We study topological properties of the correspondence of prime spectra associated to a noncommutative ring homomorphism R → S. Our main result provides criteria for the adjointness of certain functors between the categories of Zariski closed subsets of SpecR and SpecS; these functors arise naturally from restriction and extension of scalars. When R and S are left noetherian, adjointness occurs ...

Given a matroid M represented by a linear subspace L ⊂ C (equivalently by an arrangement of n hyperplanes in L), we define a graded ring R(L) which degenerates to the Stanley-Reisner ring of the broken circuit complex for any choice of ordering of the ground set. In particular, R(L) is Cohen-Macaulay, and may be used to compute the h-vector of the broken circuit complex of M . We give a geometr...

Let p be a prime. In this note we make explicit some results on the canonical subgroup of an elliptic curve E over the ring of integers Rp of Cp implicit in [K-pPMF]. In particular, if ω generates Ω1E/Rp and E has a canonical subgroup CE , knowlege of the Hasse invariant of the reduction of (E,ω) modulo p is equivalent to knowledge of the pair (CE , ω|CE). 1. Group Schemes of order p. Let μ den...

Abstract. Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-module...

In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. This results from the main theorem of this paper, which asserts that an automorphism in any dimension n is stably tame if said condition holds point-wise over SpecR. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-...

Let R be a regular ring, essentially of finite type over a perfect field k. An R–module M is called a unit R[F ]–module if it comes equipped with an isomorphism F M −→ M, where F denotes the Frobenius map on SpecR, and F e∗ is the associated pullback functor. It is well known that M then carries a natural DR–module structure. In this paper we investigate the relation between the unit R[F ]–stru...

Let R be a regular ring essentially of finite type over a perfect field k. An R–module M is called a unit R[F ]–module if it comes equipped with an isomorphism F M −→ M where F denotes the Frobenius map on SpecR, and F e∗ is the associated pullback functor. It is well known that M then carries a natural DR–module structure. In this paper we investigate the relation between the unit R[F ]–struct...

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and – we feel – simpler proofs of theorems of L. Claborn and C.R. Leedham-Green. Luther Claborn received his PhD from U. Mi...

In this note I show that for most F–finite regular rings R of positive characteristic the localization Rf at an element f ∈ R is generated by f as a DR–module. This generalizes and gives an alternative proof of te results in [1] where the result is proven for the polynomial ring, thereby answering questions raised in [1] affirmatively. The proof given here is a surprisingly simple application o...

In this note we prove the following special case of Serre’s conjecture on Intersection Multiplicity: Let (R,m) be a regular local ring and let P , Q be two prime ideals such that (R/(P + Q)) < ∞, dimR/P + dimR/Q = dimR and dimension of Gm(R/P ) ⊗Gm(R) Gm(R/Q) < 2. Then χ(R/P, R/Q) ≥ em(R/P )em(R/Q); here em(T ) denotes the Hilbert-Samuel multiplicity for any finitely generated module T with res...