Narrow Results

The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras, which we developed in a previous paper [T. Bar-On, Sh. Gilat, E. Matzri and U. Vishne, Locally finite central simple algebras, Algebras Represent. Theory26(2) (2023) 553–607]. This algebra has appeared under various names before, and deserves further study. Supernatural matrices are a minimal solution to the equation of unital algebras ${\mathrm{\text{M}}}_n(X)\cong X$, which we compare to several similar conditions involving cancellation of matrices. Viewing a natural representation of this algebra, we show that supernatural matrices generalize both McCrimmon’s deep matrices algebra and $m$-petal Leavitt path algebra. We also study their simple representations.

We deduce an analogue of Quillen–Suslin’s local-global principle for the transvection subgroups of the general quadratic (Bak’s unitary) groups. As an application, we revisit the result of Bak–Petrov–Tang on injective stabilization for the ${\mathrm{\text{K}}}_1$-functor of the general quadratic groups.

We investigate the almost Cohen–Macaulay property and the Serre-type condition $(C_n),n\in \mathbb{N},$ for noetherian algebras and modules. More precisely, we find permanence properties of these conditions with respect to tensor products and direct limits.

We give explicit examples of pairs of one-ended, open $4$-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.