The Gelfand transform in commutative algebra. (English) Zbl 1106.13006

Buchstaber, V.; Lazarev, A.

Mosc. Math. J. 5, No. 2, 311-327 (2005).

The main purpose of this paper is the study of the evaluation map \(\text{ev}: A \to(A(K),K)\) and its dual map \(\text{ev}^*: K[ (A(K),K)] \to A^*\) , where \(A\) is a commutative algebra over a commutative ring \(A\), \(A(K)\) is the set of all \(K\)-algebra homomorphism \(A \to K\) (called \(K\)-points), \(K[A(K)]\) is the free \(K\)-module on the set \(A(K)\). Both ev and ev* have a topological \(K\)-module as a codomain and the paper investigates the cases in which the image of each map is dense. More generally, if \(K\) is an infinite subring in an integral domain \(L\) and \(A\) is the \(K\) algebra which is free as a \(K\)-module, then \(\text{ev}_{L/K}: A\otimes L \to \text{Map}(A(K),L)\) is defined by \( \text{ev}(a \otimes l)(x) = lx(a)\). In this situation, two objects are introduced: the ring of numerical functions \(\text{Num}_{L/K}(A)\) and the \(L\)-module \(\underline {\text{Num}}_{L/K}(A)\) of numerical functionals. In the case \(A=K[x]\), the authors prove that \(\text{ev}_{L/K}\) has a dense image iff \(\underline {\text{Num}}_{L/K}\) is the free \(L\)-module on \(K\). A similar dual result holds for ev* when \(L\) is a D.V.R.. Finally, applications are given to investigate the structure of some algebras of cohomology operations.

Reviewer: Carla Massaza (Torino)

MSC:

13B25	Polynomials over commutative rings
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology
13A99	General commutative ring theory
13J99	Topological rings and modules

Keywords:

linear topology; Gelfand transform; rings of divided powers; numerical polynomials; Landweber-Novikov algebra; Steenrod algebra