The singularity space consists of all germs (X, x), with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a Polish space for the metric given by the order to which infinitesimal neighborhoods agree after base change. In other words, the classification of singularities up to analytic extensions of scalar...

The singularity space consists of all germs (X, x), with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a Polish space for the metric given by the order to which infinitesimal neighborhoods agree after base change. In other words, the classification of singularities up to analytic extensions of scalar...

A result of Artin, Small, and Zhang is used to show that a noetherian algebra over a commutative, noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, noetherian associated graded ring. This result is extended to show that if an algebra over a commutative noetherian ring has a locally finite, noetherian associated graded ring, then the intersection of the powers ...

Let k be an uncountable algebraically closed field and let A be a countably generated left Noetherian k-algebra. Then we show that A⊗k K is left Noetherian for any field extension K of k. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over k are finitely generated extensions of k. We give examples which show that A⊗k K need not re...

In this paper we study group actions on quasi-median graphs, or 'CAT(0) prism complexes', generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a graph $X$ and define contact $\mathcal{C}X$ for these hyperplanes. show that is always quasi-isometric to tree, result Hagen, under certain conditions action $G \curvearrowright X$ induces an acylindrical \mathcal{C}X$, giving a...

We consider a filtration of the K-theory space for a regular noetherian ring proposed by Goodwillie and Lichtenbaum and show that its successive quotients are geometric realizations of explicit simplicial abelian groups. The filtration in weight t involves t-tuples of commuting automorphisms of projective R-modules. It remains to show that the Adams operations act appropriately on the filtration.

Let (A,1) be a partially ordered set. The relation a 1 b can be read as “a precedes b”. For elements a and b in A, we write a ≺ b if and only if a 1 b and a 6= b holds. For notational convenience, we also write a o b if and only if a 1 b holds, and a  b if and only if a ≺ b holds. We call (A,1) a well-founded set if and only if every non-empty subset M of A contains at least one minimal elemen...

 ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $Bbb Z_{p^{infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dim...