Remark 4.4 . [0398] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Remark 4.4 .
(i)
We have
τ ( 𝔞 ∙ λ ) = τ ( 𝔞 m λ / m ) \tau({\mathfrak{a}}^{\lambda}_{\bullet})=\tau({\mathfrak{a}}_{m}^{\lambda/m}) m ∈ ℕ m\in\mathbb{N} [Mus13 , p. 541] .
(ii)
For all m ∈ ℕ m\in\mathbb{N} [Mus13 , p. 541, l. 4]
(4.6) τ ( 𝔞 m ) ⊆ τ ( 𝔞 ∙ m ) . \tau({\mathfrak{a}}_{m})\subseteq\tau({\mathfrak{a}}^{m}_{\bullet}).
(iii)
For all m ∈ ℕ m\in\mathbb{N} Subadditivity Property [Mus13 , Prop. 3.1(ii)]
(4.7) τ ( 𝔞 ∙ m λ ) ⊆ τ ( 𝔞 ∙ λ ) m . \tau({\mathfrak{a}}_{\bullet}^{m\lambda})\subseteq\tau({\mathfrak{a}}_{\bullet}^{\lambda})^{m}.