The K-theory of toric varieties in positive characteristic

We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic 0. The affine case of our result was conjectured by Gubeladze.Comment: Companion paper to arXiv:1106.138

Topology and monoid representations

The goal of this paper is 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. For von Neumann regular monoids in which Green's $\mathscr L$- and $\mathscr J$-relations coincide (e.g., left regular bands), the computation of $\mathrm{Ext}$ between arbitrary simple modules reduces to this case, and so our results subsume those of S. Margolis, F. Saliola, and B. Steinberg, Combinatorial topology and the global dimension of algebras arising in combinatorics, J. Eur. Math. Soc. (JEMS), 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. These generalize the results of S. Margolis, F. V. Saliola, and B. Steinberg. Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry, Mem. Amer. Math. Soc., 274, 1-135, (2021). We also 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

Moment categories and operads

A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment structures on $\Gamma$, $\Delta$ and $\Theta_n$ respectively. There is an analog of Baez-Dolan's plus construction taking a unital moment category $\mathbb{C}$ to a unital hypermoment category $\mathbb{C}^+$. Under this construction, $\mathbb{C}$-operads get identified with $\mathbb{C}^+$-monoids, i.e. presheaves on $\mathbb{C}^+$ satisfying Segal-like conditions strictly. We show that the plus construction of Segal's category $\Gamma$ embeds into the dendroidal category $\Omega$ of Moerdijk-Weiss.Comment: Introduction and Bibliography extended. Definition of reduced dendrix corrected. Proofs of Section 3 amended. Two appendices adde

Free shuffle algebras in language varieties

AbstractWe give simple concrete descriptions of the free algebras in the varieties generated by the “shuffle semirings” LΣ := (P(Σ∗),+,., ⊗, 0,1), or the semirings RΣ := (R(Σ∗),+,., ⊗,∗,0,1), where P(Σ∗) is the collection of all subsets of the free monoid Σ∗, and R(Σ∗) is the collection of all regular subsets. The operation x ⊗ y is the shuffle product

Monoids of moduli spaces of manifolds

We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_\theta$ of closed smooth (d-1)-manifolds and smooth d-dimensional cobordisms, equipped with generalised orientations specified by a fibration $\theta : X \to BO(d)$. The main result of GMTW is a determination of the homotopy type of the classifying space $BC_\theta$. The goal of the present paper is a systematic investigation of subcategories $D$ of $C_\theta$ having classifying space homotopy equivalent to that of $C_\theta$, the smaller such $D$ the better. We prove that in most cases of interest, $D$ can be chosen to be a homotopy commutative monoid. As a consequence we prove that the stable cohomology of many moduli spaces of surfaces with $\theta$-structure is the cohomology of the infinite loop space of a certain Thom spectrum. This was known for certain special $\theta$, using homological stability results; our work is independent of such results and covers many more cases.Comment: 52 pages, 5 figures; v2: extended discussion of application

Factorizations of Elements in Noncommutative Rings: A Survey

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.Comment: 50 pages, comments welcom

Toposes of monoid actions

openWe study toposes of actions of monoids on sets. We begin with ordinary actions, producing a class of presheaf toposes which we characterize. As groundwork for considering topological monoids, we branch out into a study of supercompactly generated toposes (a class strictly larger than presheaf toposes). This enables us to efficiently study and characterize toposes of continuous actions of topological monoids on sets, where the latter are viewed as discrete spaces. Finally, we refine this characterization into necessary and sufficient conditions for a supercompactly generated topos to be equivalent to a topos of this form.openInformatica e matematica del calcoloRogers, Morga

Lokalnemetode za relacione strukture i njihove slabe Krasnerove algebre

In this thesis local methods are made available as a tool to study the unary parts of clones (or, equivalently, the weak Krasner algebras). Using the language of model theory and Galois connections we develop a link between homomorphism-homogeneous relational structures and local methods, via the notion of endolocality. The theoretical results that are obtained are used to develop a systematic theory for the classification of homomorphism-homogeneous relational structures.U ovoj tezi su razvijene lokalne metode koje se mogu koristiti za izu- ˇcavanje unarnih delova klonova (ili, ekvivalentno, slabih Krasnerovih algebri). Koriˇs´cenjem jezika teorije modela i Galoovih veza uspostavljen je odnos izmedu homomorfizam-homogenih relacionih struktura i lokalnih metoda, preko pojma endolokalnosti. Dobijeni teoretski rezultati su upotrebljeni za razvoj sistematske teorije za klasifikaciju homomorfizam-homogenih struktura