Proposition 1 Sample Clauses

Proposition 1. 5.8. Let R be a left-artinian ring with ▇▇▇▇▇▇▇▇ radical J(R). Then the natural projection p : R → R/ J(R) induces a surjective map on the set of idempotents. Proof. Let E ∈ R be an idempotent. Then certainly p(E) is an idempotent in R/ J(R). Suppose e ∈ R/ J(R) is an idempotent, i.e. e2 − e ∈ J(R). What we want to find is an element satisfying x2 − x = 0 in R, which is mapped to e. Consider the polynomial F (x) = 3x2 − 2x3. Let e1 := F (e). Then 1 e2 − e1 = (3e2 − 2e3)2 − (3e2 − 2e3) = (4e2 − 4e − 3)(e2 − e)2 ∈ J(R)2, 1 so e2 − e1 ∈ J(R)2. Moreover, e1 = e − (2e − 1)(e2 − e), so e1 ≡ e mod J(R). We define ei := F (e i−1 ). By induction, we have e2 − ei ∈ J(R)2i and e ≡ e i mod J(R). Since R is left-artinian, J(R) is nilpotent, so there exists n ∈ Z≥0 such that e2 − en ∈ J(R)n = 0. Then E = en is the element we were after. Remark 1.5.9. The key to the above proof is that e2 e is nilpotent. Hence we can use the same lifting technique against any nil ideal of R.

Proposition 1. 4.2. (Birkenhake–▇▇▇▇▇ [6, Proposition 13.3.1] and ▇▇▇▇ [20, Theorem 1.4.1-(iii)]) With the notation above, the complex torus AΦ,m is an abelian variety and has a natural CM structure given by the action of OK on m. In this thesis, we use the complex torus Cg/Φ˜(m) instead of AΦ,m as a realization of an abelian variety over C conforming to the notation of Lang [20] and ▇▇▇▇▇▇▇–Taniyama [40]. For each α ∈ K, we let SΦ(α) be the matrix diag(φ1(α), . . . , φg (α)). Theorem 1.4.3. (Lang [20, Theorem 1.4.1-(ii)]) Let (K, Φ) be a CM pair and let (A, θ) be an abelian variety of type (K, Φ) with CM by K. Then there is a fractional ideal m I and an analytic isomorphism ι : Cg/Φ˜(m) → A(C) such that the diagram Cg/Φ˜(m) A(C) SΦ(α) θ(α) Cg/Φ˜(m) A(C) commutes for all α ∈ OK. Definition 1.4.4. We say that an abelian variety (A, θ) of type (K, Φ) is of type (K, Φ, m) if there is a fractional ideal m I and an analytic isomorphism ι : Cg/Φ˜(m) → A(C). Definition 1.4.5. Let (A, θ) be an abelian variety of type (K, Φ) and let X be an ample divisor on A. We say that (A, θ) is Φ-admissible with respect to the polarization ϕX if θ(K) is stable under the ▇▇▇▇▇▇ involution. Theorem 1.4.6. (Lang [20, Theorem 1.4.5-(iii)] If an abelian variety (A, θ) of type (K, Φ, m) is simple, then it is Φ-admissible with respect to every polarization.

Proposition 1. Let (A1, θ1) and (A2, θ2) be abelian varieties of prim- itive CM type (K, Φ). If an isogeny λ from (A1, θ1) onto (A2, θ2) is an a- multiplication, then λ∗ from (A∗1, θ1∗) onto (A∗2, θ2∗) is an a-multiplication.

Proposition 1. 3.1. (Serre [37]) Two divisors X1 and X2 are alge- braically equivalent if and only if ϕX1 = ϕX2 .

Proposition 1. The pullback of P by (▇▇, jv,−1) : C ×S J → J ×S J∨ together with its rigidifications at b and 0, is equal to Luniv. Let d be in Z≥0. The morphism : C(d) → J∨ = Pic0 Σ: C(d) → J, sending, for every S-scheme T, each point D in C(d)(T ) to the class of OCT (D − db) twisted by the pullback from T that makes it rigidified at b, followed by jv,−1 : J → J∨. Summarised in a diagram, with M := (id × jv,−1)vP: Luniv P M (1.6.3.3) i˜d×Σ Nd jb×j𝗁,−1 id×j𝗁,−1 id×Σ ( ) C ×S J J ×S J∨ J ×S J J ×S C d .

Proposition 1. The invertible O-module M on (J ×Zq J)Qq , with its rigidifica- tions, extends uniquely to an invertible O-module M with rigidifications on J ×Zq J . The biextension structure on M× extends uniquely to a biextension structure on M˜×.

Proposition 1. 6.22 ([58], Chapter I, Example 2.2(4), Proposition 2.13). Let R, R′ be commutative rings, α : R R′ a ring homomorphism and P, Q two finitely generated projective R-modules. Then HomR(P, Q) ⊗R R′ =∼ HomR' (P ⊗R R′, Q ⊗R R′), (P ⊗R Q) ⊗R R′ ∼= (P ⊗R R′) ⊗R' (Q ⊗R R′), as R′-modules.

Proposition 1. For each a in X, there exists a unique real-valued ga in C∞(X − {a}) such that the following properties hold:

Proposition 1. Assume that G is commutative, then we have a bijec- tion X-torsors under G up to isomorphism ←→ the group H1(X, G). Now, assume that B is an algebraic k-group, and A is a closed algebraic k-subgroup of B. Assume that the quotient X = B/A exists, and consider the projection B → X, then B is an X-torsor under A (A is said to be the structure group). The assumption that the quotient X = B/A exists is satisfied when B is affine or A is finite.