R$-module, the local cohomology module H^{i_0}_{\m}(T(R)) is isomorphic to
$^*E(n)^c,$ where $c$ is a finite number. If $\text{char}k=0,$ this question is
open even for $s=1.$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
$f(B^{2})$ represents the non-minimal coupling effect between vector fields and
gravity, the $f=0$ case is the minimal coupling case. For a clear comparison
with scalar field, we define a new function $F=\pm B^{2}/12+f(B^{2})$ where
$B^{2}=A_{\mu}A^{\mu}$, $A_{\mu}$ 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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