Statement of the main Sample Clauses

Statement of the main results Theorem 3.2.1. The following conditions are equivalent. (i) Xxxxxxxx’s Conjecture is valid at p. (ii) For all finite Galois extensions of number fields L/K, with G = Gal(L/K), and all finite sets of places Σ of K that contain all archimedean places, p-adic places and all those that ramify in L/K, the complex τ≥2RΓc,´et(OL,Σ, Zp(1)) belongs to Dperf(Zp[G]). (iii) For all L/K and Σ as in claim (ii) one has both (a) cΣ,γL generates H1(G, BΣ), and

Statement of the main results 1.1. Let Zn(t) be the family of Xxxxx sets of cardinality t contained in [n]. Theorem 2.2.1 Let 0 < σ < 1 be a real number. For any large enough n

Statement of the main results‌ At the outset, we fix data (C, λ, L, X ) as in Definition 3.1.3 and denote the associated special element by η = η(C,λ,L,F). For a finitely generated A-module X, we set X∗ := HomA(X, A). We note that Lemma 3.1.4 implies the set I(η) := {Φ(η): Φ ∈ ∧|F|H1(C)∗} (3.4) is an A-submodule of A. The main results in this section are the following. Theorem 3.2.1. Fix data (C, λ, L, X ) as in Definition 3.1.3 and set η := η(C,λ,L,F), a := |X | and Ar := AeC,(a). Then for any element y of the product ideal XxxX(Ext2 (H2(C), A)) · XxxX(Ext2 ′ (Ar ⊗A H2(C), Ar)) the following claims are valid. (i) If H2(C)r is any subquotient of H2(C) such that for each primitive idempotent of Ae(a), the (free) Ae-module e(F · H2(C)r) has rank at least a, then for any element x of A ∩ Ar one has xya · I(η) ⊆ Fita (H2(C)) ∩ XxxX(H2(C)rtor). (ii) If X is separable, then eC,(a) = 1 and both ya · I(η) ⊆ Fita (H2(C)) and Fita (H2(C)) ⊆ I(η). Remark 3.2.2. In certain cases, the elements x and y can be either omitted from, or made explicit, in the statement of Theorem 3. 2.1. For example, if e = eC,(a) ∈ A, then the element x can be taken to be e and so (since η = e · η) can be omitted from the statement. Regarding the choice of y, note for example, that if B is an R-order such that Ext1 (N, B) vanishes for all finitely generated torsion-free B-modules N (see Example 2.3.7(ii)), then Ext2 (Xr, B) vanishes for all finitely generated B-modules Xr. Remark 3.2.3. Under the conditions discussed in Remark 3.2.2 (for example when X is separable and A is Gorenstein), Theorem 3. 2.1 implies that I(η(C,λ,L,F)) = Fita (H2(C)) with a = |X | and so the special element completely determines the higher Fitting ideal. Σ