Pullback Diagram Research Articles

Circular bar construction

Let A be a CDG (commutative differential graded) algebra over a field of characteristic 0, and B(A), the bar construction of A. If C is a DG (differential graded) A @ Amodule, define B(A; C) = C BAOR B(A). If C’ and C” are DG A-modules, then B(A; C’ @ C”) coincides with the usual two sided bar construction B(C’, A, C”). (See [12].) In this sense, B(A; C) closes the two sides of the bar construction and is therefore “circular”. The geometric significance of B(A; C) lies in the case where A = A(M) and C = A(N) are the Rham complexes and the A @ A-module structure of C is induced by a differentiable map f : N-t M x M. Theorem 0.1[6] (Theorem 4.3.1[5]) asserts that, if M and N are differentiable manifolds and if M is simply connected with finite Betti numbers, then H(B(A; C)) is the real cohomology of the pullback space Ef of the following pullback diagram of the free path fibration of M:

Pullbacks in Regular Categories

Given a pair of mapsin a category, we would like to know whether they form part of a pullback diagram as follows:I am indebted to Basil Rattray for mentioning the solution of this problem for the category of sets. Here we shall solve it for any regular category in the sense of Barr [1].It will be useful to make the following definition.

Class groups of integral group rings

Let $\Lambda$ be an $R$-order in a semisimple finite dimensional $K$-algebra, where $K$ is an algebraic number field, and $R$ is the ring of algebraic integers of $K$. Denote by $C(\Lambda )$ the reduced class group of the category of locally free left $\Lambda$-lattices. Choose $\Lambda = ZG$, the integral group ring of a finite group $G$, and let $\Lambda ’$ be a maximal $Z$-order in $QG$ containing $\Lambda$. There is an epimorphism $C(\Lambda ) \to C(\Lambda ’)$, given by $M \to \Lambda ’{ \otimes _\Lambda }M$, for $M$ a locally free $\Lambda$-lattice. Let $D(\Lambda )$ be the kernel of this epimorphism; the groups $D(\Lambda ),C(\Lambda )$ and $C(\Lambda ’)$ are all finite. Our main theorem is that $D(ZG)$ is a $p$-group whenever $G$ is a $p$-group. This generalizes Fröhlich’s result for the case where $G$ is an abelian $p$-group. Our proof uses some facts about the center $F$ of $QG$, as well as information about reduced norms. We also calculate $D(ZG)$ explicitly for $G$ cyclic of order $2p$, dihedral of order $2p$, or the quaternion group. In these cases, the ring $ZG$ can be conveniently described by a pullback diagram.