arXiv subjects
Publications and source records attributed to Benjamin Steinberg.
An ample groupoid is said to be AF if it is a directed union of compact open principal subgroupoids. In this paper, we provide a complete homological characterization of these groupoids. Specifically, we prove that an ample groupoid is AF if and only if it has homological dimension zero. More generally, we characterize groupoids of homological dimension zero over a unital ring $R$.
We give a length one projective resolution of the trivial module for the groupoid of a semi-saturated partial action (in the sense of Exel) of a free group on a compact Hausdorff and totally disconnected space. As a consequence we obtain an elementary computation of the homology of these groupoids, which include transformation groupoids of free group actions and Deaconu-Renault groupoids of systems $(X,T)$ where $X$ is compact Hausdorff and totally disconnected and $T$ is a local homeomorphism with domain a clopen subset of $X$. We also show that algebra of such a partial action groupoid over a field has global dimension at most $2$ when the space is second countable.
A band is a semigroup in which each element is idempotent. In recent years, there has been a lot of activity on the representation theory of the subclass of left regular bands due to connections to Markov chains associated to hyperplane arrangements, oriented matroids, matroids and CAT(0) cube complexes. We prove here that the integral semigroup algebra of a band is isomorphic to the integral path algebra of a quiver modulo an admissible ideal. This leads to a uniform bound quiver presentation for band algebras over all fields. Also, we answer a question of Margolis, Saliola and Steinberg by proving that the integral semigroup algebra of a CW left regular band is isomorphic to the quotient of the integral path algebra of the Hasse diagram of its support semilattice modulo the ideal generated by the sum of all paths of length two. This includes, for example, hyperplane face semigroup algebras.
I give a short proof of a recent result due to Kiefer and Ryzhikov showing that a finite irreducible semigroup of $n\times n$ matrices has cardinality at most $3^{n^2}$.
The representation theory of left regular band semigroup algebras is well-studied and known to have close connections with combinatorial topology, as established in the work of Margolis--Saliola--Steinberg ('15, '21). In this paper, we investigate the representation theory of the invariant subalgebras of left regular band semigroup algebras carrying the action of a finite group through the lens of group-equivariant combinatorial topology. We characterize when the invariant subalgebra is semisimple or commutative and examine the equivariant structure of the Peirce components of the semigroup algebra. For CW left regular bands, we interpret these Peirce components in terms of the equivariant topology of intervals in the support semilattice, yielding the Cartan invariants of the invariant subalgebras of left regular bands associated to CAT(0)-cube complexes. We also give a topological formula for the Peirce components for left regular bands with hereditary algebras. Finally, in specializing to left regular bands associated to geometric lattices, we explore generalizations of the Desarm\'{e}ni\'{e}n--Wachs derangement representation and their connections to Markov chains.
Fiedorowicz suggested that it was likely that every finite simply connected CW complex is homotopy equivalent to the classifying space of a finite semigroup. We prove that every finite wedge of simply connected Moore spaces of finitely generated abelian groups is homotopy equivalent to the classifying space of a finite semigroup. Consequently, homology groups alone cannot preclude a finite simply connected CW complex from being homotopy equivalent to the classifying space of a finite semigroup.
A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi-$\mathrm{FP}_n$ if its group of units is of type $\mathrm{FP}_n$. We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of 2 and the cohomological dimension of its group of units. We apply these results to prove a Lyndon's Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 \rangle$. In particular, we show that all such monoids are of type bi-$\mathrm{FP}_\infty$. Moreover, we show that if $r$ is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$, while if $r$ is a proper power then it has infinite Hochschild cohomological dimension. For any non-special one-relator monoid $M$ with defining relation $u=v$ we show that if there is no nonempty word $w$ such that $u,v \in A^*w \cap w A^*$ then $M$ is of type bi-$\mathrm{FP}_\infty$ and has Hochschild cohomological dimension at most $2$.
We show that the $C^*$-algebra associated by Nekrashevych to a contracting self-similar group is simple if and only if the corresponding complex $\ast$-algebra is simple. We also improve on Steinberg and Szaka\'c's algorithm to determine if the $\ast$-algebra is simple. This provides an interesting class of non-Hausdorff amenable, effective and minimal ample groupoids for which simplicity of the $C^*$-algebra and the complex $\ast$-algebra are equivalent.
Nekrashevych associated to each self-similar group action an ample groupoid and a $\mathrm{C}^\ast$-algebra. We perform complete computations of the homology of the groupoid and the K-theory of the $\mathrm{C}^\ast$-algebra for a myriad of examples, including the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups, and self-similar actions of free abelian groups and lamplighter groups. The key development is the construction, for arbitrary self-similar group actions, of long exact sequences which compute the homology and K-theory in terms of the homology of the group and K-theory of the group $\mathrm{C}^\ast$-algebra via the transfer map and the virtual endomorphism. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of X.~Li, we are able to show that R\"over's simple group containing the Grigorchuk group and Thompson's group $V$ is rationally acyclic but has nontrivial Schur multiplier. We prove many more R\"over--Nekrashevych groups of self-similar groups are rationally acyclic.
