Associated graded algebra

Let $𝐴$ be a filtered algebra over $𝕂$ with filtration $(𝐹 𝑖 𝐴) \infty 𝑖 = 1$ . The associated graded algebra $𝐺 ∙ 𝐴$ is the direct sum of vector spaces $⨁ \infty 𝑖 = 1 𝐺 𝑖 𝐴$ where $𝐺 0 𝐴 = 𝐹 0 𝐴$ and $𝐺 𝑖 + 1 𝐴 = 𝐹 𝑖 + 1 𝐴 / 𝐹 𝑖 𝐴$ ^[Quotient vector space] with multiplication given by falg

(𝑥 + 𝐹 𝑖 𝐴) \cdot (𝑦 + 𝐹 𝑗 𝐴) = 𝑥 \cdot 𝑦 + 𝐹 𝑖 + 𝑗 𝐴

for $𝑥 \in 𝐹 𝑖 + 1 𝐴$ and $𝑦 \in 𝐹 𝑗 + 1 𝐴$ .

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Associated graded algebra

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