Dimension of filtered algebras and their associated graded algebras

Question

Let $F$ be a finite dimensional filtered algebra and let $G$ be the associated graded algebra. Will the dimension of $G$ and $F$ coincide or differ in general? If they differ in general then what is the simplest example where they differ. For example - in the Clifford algebra case the filtered and graded algebra have the same dimension.

Answer 1

$ \newcommand\K{\mathbb K} $

Let $\{F_i\}$ be the filtration on $F$ and $\{G_i\}$ the grading on $G$, where by definition $G_0 = F_0$ and $G_{i+1} = F_{i+1}/F_i$. Then we can define a filtration $\{G'_i\}$ on $G$ by $$ G'_i = \bigoplus_{j=0}^i G_j $$ and then $$\begin{aligned} \dim G'_i &= \dim F_0 + \sum_{j=1}^i\dim F_j/F_{j-1} \\ &= \dim F_0 + \sum_{j=1}^i(\dim F_j - \dim F_{j-1}) \\ &= \dim F_0 + \dim F_i - \dim F_0 \\ &= \dim F_i. \end{aligned}$$ Since $F$ is finite dimensional, each $F_i$ is a subspace, $F_i \subseteq F_{i+1}$, and $F = \bigcup_i F_i$, there must be some $k$ such that $F = F_k$. Thus $G = G'_k$ and $\dim F = \dim G$.