nn-abelian and nn-exact categories

We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories are $n$-abelian (resp. $n$-exact). These results allow to construct several examples of $n$-abelian and $n$-exact categories. Conversely, we prove that $n$-abelian categories satisfying certain mild assumptions can be realized as $n$-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius $n$-exact category has a natural $(n+2)$-angulated structure in the sense of Gei\ss-Keller-Oppermann. We give several examples of $n$-abelian and $n$-exact categories which have appeared in representation theory, commutative ring theory, commutative and non-commutative algebraic geometry.Comment: 58 pages. Corrected an error in Thm 3.20 and Lemma 3.22 and several typos. Accepted for publication in Mathematische Zeitschrif

Site selectivity of halogen oxygen bonding in 5- and 6-haloderivatives of uracil

Seven 5-and 6-halogenated derivatives of uracil or 1-methyluracil (halogen = Cl, Br, I) were studied by single crystal X-ray diffraction. In contrast with pure 5-halouracils, where the presence of N-H…O and C-H…O hydrogen bonds prevents the formation of other intermolecular interactions, the general ability of pyrimidine nucleobases to provide electron donating groups to halogen bonding was confirmed in three crystals and cocrystals containing uracil with the halogen atom at the C6 position. In the latter compounds, among the two nucleophilic oxygen atoms in the C=O moiety, only the urea carbonyl oxygen O1 can act as halogen bond acceptor, being not saturated by conventional hydrogen bonds. The halogen bonds in pure 6-halouracils are all rather weak, as supported by Hirshfeld surface analysis. The strongest interaction was found in the structure of 6- iodouracil, which displayed the largest (13%) reduction of the sum of van der Waals (vdW) radii for the contact atoms. Despite this, halogen bonding plays a rol

Higher Auslander algebras of type A\mathbb{A} and the higher Waldhausen S⁡\operatorname{S}-constructions

These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi Walde. In them we relate Iyama's higher Auslander algebras of type $\mathbb{A}$ to Eilenberg--Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen $\operatorname{S}$-construction from algebraic $K$-theory.Comment: 16 pages. The author's contribution to the Proceedings of the ICRA 2018, v.2 minor edits following referee repor

Equilibrium Fluctuations for a One-Dimensional Interface in the Solid on Solid Approximation

An unbounded one-dimensional solid-on-solid model with integer heights is studied. Unbounded here means that there is no a priori restrictions on the discret e gradient of the interface. The interaction Hamiltonian of the interface is given by a finite range part, pr oportional to the sum of height differences, plus a part of exponentially decaying long range potentials. The evolution of the interface is a reversible Markov process. We prove that if this system is started in the center of a box of size L after a time of order L^3 it reaches, with a very large probability, the top or the bottom of the box

Black hole nonmodal linear stability: the Schwarzschild (A)dS cases

The nonmodal linear stability of the Schwarzschild black hole established in Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative cosmological constant $\Lambda$. Two gauge invariant combinations $G_{\pm}$ of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map $[h_{\alpha \beta}] \to \left( G_- \left([h_{\alpha \beta}] \right), G_+ \left([h_{\alpha \beta}] \right) \right)$ with domain the set of equivalent classes $[h_{\alpha \beta}]$ under gauge transformations of solutions of the linearized Einstein's equation, is invertible. The way to reconstruct a representative of $[h_{\alpha \beta}]$ in terms of $(G_-,G_+)$ is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, $G_+$ and $G_-$ are bounded in the the outer static region. At large times, the perturbation decays leaving a linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter background solution. For negative cosmological constant it is shown that there is a choice of boundary conditions at the time-like boundary under which the Schwarzschild anti de Sitter black hole is unstable. The root of Chandrasekhar's duality relating odd and even modes is exhibited, and some technicalities related to this duality and omitted in the original proof of the $\Lambda=0$ case are explained in detail.Comment: Typos corrected, changes in the Introduction (including example of nonmodal instability