An Explicit, Characteristic-Free, Equivariant Homology Equivalence Between Koszul Complexes

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

Several authors have recently studied the problem of finding circumstances under which the amenability of a semigroup is inherited by a subsemigroup. It is known ([2] and [3]) that subgroups of an amenable group are amenable, but that in general subsemigroups of an amenable semigroup are not. We show here that subsemigroups of an amenbable group need not be amenable. Frey proved in [4] that subsemigroups of a group inherit amenability and the group contains no copy of the free semigroup S on two generators (S is not amenable). The problem is discussed in detail in [51. It is known [11 that a group generated by a copy of S contained in it need not be free; the question is whether it can be far enough from free to be amenable. Since solvable groups are amenable [6], our claim that subsemigroups of amenable groups need not be amenable will be established we can embed S in a solvable group. We proceed to do this. Granirer conjectured in his review [MR 35 #3423] of Wilde and Witz [7] that a group is amenable and it contains no copy of S. It will follow that the only if part of this conjecture is quite false. Let B be the free additive abelian group on the family of generators {b(m, n) }, m, n integers. Any permutation of these generators induces an automorphism of B. Let x and y be the automorphisms induced by b(m, n)F->b(m+1, n) and b(m, n)'->b(m, n+1), respectively. It is easy to see that the subgroup C of Aut B generated by x and y is the free abelian group on the generators x, y. We write C additively, and we indicate the action of an element c of C on an element b of B by be. Let be the normal product of B by C. consists of pairs [c, b] and the multiplication is given by -[c, b1i [c, b] [cl+c, b1+b]. c1->[c, 0], bF-*[O, b] are embeddings of C, B respectively into G, and B is thus a normal abelian subgroup with quotient G/B-C. Hence, is solvable and in fact the second derived group G of is trivial. The same holds for every subgroup of G. Now, let s= [x, b(O, 0)] and t= [y, b(0, 0)].

Given an N 1 -free abelian group G we characterize the class? G of all torsion abelian groups T satisfying Ext(G,T) = 0 assuming the special continuum hypothesis CH. Moreover, in Godel's constructible universe we prove that this characterizes? G for arbitrary torsion-free abelian G. It follows that there exists some ugly N 1 -free abelian groups.

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

Long colimits of topological groups II: Free groups and vector spaces

Invariant Theory, Young Bitableaux, and Combinatorics

Alternative Loop Rings with Solvable Unit Loops

We will construct an almost free non-free group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper G prime"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">G/G’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a free abelian group.

Using the wonderful compactification of a semisimple adjoint affine algebraic\ngroup G defined over an algebraically closed field k of arbitrary\ncharacteristic, we construct a natural compactification Y of the G-character\nvariety of any finitely generated group F. When F is a free group, we show that\nthis compactification is always simply connected with respect to the \\'etale\nfundamental group, and when k=C it is also topologically simply connected. For\nother groups F, we describe conditions for the compactification of the moduli\nspace to be simply connected and give examples when these conditions are\nsatisfied, including closed surface groups and free abelian groups when\nG=PGL(n,C). Additionally, when F is a free group we identify the boundary\ndivisors of Y in terms of previously studied moduli spaces, and we construct a\nfamily of Poisson structures on Y and its boundary divisors arising from\nBelavin-Drinfeld splittings of the double of the Lie algebra of G. In the\nappendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson\nstructure on a quotient of a Poisson algebraic variety by the action of a\nreductive Poisson algebraic group.\n

For a Noetherian ‐algebra , there is a canonical inclusion , and each element in the image satisfies a certain compatibility condition. We call compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver and a commutative Noetherian ring , the path algebra is compatible. In this paper, we prove that is compatible when is an extended Dynkin quiver and is either a Dedekind domain or a Noetherian semilocal normal ring of dimension 2.

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $V=L$, that there is a first-order definable free abelian group with no first-order definable basis.