Given a presentation of a monoid $M$, combined work of Pride and of Guba and Sapir provides an exact sequence connecting the relation bimodule of the presentation (in the sense of Ivanov) with the first homology of the Squier complex of the presentation, which is naturally a $\mathbb ZM$-bimodule. This exact sequence was used by Kobayashi and Otto to prove the equivalence of Pride's finite homological type (FHT) property with the homological finiteness condition bi-$\mathrm{FP}_3$. Guba and Sapir used this exact sequence to describe the abelianization of a diagram group. We prove here a generalization of this exact sequence of bimodules for presentations of associative algebras. Our proof is more elementary than the original proof for the special case of monoids.
I explicitly compute the Eilenberg-Mac Lane homology of a completely simple semigroup using topological means. I also complete Gray and Pride's investigation into the homological finiteness properties of completely simple semigroups, as well as studying their topological finiteness properties. I give a topological proof of Pride's unpublished homological lower bound for the deficiency of a monoid or semigroup.
We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is equal to $1$. We prove results which relate the finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid inherits the finiteness properties $F_n$ and $FP_n$ from its group of units. We also obtain results which relate the geometric and cohomological dimensions of such a monoid to those of its group of units. We apply these results to prove a Lyndon's Identity Theorem for one-relator monoids of the form $\langle A \mid r=1 \rangle$. In particular we show that all such monoids are of type $F_{\infty}$ (and $FP_{\infty}$), and that when $r$ is not a proper power, then the monoid has geometric and cohomological dimension at most $2$. The first of these results resolves an important case of a question of Kobayashi from 2000 on homological finiteness properties of one-relator monoids. We also show how our topological approach can be used to prove results about the closure properties of various homological and topological finiteness properties for amalgamated free products and HNN-extensions of monoids. To prove these results we introduce new methods for constructing equivariant classifying spaces for monoids, as well as developing a Bass-Serre theory for free constructions of monoids.
We prove the Effros-Hahn conjecture for groupoid algebras with coefficients in a sheaf, obtaining as a consequence a description of the ideals in skew inverse semigroup rings. We also use the description of the ideals to characterize when the groupoid algebras with coefficients in a sheaf are von Neumann regular, primitive, semiprimitive, or simple. We apply our results to the topological dynamics of actions of inverse semigroups, describing the existence of dense orbits and minimality in terms of primitivity and simplicity, respectively, of the associated algebra. Moreover, we apply our results to the usual complex groupoid algebra of continuous functions with compact support, used to build the C*-algebra associated with a groupoid, and describe criteria for its simplicity.
The goal of this paper is to use topological methods to compute $\mathrm{Ext}$ between irreducible representations of von Neumann regular monoids in which Green's $\mathscr L$- and $\mathscr J$-relations coincide (e.g., left regular bands). Our results subsume those of S.~Margolis, F.~Saliola, and B.~Steinberg, \emph{Combinatorial topology and the global dimension of algebras arising in combinatorics}, J. Eur. Math. Soc. (JEMS), \textbf{17}, 3037--3080 (2015). Applications include computing $\mathrm{Ext}$ between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered $G$-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product $G\wr S_n$). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. More generally, these results work for CW left regular bands of abelian groups. These results generalize the results of S.~Margolis, F.~V. Saliola, and B.~Steinberg. \emph{Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry}, Mem. Amer. Math. Soc., \textbf{274}, 1--135, (2021).
This paper aims to use topological methods to compute $\mathrm{Ext}$ between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the $E_2$-page over fields of good characteristic. As an application, we determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids. A sequel paper will use these results to study the representation theory Hsiao's monoid of ordered $G$-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product $G\wr S_n$).
We completely characterize when the algebra of an ample groupoid with coefficients in an arbitrary unital ring is von Neumann regular and, more generally, when the algebra of a graded ample groupoid is graded von Neumann regular. Our main application is to resolve the question, open since 1970, of when the algebra of an inverse semigroup is von Neumann regular. As applications, we recover known results on regularity and graded regularity of Leavitt path algebras, and prove a number of new results, in particular concerning graded regularity of algebras of Deaconu-Renault groupoids and Nekrashevych-Exel-Pardo algebras of self-similar groups.
Carlsen (Adv.~Math, 2018) showed that any $\ast$-homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterize the rings considered by Carlsen as precisely those rings for which every $\ast$-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.
This paper develops the fundamentals of modular representation theory for finite monoids, introducing the decomposition matrix and exploring its connection to Brauer characters. We define modular characteristic and explain how the representation theory in nonmodular positive characteristic behaves like the characteristic zero theory by showing that one can lift from nonmodular characteristic $p$ all simple and projective indecomposable modules, as well as a quiver presentation of the basic algebra. As an application of the theory developed, we give a new proof of Glover's theorem that the monoid of $2\times 2$-matrices over $\mathbb F_p$ has infinite representation type over fields of characteristic $p$. We also investigate the relationship between nonsingularity of the Cartan matrix of a monoid algebra in characterstic zero and in positive characteristic. We show that for von Neumann regular monoids the Cartan matrix is always nonsingular and we show that if a monoid has aperiodic left (or right) stabilizers, then nonsingularity in characteristic zero implies nonsingularity in positive characteristic. Florian Eisele has recently shown that the Cartan matrix of a monoid algebra can be nonsingular in characteristic zero and singular in positive characteristic, disproving a conjecture of the author in an earlier version of this paper. A new conjecture is proposed, unifying the cases of regular monoids and monoids with aperiodic stabilizers.