For a fixed prime p, let H∗(G) denote the mod p cohomology ring of a finite group G, that is, ExtFpG(Fp,Fp) interpreted algebraically and H ∗(K(G, 1); Fp) interpreted topologically. The cohomology ring is a graded commutative Noetherian ring, or equivalently, a finitely generated graded commutative Fp-algebra. Although the ring is not strictly commutative, the usual concepts from commutative ri...

In this expository note, we discuss various aspects of the theory of dimension of schemes, in particular focusing on which hypotheses are necessary in order to insure good behavior, and what sort of pathologies can otherwise arise. References are frequently exercises in disguise (or not in disguise, as the case may be). In particular, because the intended applications are geometric, we will sta...

The rank rk(R) of a ring R is the supremum of minimal cardinalities of generating sets of I as I ranges over ideals of R. Matson showed that every n ∈ Z+ occurs as the rank of some ring R. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension 0 or 1, we give four different constructions of rings of rank n (for all n ∈ Z+). Two constructions use one-dimension...

Derived decompositions of abelian categories are introduced in internal terms subcategories to construct semi-orthogonal (or Bousfield localizations, or hereditary torsion pairs) various derived categories. We give a sufficient condition for arbitrary have such and show that it is also necessary with enough projectives injectives. For bounded categories, we describe which determined by decompos...

let $d$ be an integral domain and $star$ a semistar operation stable and of finite type on it. we define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong s-domains. as an application, we give new characterizations of $star$-quasi-pr"{u}fer domains and um$t$ domains in terms of dimension ine...