Non-archimedean gauge seminorms

Non-archimedean seminorms on rings and modules provide in general a structure which is richer than the associated linear topology [3], [2]. We want to characterize Banach spaces and commutative algebras over a complete non-trivially valued nonarchimedean field K, as linearly topologized modules over the ring of integers K◦ of K, with no reference to any specific norm. This is analog to the classical theory of (non-archimedean) gauge seminorms [5, §2], and in fact applies to any locally convex K-space although we do not discuss the latter structure here. The motivation of this considerable effort (not discussed here) is that certain BarsottiWitt group and ring functors we construct elsewhere [1], apply more naturally to topological rather than to normed rings while, eventually, in view of the comparison with other theories, one needs to keep track of norms. This holds in particular for p-adic Hodge theory and Scholze’s tilting equivalence [6]. We are lead to distinguish, for any complete linearly topologized ring k, between topological k-modules M such that the scalar product k ×M →M is only continuous (for the product topology) and those for which the same scalar product is uniformly continuous. In the classical theory of k-formal schemes as developed, say, in Grothendieck’s E.G.A., based on [4, Chap. III, §4, n. 2], only the latter type of topological modules appear. In strong contrast, in functional analysis over K, [7], [5], K is complete in a K◦-linear topology, but not even the product map K×K → K is uniformly continuous. We play a lot on this dichotomy which seems to have received little attention so far. It generates two distinct notions ⊗̂uk and ⊗̂ c k of topological tensor product, where the apex “u” refers to “uniform” while the apex “c” refers to “continuous”. The monoidal structure ⊗̂ck is the completion of the one used for Fréchet spaces in [5], and there denoted ⊗K,π = ⊗K,ι, while ⊗̂ u k is the monoidal structure used in the theory of k-formal groups topologically of finite type. For any linearly topologized k-ring A, we will use −⊗̂ckA to define base change for K-Banach algebras (viewed as k = K◦-modules) via k → A. We will use instead −⊗̂ukA when restricting a group or ring functor from topological k-rings to topological A-rings. What we have done suffices for our purposes, but is also an indication that the theory should be developed more systematically by replacing topologies with uniformities all over in the definitions of topological algebraic structures. It seems that such a task has not been carried out yet.