Triangulated Matlis equivalence

This paper is a sequel to [L. Positselski, Dedualizing complexes and MGM duality, J. Pure Appl. Algebra 220(12) (2016) 3866–3909, arXiv:1503.05523 [math.CT]; Contraadjusted modules, contramodules, and reduced cotorsion modules, preprint (2016), arXiv:1605.03934 [math.CT]]. We extend the classical Harrison–Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules and contramodules over a Matlis domain. This generalizes to the case of any commutative ring [Formula: see text] with a fixed multiplicative system [Formula: see text] such that the [Formula: see text]-module [Formula: see text] has projective dimension [Formula: see text]. The latter equivalence connects complexes of [Formula: see text]-modules with [Formula: see text]-torsion and [Formula: see text]-contramodule cohomology modules. It takes a nicer form of an equivalence between the derived categories of abelian categories when [Formula: see text] consists of nonzero-divisors or the [Formula: see text]-torsion in [Formula: see text] is bounded.

On Divergence of Fourier Series with Respect to Multiplicative Systems on the Sets of Measure Zero

For any multiplicative system of bounded type and any set of measure zero there exists a bounded measurable function whose Fourier series with respect to this system diverges on this set.

A Note on Buchsbaum Rings and Localizations of Graded Domains

Let R = ⊕i ≧0Ri be a graded integral domain, and let p ∈ Proj (R) be a homogeneous, relevant prime ideal. Let R(p) = {r/t| r ∈ Ri, t ∈ Ri\p} be the geometric local ring at p and let Rp = {r/t| r ∈ R, t ∈ R\p} be the arithmetic local ring at p. Under the mild restriction that there exists an element r1 ∈ R1\p, W. E. Kuan [2], Theorem 2, showed that r1 is transcendental over R(p) andwhere S is the multiplicative system R\p. It is also demonstrated in [2] that R(p) is normal (regular) if and only if Rp is normal (regular). By looking more closely at the relationship between R(p) and R(p), we extend this result to Cohen-Macaulay (abbreviated C M.) and Gorenstein rings.

On Rings of Fractions

Let R be a commutative Noetherian ring with identity, and let M be a fixed ideal of R. Then, trivially, ring multiplication is continuous in the ilf-adic topology. Let S be a multiplicative system in R, and let j = js: R → S-1R, be the natural map. One can then ask whether (cf. Warner [3, p. 165]) S-1R is a topological ring in ihe j(M)-adic topology. In Proposition 1, I prove this is the case if and only if M ⊂ p(S), whereHence S-1R is a topological ring for all S if and only if M ⊂ p*(R), where