Differential Operators and Differential Modules
In this chapter k is a differential field such that its subfield of constants C is different from k and has characteristic 0. The skew (i.e., noncommutative) ring D :=k[∂] consists of all expressions L :=a n ∂ n + ⋯ + a1∂ + a0 dot with n ∈ Z, n ≥ 0 and al
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2.1 The Ring D = k[∂] of Differential Operators In this chapter k is a differential field such that its subfield of constants C is different from k and has characteristic 0. The skew (i.e., noncommutative) ring D := k[∂] consists of all expressions L := an ∂ n + · · · + a1 ∂ + a0 with n ∈ Z, n ≥ 0 and all ai ∈ k. These elements L are called differential operators. The degree of L deg L above is m if am = 0 and ai = 0 for i > m. In the case L = 0 we define the degree to be −∞. The addition in D is obvious. The multiplication in D is completely determined by the prescribed rule ∂a = a∂ + a . Since there exists an element a ∈ k with a = 0, the ring D is not commutative. One calls D the ring of linear differential operators with coefficients in k. A differential operator L = an ∂ n + · · · + a1 ∂ + a0 acts on k and on differential extensions of k, with the interpretation ∂(y) := y . Thus the equation L(y) = 0 has the same meaning as the scalar differential equation an y(n) + · · · + a1 y(1) + a0 y = 0. In connection with this one sometimes uses the expression order of L, instead of the degree of L. The ring of differential operators shares many properties with the ordinary polynomial ring in one variable over k. Lemma 2.1 For L 1 , L 2 ∈ D with L 1 = 0, there are unique differential operators Q, R ∈ D such that L 2 = Q L 1 + R and deg R < deg L 1 . The proof is not different from the usual division with remainder for the ordinary polynomial ring over k. The version where left and right are interchanged is equally valid. An interesting way to interchange left and right isprovided by the “involution” i : L → L ∗ of D defined by the formula i( ai ∂ i ) = (−1)i ∂ i ai . The operator L ∗ is often called the formal adjoint of L. Exercise 2.2 The term “involution” means that i is an additive bijection, i 2 = id and i(L 1 L 2 ) = i(L 2 )i(L 1 ) for all L 1 , L 2 ∈ D. Prove that i, as defined above, has these properties. Hint: Let k[∂] denote the additive group k[∂] made into a ring by the opposite multiplication given by the formula L 1 L 2 = L 2 L 1 . Show that k[∂] is also a skew polynomial ring over the field k and with variable −∂. Observe that (−∂) a = a (−∂) + a . M. van der Put. et al., Galois Theory of Linear Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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2 Differential Operators and Differential Modules
Corollary 2.3 For any left ideal I ⊂ k[∂] there exists an L 1 ∈ k[∂] such that I = k[∂]L 1 . Similarly, for any right ideal J ⊂ k[∂] there exists an L 2 ∈ k[∂] such that J = L 2 k[∂]. From these results one can define the least common left multiple, LCLM(L 1 , L 2 ), of L 1 , L 2 ∈ k[∂] as the unique monic generator of k[∂]L 1 ∩ k[∂]L 2 and the greatest common left divisor, GCLD(L 1 , L 2 ), of L 1 , L 2 ∈ k[∂] as the unique monic generator of L 1 k[∂] + L 2 k[∂] . The least common right multiple of L 1 , L 2 ∈ k[∂], LCRM(L 1 , L 2 ) and the greatest common right divisor of L 1 , L 2 ∈ k[∂], GCRD(L 1 , L 2 ) can be defined similarly. We note that a modified version of the Euclidean
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