# Skew-symmetric multi-derivations of $k[x_1,\ldots,x_n]/I$

Let $$I = \langle f_1, \ldots f_r \rangle$$ be an ideal in $$R=k[x_1,\ldots,x_n]$$ where $$k$$ is a field, and put $$A = R/I$$.

(If $$k$$ is algebraically closed and $$I$$ is radical then $$A$$ is the coordinate ring of an affine variety.)

Let $$\mathfrak{X}^p(A) = \operatorname{Der}_k(\wedge^p A, A)$$ be skew-symmetric $$p$$-derivations (derivation in each argument) of $$A$$.

Question: Is there an algorithm to calculate $$\mathfrak{X}^p(A)$$ in terms of $$f_1,\ldots,f_r$$? I am interested in $$p=1,2,3$$.

What I have tried/observed so far:

1. The condition that $$P \in \mathfrak{X}^p(R)$$ descends to (is well-defined on) $$\mathfrak{X}^p(A)$$ is a system of (ideal membership) equations $$P(x_{i_1}, \ldots, x_{i_{p-1}}, f_j) \in I \quad \text{for each } 1\leq i_1<\cdots (Note also every element of $$\mathfrak{X}^p(A)$$ has a lift to $$\mathfrak{X}^p(R)$$.)

2. Letting $$P_{i_1,\ldots,i_p} = P(x_{i_1},\ldots,x_{i_p}) \in R$$ be the (sought for) coefficients of $$P$$, the system becomes $$\sum_k P_{i_1,\ldots,i_{p-1},k} \frac{\partial f_j}{\partial x_k} \in I \quad \text{for each } 1\leq i_1<\cdots

3. Here one can restrict to the sum over $$k \in \{1,\ldots,n\} \setminus \{i_1,\ldots i_{p-1}\}$$. Note the equations are not independent, due to skew-symmetry.

4. The equations above for fixed $$j$$ suggest to consider the intersection $$I \cap \langle \frac{\partial f_j}{\partial x_1}, \ldots \frac{\partial f_j}{\partial x_n} \rangle$$ which can be computed using Groebner bases. I am not sure what we can conclude about the coefficients $$P_{i_1,\ldots,i_{p-1},k}$$. Furthermore, I do not see how one would combine these results for different $$(i_1,\ldots,i_{p-1})$$'s and $$j$$'s.

5. $$\mathfrak{X}^p(A)$$ is an $$A$$-module. I think it is finitely generated. (I am interested in the case where it is.)

6. $$\mathfrak{X}^p(A) \cong \operatorname{Hom}_A(\Omega^p(A), A)$$ where $$\Omega^p(A)$$ are Kähler $$p$$-forms.

7. $$\mathfrak{X}^2(A) \neq \wedge^2 \mathfrak{X}^1(A)$$ in general: if $$I = \langle yx, yz, y^2 \rangle$$ in $$R=\mathbb{C}[x,y,z]$$ then $$0 \neq y \frac{\partial}{\partial y} \wedge \frac{\partial}{\partial z}$$ is in $$\mathfrak{X}^2(A)$$ and not in $$\wedge^2 \mathfrak{X}^1(A)$$.

I would also be interested in (classes of) examples where one can calculate $$\mathfrak{X}^p(A)$$ explicitly for $$p=1,2,3$$.

Put $$N_p =\binom{n}{p}$$ and consider the sought-for $$R$$-submodule $$M_p \subset R^{N_p}$$ consisting of $$(P_{i_1,\ldots,i_p})$$ with $$1 \leq i_1<\ldots< i_p \leq n$$ such that $$\sum_{1 \leq i_1<\ldots< i_p \leq n} P_{i_1,\ldots,i_p} \frac{\partial}{\partial x_{i_1}} \wedge \cdots \wedge \frac{\partial}{\partial x_{i_p}} \in \mathfrak{X}^p(A).$$

Define $$P_{i_1,\ldots,i_p}$$ for unsorted $$(i_1,\ldots,i_p)$$ in the usual way, by using the appropriate sign.

The defining equations are (as in the question): for every $$1 \leq j_1 < \ldots < j_{p-1} \leq n$$ and $$1 \leq j \leq r$$, $$\sum_{k=1}^n P_{j_1,\ldots,j_{p-1},k}\frac{\partial f_j}{\partial x_k} \in I. \tag{1}$$

Here the coefficients are unsorted in general, so we define a $$R$$-linear map $$\Phi^p_{j_1,\ldots,j_{p-1}}:R^{N_p}\to R^n$$ by $$\Phi^p_{j_1,\ldots,j_{p-1}}((P_{i_1,\ldots,i_p}))_k = \begin{cases} (-)^\sigma P_{i_1,\ldots,i_{p}} & \text{ if } \exists \sigma \in S_{p} \text{ such that } \sigma(i_1,\ldots,i_p) = (j_1,\ldots,j_{p-1},k)\\ 0 & \text { otherwise. }\end{cases}$$ which gives the coefficients of the equation $$(1)$$ in terms of the sorted ones.

(For example, $$\Phi_{1}^2((P_{12},P_{13},P_{23})) = (0, P_{12}, P_{13})$$ and $$\Phi_{2}^2((P_{12},P_{13},P_{23})) = (-P_{12},0,P_{23})$$.)

Put $$T_j = \{ c \in R^n | \sum c_k \frac{\partial f_j}{\partial x_k} \in I\}$$ and let $$\pi_j : R^n \to R^n/T_j$$ be the quotient map. Note $$T_j$$ can be computed.

Then we have constructed the $$R$$-module as $$M_p = \bigcap_{\substack{1\leq j_1 < \ldots < j_{p-1} \leq n\\1 \leq j \leq r}} \ker(\pi_j \circ \Phi^p_{j_1,\ldots,j_{p-1}})$$ which can be computed because kernels of $$R$$-linear maps and intersections of $$R$$-modules can be computed.

(For example in Singular using modulo and intersect respectively.)

Of course the coefficients can be taken modulo $$I$$, so we can consider $$M_p/IR^{N_p}$$. In practice it means we can throw away some of the generators of $$M_p$$ which we obtain, in the same way as in the answer that was just linked.