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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. †

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 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 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 (d 1 , d 2 , ..., d K ) achievable through linear interference alignment. For a symmetric systemfor 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.

Abstract. 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 0 * E = 0 f g * E in every local ring of R.

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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 CohenMacaulay 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.