A Jacobian identity in positive characteristic. (English) Zbl 1342.13008

Lang, Jeffrey

J. Commut. Algebra 7, No. 3, 393-409 (2015).

In this note, the author presents us with a Jacobian identity: \[ \sum_{i_1=1}^{p-1}\cdots\sum_{i_n=1}^{p-1} f_1^{i_1}\cdots f_n^{i_n} D_1^{p-1}\cdots D_n^{p-1} (f_1^{p-1-i_1}\cdots f_n^{p-1-i_n})=(-1)^n (J(f_1,\dots,f_n))^{p-1} \] where \(f_i \in k(x_1, \dots, x_n)\), \(k\) a field of characteristic \(p\), \(D_i=\partial/\partial x_i\) and \(J(f_1,\cdots,f_n)\) is the Jacobian of \(f_1, \dots, f_n\) with respect to the variables \(x_1, \dots, x_n\). A nice consequence of this identity is that \[ k[x_1, \dots, x_n][1/J(f_1, \dots, f_n)]=k[x_1^p, \dots, x_n^p,f_1, \dots, f_n][1/J(f_1, \ldots, f_n)] \] for \(f_1, \dots f_n \in k[x_1, \dots, x_n]\). The article concludes with a discussion of the two variable Jacobian conjecture in characteristic \(p\).

Reviewer: Janet Vassilev (Albuquerque)

MSC:

13A35	Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13N15	Derivations and commutative rings

Keywords:

Jacobian; derivation; positive characteristic; nilpotent derivations; points at infinity

References:

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[4]	Pierre Samuel, Lectures on unique factorization domains , Tata Lect. Notes, 1964. · Zbl 0184.06601