# The system of regular parameters form a basis of the module of relative differential forms

The problem comes from Serre's "Groupes algébriques et corps de classes", chapter II, section 7. Let $$k$$ be a field, and $$A$$ be a $$k$$-algebra. $$A$$ is a Noetherian regular ring (every localization at prime is regular) and $$\dim A = r$$. Let $$P$$ be a prime of $$A$$ and $$P_P = (t_1, \dots, t_r)$$, where $$t_1, \dots, t_r$$ is a system of regular parameters.

Prove that $$\Omega_{A_P/k}$$, the module of relative differential forms, has a basis $$\rm{d}t_1, \dots, \rm{d}t_r$$.

What I tried

I know that the polynomial rings over $$k$$ have the above properties. But I am not sure if it holds for general regular local rings.

I also know that if $$k$$ is algebraically closed and $$P$$ is a maximal ideal of $$A$$. Then $$A/P = k$$. Actually, in this special case over varieties, $$k \subset A$$ and this inclusion induces $$k \cong A/P$$. Since $$A_P$$ is a Noetherian regular local ring, $$G_P(A_P) = \oplus_{i \geq 0} P^i_P/P^{i+1}_P \cong k[X_1,\dots, X_r]$$. This means that although every element in $$P^i_P$$ may have various forms, it must be like $$\sum_{j_1 \leq \dots \leq j_i} c_{j_1, \dots,j_i} t_{j_1} \dots t_{j_i}$$ in $$P^i_P/P^{i+1}_P$$, $$c_{j_1, \dots,j_i} \in k$$.

This is proved in Hartshorne's Algebraic Geometry II, 8.8. Only need to assume $$k$$ is perfect. Then $$\Omega_{A_P/k}$$ is a free $$A_P$$-module of rank $$r$$ iff $$A_P$$ is a regular local ring. We know that $$A_P$$ is a regular local ring. So $$\Omega_{A_P/k}$$ is a free $$A_P$$-module of rank $$r$$. (8.7) in that book also shows $$dt1,…,dtr$$ is a basis.