Galois extensions of differential algebras (Q1382070)

Let \(A\) be a commutative ring containing a field of characteristic zero, and let \(T_A\) be a finitely generated \(A\) submodule of \(\text{Der}(A,A)\) closed under Lie bracket. The pair \((A,T_A)\) is then called a differential algebra. An ideal of \(A\) is a differential ideal if it is preserved by every derivation in \(T_A\), and \((A,T_A)\) is said to be simple if its only differential ideals are \(0\) and \(A\). Let \(\mathcal D_A\) be the algebra of differential operators on \(A\) (the algebra generated by \(A\) and \(T_A\)). \(\mathcal D_A\) is naturally filtered and, under appropriate hypotheses, called regularity, \(T_A\) is projective and \(\text{gr} \mathcal D_A\) is the symmetric algebra on it. If \(M\) is a \(\mathcal D_A\) module, \(C(M) = \{ m \in M \mid \partial m =0 \quad \forall \partial \in T_A \}\). If \((A,T_A)\) is simple then the natural map \(A \otimes_{C(A)} C(M) \to M\) is injective; if it is bijective then \(M\) is called trivial. A morphism \((A,T_A) \to (B,T_B)\) of differential algebras is a pair \((\phi,\phi')\) where \(\phi: A \to B\) is a ring homomorphism, \(\phi': T_A \to T_B\) is an \(A\) module homomorphism, satisfying compatibility conditions and such that \(\phi(C(A)) \subset C(B)\). \((B,T_B)\) is a differential algebra over \((A,T_A)\) if further \(T_B= B\phi'(T_A)\); it is a trivial differential algebra if the canonical morphism \(A \otimes_{C(A)} C(B) \to B\) is an isomorphism. Now assume \((A,T_A)\) is simple. A left ideal \(I\) of \(\mathcal D_A\) is called Picard-Vessiot (P.V.) if \(\mathcal D_A/I\) is \(A\) finitely generated and projective. If \((B,T_B)\) is a differential algebra over \((A,T_A)\) and \(B \otimes_A \mathcal D_A/I\) is trivial, then \(I\) is said to be completely solvable in \(B\). For any \(I\), \(\text{Sol}(I,B)\) denotes \(C(\text{Hom}_A(\mathcal D_A/I, B))\). The author proves that if \(M\) is a finite \(A\) rank \(A\) free \(\mathcal D_A\) module, and \(C(A)\) is an algebraically closed field, then there exists a unique minimal simple differential \((A,T_A)\) algebra \((B,T_B)\), called a Joyal algebra, such that \(C(B)=C(A)\) and \(B \otimes_A M\) is \(B\) trivial. A differential algebra \((B,T_B)\) over \((A,T_A)\) is called Galois if \(B\) is finitely generated as an \(A\) algebra, \(C(B)=C(A)\), and \(B \to B \otimes_A B\) by \(b \mapsto b \otimes 1\) is trivial. Assume that \(C(A)=k\) is algebraically closed. The author shows that a Joyal algebra \(B\) is a Galois extension, that \(C(B \otimes_A B)\) is the coordinate ring of an affine algebraic group \(G\) over \(k\), and that \(G\) can be identified with the group \(\text{Gal}(B/A)\) of differential automorphisms of \(B\) over \(A\). Now let \((K, T_K)\) be a differential field and let \((B,T_B)\) be a Galois extension. Assume that \(C(A)=k\) is algebraically closed and that \(B\) is an integral domain. Then as above \(G=\text{Gal}(B/K)\) is an affine algebraic group over \(k\), and \(B^G=K\). Let \(L\) be the fraction field of \(B\), and let \(PV(L/K)\) be the set of elements \(f \in L\) such that \(\mathcal D_K \cdot f\) is finite dimensional over \(K\). It turns out that \(PV(L/K)=B\) and \(G=\text{Gal}(L/K)\). A differential field extension \((L,T_L)\) of \((K,T_K)\) is called P.V. normal if \(L\) is finitely generated over \(K\), \(C(L)=C(K)=k\) is algebraically closed, \(L\) is the quotient field of \(PV(L/K)\) and for every \(f \in PV(L/K)\), its annihilator is a completely solvable ideal in \(L\). Then the author shows that \(PV(L/K)\) is a Galois extension of \(K\). More generally, if \((A,T_A)\) is a simple differential algebra with \(C(A)\) algebraically closed and \(A\) local and Noetherian, and \((B,T_B)\) is a Galois differential \(A\) which is an integral domain and such that \(B=PV(B/A)\) then the author shows that \(B\) is a Joyal extension for a suitable module. The proofs of the major results in the paper are outlines, and most of the proofs of the supporting lemmas and propositions are omitted.