Graded FF-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