An Algorism for Differential Invariant Theory by Glenn O. E.

By Glenn O. E.

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Generalities 644 Recall that, for any subset X of FG, 1 ( X ) denotes the left annihilator of X and X I is defined by X* = E FGltr(Xu) = 0) {U We shall refer t o the ideal Z(FG) n Soc(FG) of Z ( F G ) as the Reynolds ideal of Z(FG) and denote it by Rey(FG). 5 (iii), - Soc(FG) = FG Reg( FG) Our first aim is to determine Rey(FG) exclusively in terms of F and G which will give us the original ideal contructed by Reynolds (1972). Let us fix some notation. For any subset X of G , we put x-l = {x-+ EX} and denote by F X the F-linear span of X .

5. Let A be an R-algebra and let V be an A-module which is R-free of finite rank. Then EndA(V) is a direct summand of the R-module EndR( V ) . Proof. It suffices to show that EndA(V)is a pure submodule of EndR(V). The latter will follow provided we show that for any 0 # r E R, To this end, assume that f E EndA(V) is such that f = r p for some cp E EndR(V). Then, for all w E V , a E A , we have whence r(cp(av)- ap(v)) = 0. Since V is R-free and r # 0, we conclude that cp(aw) - acp(v) = 0 for all a E A , w E V .

2, E v @ F Ew is a division algebra. Hence, by (b), V '@RW is indecomposable, proving (c). (iii) If V @ R W is indecomposable, then by (b) Ev @IFEw is a division algebra. Hence both Ev and Ew are division algebras and so V ,W are indecomposable. 6 (ii), Ev, Ew are division algebras over the finite field F . It follows from Wedderburn's theorem that Ev and Ew are finite fields. By (b), V '@RW is indecomposable if and only if Ev ' 8 Ew ~ is Generalities 658 a field. 4. Using conventions and notation introduced at the begining of this section we now prove the following result, in which R denotes a principal ideal domain and a complete local ring and F = R / J ( R ) .

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