Configurations in abelian categories—I: Basic properties and moduli stacks

This is the first in a series of papers on configurations in an abelian category A. Given a finite partially ordered set (I,?), an (I,?)-configuration(σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or π(J,K):σ(J)→σ(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects, and are useful for studying stability conditions on A.We define and motivate the idea of configurations, and explain some natural operations upon them—subconfigurations, quotient configurations, substitution, refinements and improvements. Then we study moduli spaces of (I,?)-configurations in A, and natural morphisms between them, using the theory of Artin stacks. We prove well-behaved moduli stacks exist when A is the abelian category of coherent sheaves on a projective scheme P, or of representations of a quiver Q.In the sequels, given a stability condition (τ,T,?) on A, we will show the moduli spaces of τ-(semi)stable objects or configurations are constructible subsets in the moduli stacks of all objects or configurations. We associate infinite-dimensional algebras of constructible functions to a quiver Q using the method of Ringel-Hall algebras, and define systems of invariants of P that ‘count’ τ-(semi)stable coherent sheaves on P and satisfy interesting identities.     

Generating all linear transformations

For an n×n Boolean matrix R, let A_R={n×n matrices A over a field F such that if r_ij=0 then a_ij=0}. We show that a collection A_R〈1〉,…,A_R〈k〉 generates all n×n matrices over F if and only if the matrix J all of whose entries are 1 can be expressed as a Boolean product of Hall matrices from the set {R〈1〉,…,R〈k〉}. We show that J can be expressed as a product of Hall matrices R〈i〉 if and only if ΣR〈i〉?R〈i〉 is primitive.     

Ω-compact semigroups having congruence extension property

The congruence extension property (CEP) of semigroups has been extensively studied by a number of authors. We call a compact semigroup S an Ω-compact semigroup if the set of all regular elements of S forms an ideal of S. In this note, we characterize the Ω-compact semigroup having (CEP). Our result extends a recent result obtained by X.J. Guo on the congruence extension property of strong Ω-compact semigroups which is a semigroup containing precisely one regular D-class.     

Algebraic isomorphisms and strongly double triangle subspace lattices

Let D={{0},K,L,M,X} be a strongly double triangle subspace lattice on a non-zero complex reflexive Banach space X, which means that at least one of three sums K + L, L + M and M + K is closed. It is proved that a non-zero element S of AlgD is single in the sense that for any A,B∈AlgD, either AS = 0 or SB = 0 whenever ASB = 0, if and only if S is of rank two. We also show that every algebraic isomorphism between two strongly double triangle subspace lattice algebras is quasi-spatial.     

Almost disjoint large subsets of semigroups

There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick, central, syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many pairwise disjoint central sets. We extend this result to a large class of semigroups (including (N,+)) by showing that if S is a semigroup in this class which has cardinality κ then any central set can be partitioned into κ many pairwise disjoint central sets. We also show that for this same class of semigroups, if there exists a collection of μ almost disjoint subsets of any member S, then any central subset of S contains a collection of μ almost disjoint central sets. The same statement applies if “central” is replaced by “thick”; and in the case that the semigroup is left cancellative, “central” may be replaced by “piecewise syndetic”. The situation with respect to syndetic sets is much more restrictive. For example, there does not exist an uncountable collection of almost disjoint syndetic subsets of N. We investigate the extent to which syndetic sets can be split into disjoint syndetic sets.     

γ-Bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C^∗(G)→B(X) on the group C^∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α).     

Configurations in abelian categories. IV. Invariants and changing stability conditions

This is the last in a series on configurations in an abelian category A. Given a finite poset (I,?), an (I,?)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using Artin stacks. It showed well-behaved moduli stacks ObjA,MA(I,?) of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.The third introduced stability conditions(τ,T,?) on A, and showed the moduli space of τ-semistable objects in class α is a constructible subset in ObjA, so its characteristic function is a constructible function. It formed algebras , , , of constructible and stack functions on ObjA, and proved many identities in them.In this paper, if (τ,T,?) and are stability conditions on A we write in terms of the , and deduce the algebras are independent of (τ,T,?). We study invariants or Iss(I,?,κ,τ) ‘counting’ τ-semistable objects or configurations in A, which satisfy additive and multiplicative identities. We compute them completely when A=mod-KQ or A=coh(P) for P a smooth curve. We also find invariants with special properties when A=coh(P) for P a smooth surface with nef, or a Calabi-Yau 3-fold.     

Nonmetrizable ANR's admitting a group structure are manifolds

It is shown that a nonmetrizable ANR-space of weight τ>ω, admitting a group structure, is (topologically) an Rτ-manifold.     

