Gennady Lyubeznik

On the Degrees of Freedom Achievable Through Interference Alignment in a MIMO Interference Channel

Consider a K-user flat fading MIMO interference channel where the kth transmitter (or receiver) is equipped with Mk (respectively Nk) 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 Mk and Nk 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 (d1. d2....dK) achievable through linear interference alignment. For a symmetric system with Mk = M, Nk = N, dk = 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.

Finiteness properties of local cohomology modules (an application of D-modules to Commutative Algebra)

SummaryThe main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by C. Huneke and R. Sharp in characteristicp>0. Our method, based on the theory of algebraicD-modules, seems to be the first application ofD-modules to Commutative Algebra.

F-modules: applications to local cohomology and D-modules in characteristic p>0.

Let F: Ä-mod -* Ä-mod be the Frobenius functor of Peskine-Szpiro [PSz], 1.1.2. An F-module, or, more accurately, an FÄ-module is a pair (Jt, 0), where M is an Ä-module and : M » F ( J f ) is an -module isomorphism. F-modules form an abelian category (Definition 1.1). It should be noted that our notion of F-module is, in a sense, dual to the notion of level (R, F)-module of Hartshorne-Speiser [HaSp], Sec.l. It has been known that local cohomology modules of R with support in any ideal / c R have the property that they are isomorphic to their own images under the Frobenius functor and this fact has been used, for example, by Hartshorne-Speiser [HaSp], Huneke-Sharp [HuSh], Peskine-Szpiro [PSz] and Sharp [Sh] to study local cohomology, but a systematic theory of modules having this property has not been constructed. One of our main goals in this paper is to develop such a theory.

On the degrees of freedom achievable through interference alignment in a MIMO interference channel

Consider a K-user flat fading MIMO interference channel where the k-th transmitter (or receiver) is equipped with M k (respectively N k ) antennas. If a large number of statistically independent channel extensions are allowed either across time or frequency, the recent work [1] suggests that the total achievable degrees of freedom (DoF) can be maximized via interference alignment, resulting in a total DoF that grows linearly with K 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 achievable through linear interference alignment. When M k = M and N k = N for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than M + N −1. If, in addition, all users have the same DoF d = 1, then this upper bound on the total DoF is actually tight for almost all MIMO interference channels.

On the commutation of the test ideal with localization and completion

Let R be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of R commutes with localization and, if R is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull E of the residue field of every local ring of R is equal to the finitistic tight closure of zero in E. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each R-module is introduced and studied. This theory gives rise to an ideal of R which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of R in general, and shown to equal the test ideal under the hypothesis that 0E = 0 fg∗ E in every local ring of R.

On the arithmetical rank of monomial ideals

The arithmetical rank of monomial ideals has been studied by several authors. In [ 12, 131, for example, the result of the above theorem is proved in some special cases. On the other hand in [8] we stated the conjecture to the effect that every algebraic set l’c P” of pure codimension t can be set-theoretically cut out by n [n/t] + 1 hypersurfaces. Supporting evidence for this conjecture is as follows:

Finiteness properties of local cohomology modules for regular local rings of mixed characteristic: the unramified case

In this note we partially extend results on the finite-ness properties of local cohomology modules from the case of a regular local ring containing a field to the unramified case of a regular local ring of mixed characteristic.

A new explicit finite free resolution of ideals generated by monomials in an R-sequence

Abstract This paper gives a new explicit finite free resolution for ideals generated by monomials in an R -sequence (where R is a local commutative ring), which is more efficient than the standard Taylor resolution.

On the vanishing of local cohomology in characteristic p > 0

Let R be a d -dimensional regular local ring of characteristic p > 0 with maximal ideal

Finiteness properties of local cohomology modules: a characteristic-free approach ☆

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