99,593 research outputs found

    Graded FF-modules and Local Cohomology

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    Let R=k[x1,...,xn]R=k[x_1,..., x_n] be a polynomial ring over a field kk of characteristic p>0,p>0, let \m=(x_1,..., x_n) be the maximal ideal generated by the variables, let βˆ—E^*E be the naturally graded injective hull of R/\m and let βˆ—E(n)^*E(n) be βˆ—E^*E degree shifted downward by n.n. We introduce the notion of graded FF-modules (as a refinement of the notion of FF-modules) and show that if a graded FF-module \M has zero-dimensional support, then \M, as a graded RR-module, is isomorphic to a direct sum of a (possibly infinite) number of copies of βˆ—E(n).^*E(n). As a consequence, we show that if the functors T1,...,TsT_1,...,T_s and TT are defined by Tj=HIjij(βˆ’)T_{j}=H^{i_j}_{I_j}(-) and T=T1∘...∘Ts,T=T_1\circ...\circ T_s, where I1,...,IsI_1,..., I_s are homogeneous ideals of R,R, then as a naturally graded RR-module, the local cohomology module H^{i_0}_{\m}(T(R)) is isomorphic to βˆ—E(n)c,^*E(n)^c, where cc is a finite number. If chark=0,\text{char}k=0, this question is open even for s=1.s=1.Comment: Revised result in section

    The Slow-Roll and Rapid-Roll Conditions in the Space-like Vector Field Scenario

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    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(B2)f(B^{2}) represents the non-minimal coupling effect between vector fields and gravity, the f=0f=0 case is the minimal coupling case. For a clear comparison with scalar field, we define a new function F=Β±B2/12+f(B2)F=\pm B^{2}/12+f(B^{2}) where B2=AΞΌAΞΌB^{2}=A_{\mu}A^{\mu}, AΞΌA_{\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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