Consider a K-user flat fading MIMO interference channel where the kth transmitter (or receiver) is equipped with M k (respectively N k ) antennas. If an exponential (in K) number of generic channel ...extensions are used either across time or frequency, Cadambe and Jafar 1 showed that the total achievable degrees of freedom (DoF) can be maximized via interference alignment, resulting in a total DoF that grows linearly with A even if M k and N k are bounded. In this work we consider the case where no channel extension is allowed, and establish a general condition that must be satisfied by any degrees of freedom tuple (d 1 . d 2 ...d K ) achievable through linear interference alignment. For a symmetric system with M k = M, N k = N, d k = d for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than K(M + N)/(K + 1). We also show that this bound is tight when the number of antennas at each transceiver is divisible by d, the number of data streams per user.
Let (R,\mathfrak{m}) be a Noetherian regular local ring of characteristic p>0 and let I be a nonzero ideal of R. Let D(-)= \operatorname {Hom}_R(-, E) be the Matlis dual functor, where E = ...E_R(R/{\mathfrak{m}}) is the injective hull of the residue field R/{\mathfrak{m}}. In this short note, we prove that if {H}^i_I(R)\neq 0, then \operatorname {Supp}_R(D({H}^i_{I}(R)))=\operatorname {Spec}(R).
Let
X
be a closed equidimensional local complete intersection subscheme of a smooth projective scheme
Y
over a field, and let
X
t
denote the
t
-th thickening of
X
in
Y
. Fix an ample line bundle
O
Y
...(
1
)
on
Y
. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer
c
, such that for all integers
t
⩾
1
, the cohomology group
H
k
(
X
t
,
O
X
t
(
j
)
)
vanishes for
k
<
dim
X
and
j
<
-
c
t
. Note that there are no restrictions on the characteristic of the field, or on the singular locus of
X
. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant
c
is unbounded, even in a fixed dimension.
An extension of a theorem of Hartshorne Katzman, Mordechai; Lyubeznik, Gennady; Zhang, Wenliang
Proceedings of the American Mathematical Society,
03/2016, Letnik:
144, Številka:
3
Journal Article
Recenzirano
Odprti dostop
We extend a classical theorem of Hartshorne concerning the connectedness of the punctured spectrum of a local ring by analyzing the homology groups of a simplicial complex associated with the minimal ...primes of a local ring.
Let
R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke M. Hochster, C. Huneke, Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math. 135 ...(1992) 53–89 states that if
R is excellent, then the absolute integral closure of
R is a big Cohen–Macaulay algebra. We prove that if
R is the homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps to zero in a finite extension of the ring. As a result there follow an extension of the original result of Hochster and Huneke to the case in which
R is a homomorphic image of a Gorenstein local ring, and a considerably simpler proof of this result in the cases where the assumptions overlap, e.g., for complete Noetherian local domains.
Let R be a d-dimensional regular local ring of characteristic p > 0 with maximal ideal $\mathfrak m$, let I be an ideal of R and let A = R/I. We describe some properties of the local cohomology ...module HiI(R), in particular its vanishing, in terms of the Frobenius action on the local cohomology module $H^{d-i}_{\mathfrak m}(A)$.
We prove that the
F-jumping coefficients of a principal ideal of an excellent regular local ring of characteristic
p
>
0
are all rational and form a discrete subset of
R
.
PDF
