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We show that certain tiled R-orders have periodic projective resolutions and hence we determine the Hochschild cohomology ring of . This generalises, reproves and connects several results in the literature. Introduction.Hochschild cohomology has been intensively studied for various classes of algebras, commutative and non-commutative, artinian and non-artinian ones.For example, Larsen and Lindenstrauss [8] computed the Hochschild cohomology of the integers in algebraic number fields. Larsen [7] generalised this result to the noncommutative number-theoretic setup of maximal orders, and Sanada [10] went one step further and computed the Hochschild cohomology of hereditary orders. Although all of these orders have finite global cohomological dimension, their Hochschild cohomology turns out to be periodic.In a different context, (generalised) periodicity results for Hochschild cohomology of finite-dimensional algebras have been obtained for self-injective algebras of finite type [1, 2, 5], for some preprojective algebras [3] and for other algebras of infinite global dimension.This paper computes the Hochschild cohomology of a class of orders containing examples of finite and of infinite global dimension. In particular, we will thus generalise and reprove (by a different method) the main result of [10]. Moreover, we show that by reducing some of these orders modulo a prime we recover some of the self-injective algebras studied in [1], thus explaining why the same periodicity appears in both situations.The orders we are studying are tiled orders. This class of orders is known to have a sophisticated cohomological structure. For example, Tarsy's conjecture [11] on finiteness of global dimension has been disproved, first by Fujita [4] and then, giving more drastic examples, by Jansen and Odenthal [6]. Moreover, Rump [9] proved that the global

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom K (A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the N-graded second Hochschild homology group HH 2 (A) in the sense of Sköldberg.2010 Mathematics Subject Classification. 16E40, 16G20, 16L60.