Statement of the main Sample Clauses

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

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