Pure morphisms of commutative rings are effective descent morphisms for modules – a new proof

We study the homology of the dual de Rham complex as functors on the category of abelian groups. We give a description of homology of the dual de Rham complex up to degree 7 for free abelian groups and present a corrected version of the proof of Jean’s computations of the zeroth homology group.

An endomorphism f of an Abelian group A is said to be inessential (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains A as a subgroup. In this paper we show that f is as above if and only if (f − v idA)(A) is contained in the maximal divisible subgroup of A for some v ∈ Z.

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

Let G-ZFIDCW denote the category of G-spaces having the G-homotopy type of a finitely dominated G-CW complex for a compact Lie group G. Luck [8] has introduced a functor Wa from G-EF^C^ into the category of abelian groups and has realized the equivariant finiteness obstriction as the element w(X) in Wa(X). That is, a finitely dominated G-CW complex X is G-homotopy equivalent to a finite G-CW complex if and only if w(X)=Q. When G is the trivial group, there is an isomorphism from Wa(X) to the reduced projective group KQ(Z[πι(X)]) which sends the element w (X) to the WalΓs finiteness obstruction ([14]). Anderson [1] and Ehrlich [4] have studied a sufficient condition for w(E)= 0 for some fibration E-^B with fiber S. Munkholm, Pedresen [11], Luck [6, 7, 9] and others have studied the transfer map KQ(Z[^(B)])-^K:Q(Z[^(E)]). The purpose of this paper is to get a sufficient condition for w(X)=0 for a S^space X and a finite cyclic group L. We call G-maps /0: YQ-^X and /0: Y4-*X equivalent if there exists a commutative diagram

Given a pair of short exact sequences in an abelian category A, extending results due to Pressman [7]. We construct nine bivariant long exact sequences that involve the groups extn(δ,β), extn(δ,α) and extn(γ,α) that interlock with the univariant hom-ext sequences determined by (i). Two applications of these sequences in the category of abelian groups are given. Firstly we describe an isomorphism that yields a “constructive” way to compute the adjusted cotorsion part of a reduced cotorsion group and secondly, given a torsion-free, reduced group G with no free summands, we indicate how to determine the p-ranks of the divisible group Ext(G,Z).

We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën–Vaquié–Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

G(A, X) consists of all homotopy classes of cyclic maps from a space A to another space X. If A is an H-cogroup, then G(A, X) is a group. G(A, X) preserves products in the second variable and is a contravariant functor of A from the full subcategory of H-cogroups and maps into the category of abelian groups and homomorphisms. If X is an H-cogroup, then G(X, X) is a ring.

This chapter is an introduction to module theory in the category \({\mathtt {Sup}}\) of complete lattices and join preserving maps — these are modules on unital quantales. In this context, an interesting class of modules on involutive and unital quantales is induced by irreducible representations of \(C^*\)-algebras. Since \({\mathtt {Sup}}\) is \(*\)-autonomous, there exist constructions in the connection with modules on unital quantales which are not possible in the traditional theory of modules in the category of abelian groups. As an important example, this chapter explains the details of the equivalence between right \({\mathfrak {Q}}\)-modules and join-complete \({\mathfrak {Q}}\)-valued lattices where \({\mathfrak {Q}}\) is a unital quantale. Finally, a sketch of automata in \({\mathtt {Sup}}\) is given showing that every automaton in \({\mathtt {Sup}}\) gives rise to a right module on the free unital quantale generated by the corresponding input alphabet.

Let G be an abelian group, R a commutative ring of prime characteristic p with identity and R t G a commutative twisted group ring of G over R. Suppose p is a fixed prime, G p and S(R t G) are the p-components of G and of the unit group U(R t G) of R t G, respectively. Let R* be the multiplicative group of R and let f α(S) be the α-th Ulm-Kaplansky invariant of S(R t G) where α is any ordinal. In the paper the invariants f n (S), n ∈ ℕ ∪ {0}, are calculated, provided G p = 1. Further, a commutative ring R with identity of prime characteristic p is said to be multiplicatively p-perfect if (R*)p = R*. For these rings the invariants f α(S) are calculated for any ordinal α and a description, up to an isomorphism, of the maximal divisible subgroup of S(R t G) is given.

Torsion theories over commutative rings

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

Endoregular modules

Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.

Introduction.M. M. Day has shown that an abelian group has only one invariant mean if, and only if, the group is finite [3, Theorem 4, p. 534].Knowing this we now ask to what extent the invariant means on a given abelian group are alike.In this paper we shall show that they all have the same "measure-theoretic character."An invariant mean on an abelian group G is a normalized, positive, translation-invariant linear functional defined on the space of all bounded complex-valued functions on G.There is a canonical way of associating with the invariant mean L a (unique) positive measure (which is invariant in a sense that will be made clear) defined on the Stone-Cech compactification of G (discrete topology).Using this one-to-one correspondence, we shall ascribe to the invariant mean L the measure-theoretic properties of .Thus the measure algebra of the invariant mean L is simply the measure algebra of the associated measure .Our main theorem (Theorem 2) is that the measure algebras of any two invariant means on G are isomorphic.A corollary extends this result to the measure algebras of invariant means on a subsemigroup of an abelian group.An essential lemma in the proof of the main theorem is derived in 2.We show that if m is the cardinality of G and is assumed to be infinite, then 2m is the dimension of the S2 space associated with an invariant mean on G.Because we believe this result to be of independent interest, we state it as Theorem 1.A corollary to Theorem 1 gives a similar result for invariant means on certain abelian semigroups.In particular, this corollary is applicable to the semigroup of positive integers.An invariant mean on this semigroup is called a Banach mean.The corollary shows that the S2 space associated with a Banach mean is not separable.The question of the separability of this S2 space was raised by P. Porcelli.As a final topic we shall discuss (in 4) some related problems.This discussion amounts to an exploration of further possibilities in the directions of Theorems 1 and 2, and we shall raise several questions in this connection.

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.