On the connectedness of self-affine attractors

Let T = T(A, D) be a self-affine attractor in defined by an integral expanding matrix A and a digit set D. In the first part of this paper, in connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers . It is shown that in and , for any integral expanding matrix A, T(A, D) is connected. In the second part, we study connectedness of Pisot dual tiles, which play an important role in the study of -expansions, substitutions and symbolic dynamical systems. It is shown that each tile of the dual tiling generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. We even give a simple necessary and sufficient condition of connectedness of the Pisot dual tiles of degree 4. Detailed proofs will be given in [4]. Received: 2 March 2003     

Sets of periods for automorphisms of compact Riemann surfaces

Let G=〈f〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g≥2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x∈S, x is fixed by f^d but not by any smaller power of f. For an arbitrary subset A of proper divisors of N, there is always an associated action and, if g_A denotes the minimal genus for such an action, an algorithm is obtained here to determine g_A. Furthermore, a set A_max is determined for which g_A is maximal.     

Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

Note on separate continuity and the Namioka property

A pair 〈B,K〉 is a Namioka pair if K is compact and for any separately continuous , there is a dense A⊆B such that f is ( jointly) continuous on A×K. We give an example of a Choquet space B and separately continuous such that the restriction fΔ_| to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous and for any Baire subspace F of T×K, the set of points of continuity of is dense in F. We say that 〈B,K〉 is a weak-Namioka pair if K is compact and for any separately continuous and a closed subset F projecting irreducibly onto B, the set of points of continuity of fF_| is dense in F. We show that T is a Baire space if the pair 〈T,K〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈B,K〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous which has no dense set of continuity points; in fact, f does not even have the Baire property.     

Relative uniform completeness and order-convex representations of archimedean ?-groups and f-rings

Results of Henriksen and Johnson, for archimedean f-rings with identity, and of Aron and Hager, for archimedean ?-groups with unit, relating uniform completeness to order-convexity of a representation in a D(X) (the lattice of almost real continuous functions on the space X) are extended to situations without identity or unit. For an archimedean ?-group, G, we show: if G admits any representation G?D(X) in which G is order-convex, then G is divisible and relatively uniformly complete. A converse to this would seem to require some sort of canonical representation of G, which seems not to exist in the ?-group case. But for a reduced archimedean f-ring, A, there is the Johnson representation A?D(XA), and we show: A is divisible, relatively uniformly complete and square-dominated if and only if A is order-convex in D(XA) and square-root-closed. Also, we expand on the situation with unit, where we have the Yosida representation, G?D(YG): if G is divisible, relatively uniformly complete, and the unit is a near unit, then G is order-convex in D(YG).     

Minimal KC spaces are compact

We show that every KC space (X,τ), such that τ is minimal among the KC topologies on X, must be compact (not necessarily T₂). This solves a long-standing question, first raised by R. Larson in 1973.     

Subcompactness and domain representability in GO-spaces on sets of real numbers

In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R, then (R,τ) is subcompact, and so is any Gδ-subspace of (R,τ). We also show that if (X,τ) is a subcompact GO-space constructed on a subset X⊆R, then X is a Gδ-subset of any space (R,σ) where σ is any GO-topology on R with τ=σX_|. It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to Gδ-subsets. In addition, it follows that if (X,τ) is a subcompact GO-space constructed on any set of real numbers and if τS is the topology obtained from τ by isolating all points of a set S⊆X, then (X,τS) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known.We use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R. For example, examples show that there are subcompact GO-spaces constructed on subsets X⊆R where X is not a Gδ-subset of the usual real line. However, if (X,τ) is a dense-in-itself GO-space constructed on some X⊆R and if (X,τ) is subcompact (or more generally domain-representable), then (X,τ) contains a dense subspace Y that is a Gδ-subspace of the usual real line. It follows that (Y,τY_|) is a dense subcompact subspace of (X,τ). Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of (X,τ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: “Is there a domain-representable GO-space constructed on a subset of R that is not subcompact”? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.     

Nearest southeast submatrix that makes multiple a prescribed eigenvalue. Part 1

Given four complex matrices A,B,C and D, where A∈C^n×n and D∈C^m×m, and given a complex number z₀: What is the (spectral norm) distance from D to the set of matrices X∈C^m×m such that z₀ is a multiple eigenvalue of the matrix

     

A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST₄. The result says that if A=〈A₀,A₁,A₂,A₃〉 is a standard transitive and rich model of TST₄, then A satisfies the 〈0,0,n〉-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in Tzouvaras (2007) [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras (2009) [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras (2007) [5, Theorem 3.6] which is just equivalent to the 〈0,0,2〉-property.     

A solution to Dilworth's congruence lattice problem

We construct an algebraic distributive lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R.P. Dilworth, from the forties. The lattice D has a compact top element and ℵω+1 compact elements. Our results extend to any algebra possessing a congruence-compatible structure of a join-semilattice with a largest element.