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First Principles

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Algebraic Theory of Locally Nilpotent Derivations

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 136.3))

Abstract

Throughout this chapter, assume thatB is an integral domain containing a fieldk of characteristic zero.B is referred to as ak-domain.B ∗ denotes the group of units ofB and frac(B) denotes the field of fractions ofB. Further, Aut(B) denotes the group of ring automorphisms ofB, and Aut k (B) denotes the group of automorphisms ofB as ak-algebra. IfA ⊂B is a subring, then tr.deg A B denotes the transcendence degree of frac(B) over frac(A). The ideal generated byx 1, …, x n ∈B is denoted by either (x 1, …, x n ) orx 1 B + ⋯ +x n B. The ring ofm ×n matrices with entries inB is indicated by \(\mathcal{M}_{m\times n}(B)\) and the ring ofn ×n matrices with entries inB is indicated by \(\mathcal{M}_{n}(B)\). The transpose of a matrixM isM T.

Why should we study derivations? Answer: they occur naturally all over the place. Irving Kaplansky [ 242 ]

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Notes

  1. 1.

    The termplinth commonly refers to the base of a column or statue.

  2. 2.

    Note that the termregular action is also used in the literature to indicate an algebraic action . A regular action is thus distinguished from arational action , which refers to the action of a group by birational automorphisms.

  3. 3.

    Vasconcelos’s definition of “locally finite” is the same as the present definition of locally nilpotent. Apparently, the terminology at the time (1969) was not yet settled.

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  1. Department of Mathematics, Western Michigan University, Kalamazoo, Michigan, USA

    Gene Freudenburg

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Freudenburg, G. (2017). First Principles. In: Algebraic Theory of Locally Nilpotent Derivations. Encyclopaedia of Mathematical Sciences, vol 136.3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55350-3_1

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