Theorem. The graded pieces are exterior powers [chevalley1954algebraic, II.1.6, pp. 41--42] [fcap-000I]
Theorem. The graded pieces are exterior powers [chevalley1954algebraic, II.1.6, pp. 41--42] [fcap-000I]
Let \(R\), \(M\), and \(Q\) be as in Lemma [fcap-000H]. Chevalley's finite-dimensional field argument motivates the identification. TauCeti proves the following coordinate-free commutative-ring and module equivalence. For every \(n\ge 0\), the leading-symbol map gives \[F_n\mathcal {C}\kern -2pt\ell (Q)/F_{n-1}\mathcal {C}\kern -2pt\ell (Q)\simeq \bigwedge ^n_R M.\] At \(n=0\) this is the scalar equivalence \(F_0/0\simeq R=\bigwedge ^0_RM\). For \(n>0\), the inverse sends a decomposable exterior product to the class of the corresponding Clifford word.
The point of the quotient is that every contraction term has already fallen into a lower filtration step. Alternation is therefore exact in the graded piece even though it is not exact in the Clifford algebra itself.