TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

The strong nil-cleanness of semigroup rings

In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring {R}_{0}{[}M] is strongly nil-clean if and only if either |I|=1 or |\text{Λ}|=1 , and R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R{[}S] is strongly nil-clean if and only if R{[}{S}_{\alpha }] is strongly nil-clean for each \alpha \in Y .

Two semigroup rings associated to a finite set of meromorphic functions

We fix z0 ∈ ℂ and a field 𝔽 with ℂ ⊂ 𝔽 ⊂ 𝓜z0 := the field of germs of meromorphic functions at z0. We fix f1, …, fr ∈ 𝓜z0 and we consider the 𝔽-algebras S := 𝔽[f1, …, fr] and $\begin{array}{} \overline S: = \mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]. \end{array} $ We present the general properties of the semigroup rings$$\begin{array}{} \displaystyle S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\\overline S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{array} $$and we tackle in detail the case 𝔽 = 𝓜<1, the field of meromorphic functions of order < 1, and fj’s are meromorphic functions over ℂ of finite order with a finite number of zeros and poles.

Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

Unique factorization property of non-unique factorization domains

Let [Formula: see text] be an integral domain, [Formula: see text] be the polynomial ring over [Formula: see text], [Formula: see text] be the so-called [Formula: see text]-operation on [Formula: see text], and [Formula: see text]-Spec[Formula: see text] be the set of prime [Formula: see text]-ideals of [Formula: see text]. A nonzero nonunit of [Formula: see text] is said to be homogeneous if it is contained in a unique maximal [Formula: see text]-ideal of [Formula: see text]. We say that [Formula: see text] is a homogeneous factorization domain (HoFD) if each nonzero nonunit of [Formula: see text] can be written as a finite product of pairwise [Formula: see text]-comaximal homogeneous elements. In this paper, among other things, we show that (1) a Prüfer [Formula: see text]-multiplication domain (P[Formula: see text]MD) [Formula: see text] is an HoFD if and only if [Formula: see text] is an HoFD (2) if [Formula: see text] is integrally closed, then [Formula: see text] is a P[Formula: see text]MD if and only if [Formula: see text]-Spec[Formula: see text] is treed, and (3) [Formula: see text] is a weakly Matlis GCD-domain if and only if [Formula: see text] is an HoFD with [Formula: see text]-Spec[Formula: see text] treed. We also study the HoFD property of [Formula: see text] constructions, pullbacks, and semigroup rings.

Hausdorff series in a semigroup ring

Let [Formula: see text] and [Formula: see text] be the semigroup rings spanned on the right zero semigroup [Formula: see text], and on the left zero semigroup [Formula: see text], respectively, together with the identity element [Formula: see text]. We suggest a closed formula solving the equation [Formula: see text] which is the evolution of the Campbell–Baker–Hausdorff formula given by the Hausdorff series [Formula: see text] where [Formula: see text], in the algebras [Formula: see text] and [Formula: see text].