Pure injective and absolutely pure sheaves

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally we introduce the class of locally absolutely pure (quasi--coherent) sheaves, with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.Comment: Updated version. To appear in Proc. Edinburgh Math. So

Pure Natural Inflation

We point out that a simple inflationary model in which the axionic inflaton couples to a pure Yang-Mills theory may give the scalar spectral index (n_s) and tensor-to-scalar ratio (r) in complete agreement with the current observational data.Comment: 4 pages, 3 figures, published versio

Modules with Pure Resolutions

We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.Comment: 9 page

Generalized Pure Lovelock Gravity

We present a generalization of the n-dimensional (pure) Lovelock Gravity theory based on an enlarged Lorentz symmetry. In particular, we propose an alternative way to introduce a cosmological term. Interestingly, we show that the usual pure Lovelock gravity is recovered in a matter-free configuration. The five and six-dimensional cases are explicitly studied.Comment: v2, 16 pages, references and comments adde

Subsystem Pseudo-pure States

A critical step in experimental quantum information processing (QIP) is to implement control of quantum systems protected against decoherence via informational encodings, such as quantum error correcting codes, noiseless subsystems and decoherence free subspaces. These encodings lead to the promise of fault tolerant QIP, but they come at the expense of resource overheads. Part of the challenge in studying control over multiple logical qubits, is that QIP test-beds have not had sufficient resources to analyze encodings beyond the simplest ones. The most relevant resources are the number of available qubits and the cost to initialize and control them. Here we demonstrate an encoding of logical information that permits the control over multiple logical qubits without full initialization, an issue that is particularly challenging in liquid state NMR. The method of subsystem pseudo-pure state will allow the study of decoherence control schemes on up to 6 logical qubits using liquid state NMR implementations.Comment: 9 pages, 1 Figur

Right orthogonal class of pure projective modules over pure hereditary rings

Let $\mathcal{W}$ be the class of all pure projective modules. In this article, $\mathcal{W}$-injective modules is defined via the vanishing of cohomology of pure projective modules. First we show that every module has a $\mathcal{W}$-injective coresolution over an arbitrary ring and the class of all $\mathcal{W}$-injective modules is coresolving over a pure-hereditary ring. Further, we analyze the dimension of $\mathcal{W}$-injective coresolution over a pure-hereditary ring. It is shown that \Fcor_{\mathcal{W}^{\bot}}(R) = \sup\{\pd(G) \colon G is a pure projective $R$-module\} = \sup\{\cores_{\mathcal{W}^{\bot}}(M) \colon M is an $R$-module$\}.$ Finally, we give some equivalent conditions of $\mathcal{W}$-injective envelope with the unique mapping property. The dimension has desirable properties when the ring is semisimple artinian.Comment: 17 pages, 9 figure