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

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 $$I$$ is prime then $$A$$ is the coordinate ring of an irreducible 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 find (generators and relations for) $$\mathfrak{X}^p(A)$$ in terms of $$f_1,\ldots,f_r$$?

I asked this question on Math.SE and eventually only partially answered it myself.

The algorithm in my partial answer finds only the module consisting of those elements of $$\mathfrak{X}^p(A)$$ which lift to $$\mathfrak{X}^p(R)$$. Is this algorithm (or any alternative algorithm) known and/or written down in the literature somewhere? Is there any algorithm that works more generally?

• I think every (multi)derivation of $A$ lifts to one on $R$. Let $y_i = x_i \mod I$ be the images of the generators. Then a derivation of $A$ is completely determined by how it acts on each $y_i$. Suppose that $Z : A \to A$ is a derivation and let $Z_i = Z(y_i) \in A$. Now choose lifts $\tilde Z_i \in R$ of these elements and set $\tilde Z = \sum_i \tilde Z_i \partial_{x_i}$. Then $\tilde Z$ is a derivation of $R$ that lifts $Z$. The case $p > 1$ is similar. – Brent Pym Nov 11 '18 at 8:06
• This identifies $Der(A)$ with the subquotient $Der(R,I)/I$ of $Der(R)$, where $Der(R,I) < Der(R)$ is the module of derivations preserving $I$. Note that $Der(R,I)$ is the kernel of the natural $R$-module map $Der(R) \to Hom_R(I/I^2,A)$ defined by evaluating derivations on elements of the ideal. This makes it fairly straightforward to get Macaulay2 to calculate $Der(A)$ in terms of generators and relations as you suggest; it it probably equivalent to the algorithm you described. – Brent Pym Nov 11 '18 at 8:24