The prime spectrum of commutative differential algebras
Let \(A\) be a commutative algebra over a field with characteristic zero and \(D\) be a derivation on \(A\). Let \(p_1\) and \(p_2\) be distinct prime ideals of \(A\) such that \(p_1\subset p_2\) with \(D(p_1) \not\subset p_2\) and let \(q_1= \{x: D^k x\in p_1\) for all \(k\geq 0\}\). It is shown that for each \(z\in p_2- p_1\) there exists a prime ideal \(p_3\) of \(A\) such that a) \(z\in p_3\), \(q_1 \subset p_3 \subset p_2\), \(p_1\not\subset p_3\) and \(p_3\not\subset p_1\) and b) \(q_1 \subset p_3\) and \(x^2+ z^2\in p_3\), but \(x,z\not\in p_3\). It is known that the prime spectrum of the algebra \(C^k ([0, 1], \mathbb{C})\) of \(k\) times continuously differentiable \(\mathbb{C}\)-valued functions on \([0, 1]\) is a tree (that is for each prime ideal \(p\) of \(A\) the set \(\{q: q\) is a prime ideal of \(A\) with \(p\subset q\}\) is totally ordered by inclusion). Using results above it is possible to show that the prime spectrum of the algebra \(C^\infty ([0, 1])\) is not a tree. As an application it is given an alternative proof for the following well-known result of Bade and Curtis: there are no non-trivial derivations on \(C^k ([0, 1], \mathbb{C})\).
scientific article; zbMATH DE number 850619