Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$.
By Koszul resolution I mean
$$\ldots \to A \otimes \Lambda^k \mathbb{C}^n \xrightarrow{\partial_k} \ldots \to \mathbb{C}[x_1, \ldots, x_n] \to \mathbb{C},$$
where $\partial_k(z \otimes e_{i_1} \wedge \ldots \wedge e_{i_k}) = \sum \limits_{j=1}^n (-1)^{j-1}x_jz \otimes e_{i_1} \wedge \hat{\ldots e_{i_j}} \ldots \wedge e_{i_k}$.
For $A_q$ it is almost the same but with $(-q)^j$.
So I know a contracting homotopy for both $A$ and $A_q$ (they are pretty similar) but it is not suitable for my purposes. Here's an explicit formula for quantum case $n = 2$.
$s_{-1}(1) = 1$,
$s_0(x^iy^j) = \frac{1}{i+j}(q^jix^{i-1}y^j, jx^iy^{j-1}) $ \quad ($s_0(1) = 0$, of course),
$s_1((x^iy^j, 0)) = - \frac{j}{(i+j + 1)q} x^iy^{j-1} $, $s_1((0, x^iy^j)) = \frac{i q^j}{(i+j + 1)q} x^{i-1}y^{j} $.
Does anyone know another homotopies? The more of them the better.
