Divisor class group and theory of quasi-divisors

1993

Czechoslovak Mathematical Journal

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Acta Mathematica et Informatica Universitatis …, 2000

In this paper we study, using r-ideal systems, how some properties of the directed groups Gi,i = 1,2, can be transferred into their cartesian product G, and vice-versa. In particular, beginning from the structures (Gi,ri) and (G2,r2), we construct the r\ ®r2ideal system in G and we prove that (Gt,r{) are rf-Priifer groups, i = 1,2, if and only if (G, n <8>r2) is an r\ (g)r2-Priifer group. Our main result is that the group G admits a theory of divisors, theory of quasi-divisors or strong theory of quasi-divisors, if and only if the groups Gi,i = 1,2, admit such a theory, respectively Finally, when the groups Gi,G2 and G admit theories of quasi-divisors, we investigate the relation between their corresponding Lorenzen t-groups and divisor class groups.

2021

We revisit, in an elementary way, the classical statement of various "Main Conjectures" for p-class groups H K and p-ramified torsion groups T K of abelian fields K, in the non semisimple case p | [K : Q]. The classical "algebraic" definition of the p-adic isotopic components, H alg K,ϕ , used in the literature, is inappropriate with respect to analytical formulas. For that reason we have introduced, in the 1970's, an "arithmetic" definition, H ar K,ϕ , in perfect correspondence with all analytical formulas and giving a natural "Main Conjecture", still unproved for real fields in the non semi-simple case. The two notions coincide for relative class groups H − K and groups T K since, in p-extensions, transfer maps are injective for these groups but not necessarily for real class groups. Numerical evidence of the gap between the two notions is given (Examples A.2.2, A.2.3) and PARI calculations corroborate that the true Real Main Conjecture for K writes on the form #H ar K,ϕ = #(E K / E K F K)ϕ, in terms of units E K , E K (units of the strict subfields) and F K (Leopoldt's cyclotomic units). A recent approach, conjecturing the capitulation of H K in some auxiliary cyclotomic extensions K(µ ℓ), proves the difficult real case. Contents Foreword and preliminary remarks ϕ 3.3. Comparison with classical definitions of θ-components 3.4. Arithmetic factorization of #M K and #M K 4. Semi-simple decomposition of A χ := Z p [G χ ]/(P χ (σ χ)) 4.1. Semi-simple decomposition of the A χ-modules M alg χ 4.2. Semi-simple decomposition of the A χ-modules M ar χ 4.3. Summary of the properties of the G-families M alg , M ar 5. Application to relative class groups 5.1. Arithmetic definition of relative class groups 5.2. Proof of the equality H ar χ = H alg χ , for all χ ∈ X −

Journal of Algebra, 1985

Let R be a normal noetherian domain and let I be an ideal whose associated graded ring gr,(R) is reduced. Barshay [2] showed that under these conditions the Rees-algebra S = R [It] = R 0 It 0 IZt2 @ ..a E R [t] is normal. A much simpler proof was given later by Huneke [ 111: Let P= R[It, t-'] be the extended Rees-algebra, then Pt-, E R[t, t-'1 is normal and P/tP E gr,(R) is reduced. It follows that P itself is normal. This implies easily the normality of S as well.

Communications in Algebra, 2005

In this paper we study the possible torsion in even dimensional higher class groups Cl 2n (Λ) (n ≥ 1) of an order Λ in a semi-simple algebra A over a number field F with ring of integers O F . We show that for certain orders − called generalized Eichler orders − p-torsion in Cl 2n (Λ) can only occurs for primes p dividing prime ideals ℘ of O F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.

Tbilisi Mathematical Journal, 2010

The divisor class group of the Quot scheme parameterizing quotients of fixed degree and rank of a trivial bundle on a curve is computed. The method used is a reduction to the case of rank 0 quotients.

2002

For an order embedding G h → Γ of a partly ordered group G into an l-group Γ a topology T W is introduced on Γ which is defined by a family of valuations W on G. Some density properties of sets h(G), h(X t) and (h(X t) \ {h(g 1),. .. , h(gn)}) (X t being t-ideals in G) in the topological space (Γ, T W) are then investigated, each of them being equivalent to the statement that h is a strong theory of quasi-divisors.

Journal of Pure and Applied Algebra, 1987

3.6)).

Bulletin of the American Mathematical Society, 1967