Given a field $ \mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$ , some natural numbers $ q,n,m$ with $ q \leq n$ and $ m < n$ and a polynomial algebra $ S := \mathbb{K}[\eta_1, \dots, \eta_n]$ . We can consider $ \otimes_\mathbb{K}^q S$ an $ S$ -module via $ $ \eta_i \cdot p_1 \otimes \dots \otimes p_q := \sum_{j=1}^q p_1 \otimes \dots \otimes \eta_i \cdot p_j \otimes \dots \otimes p_q \quad \forall p_1, \dots, p_q \in S$ $

Fuks claims that the following statement on pages 85 to 86 of his book “Cohomology of infinite-dimensional Lie algebras” is obvious:

If there is a family of tensors $ \{c_{i_1 \dots i_m} \in \otimes_{\mathbb{K}}^q S \mid i_1, \dots, i_m = 1, \dots, n\}$ which is skew-symmetric with respect to its indices, with the property that

$ $ \sum_{r=1}^{m + 1} (-1)^r \eta_{i_r} \cdot c_{i_1 \dots \hat{i_r} \dots i_{m+1}} = 0 \quad \forall i_1, \dots, i_{m+1},$ $

then it follows that there exists some family of tensors $ \{b_{j_1 \dots j_{m-1}} \in \otimes_{\mathbb{K}}^q S \mid j_1, \dots, j_{m-1} = 1, \dots, n\}$ , also skew-symmetric with respect to its indices, with the property that

$ $ \sum_{r=1}^{m} (-1)^r \eta_{i_r} \cdot b_{i_1 \dots \hat{i_r} \dots i_{m}} = c_{i_1 \dots i_m} \quad \forall i_1, \dots, i_{m}.$ $

If one tries to write it out in coefficients, one can sort of see why this might be true, and clearly this looks like it arises from exactness of some “complex of tensor families”, but I’m unsure how to write down any of these approaches in a way that one could call “obvious”. I am wondering if I am missing a point of view from which this indeed is an obvious statement. Thanks in advance!

(The context of this statement is the triviality of the Lie algebra cohomology of $ n$ -dimensional formal vector fields in order $ q = 1, \dots, n$ . While I did find another source that proves this, the approach in Fuks’ book still confuses me, and I’d be happy to understand it!)