Blueprint for the Liquid Tensor Experiment

1.2 Variants of normed groups

Normed groups are well-studied objects. In this text it will be helpful to work with the more general notion of semi-normed group. This drops the separation axiom \(\| x\| = 0 \iff x = 0\) but is otherwise the same as a normed group.

The main difference is that this includes “uglier” objects, but creates a “nicer” category: semi-normed groups need not be Hausdorff, but quotients by arbitrary (possibly non-closed) subgroups are naturally semi-normed groups.

Nevertheless, there is the occasional use for the more restrictive notion of normed group, when we come to polyhedral lattices below (see Section 1.6).

In this text, a morphism of (semi)-normed groups will always be bouned. If the morphism is supposed to be norm-nonincreasing, this will be mentioned explicitly.

Definition 1.2.1

Let \(r {\gt} 0\) be a real number. An \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-module is a semi-normed group \(V\) endowed with an automorphism \(T \colon V \to V\) such that for all \(v \in V\) we have \(\| T(v)\| = r\| v\| \).

The remainder of this subsection sets up some algebraic variants of semi-normed groups.

Definition 1.2.2

A pseudo-normed group is an abelian group \((M,+)\), together with an increasing filtration \(M_c \subseteq M\) of subsets \(M_c\) indexed by \(\mathbb R_{\ge 0}\), such that each \(M_c\) contains \(0\), is closed under negation, and \(M_{c_1} + M_{c_2} \subseteq M_{c_1 + c_2}\). An example would be \(M=\mathbb {R}\) or \(M=\mathbb {Q}_p\) with \(M_c :=\{ x\, :\, |x|\leq c\} \).

A pseudo-normed group \(M\) is exhaustive if \(\varinjlim _c M_c = M\).

All pseudo-normed groups that we consider will have a topology on the filtration sets \(M_c\). The most general variant is the following notion.

A pseudo-normed group \(M\) is CH-filtered if each of the sets \(M_c\) is endowed with a topological space structure making it a compact Hausdorff space, such that following maps are all continuous:

  • the inclusion \(M_{c_1} \to M_{c_2}\) (for \(c_1 \le c_2\));

  • the negation \(M_c \to M_c\);

  • the addition \(M_{c_1} \times M_{c_2} \to M_{c_1 + c_2}\).

The pseudo-normed group \(M\) is profinitely filtered if moreover the filtration sets \(M_c\) are totally disconnected, making them profinite sets.

Remark 1.2.4

The topologies on the filtration sets \(M_c\) will induce a topology on \(M\): the colimit topology. If \(M\) is some sort of normed group, then this topology is typically genuinely different from the norm topology.

Definition 1.2.5

A morphism of CH-filtered pseudo-normed groups \(M \to N\) is a group homomorphism \(f \colon M \to N\) that is

  • bounded: there is a constant \(C\) such that \(x \in M_c\) implies \(f(x) \in N_{Cc}\);

  • continuous: for one (or equivalently all) constants \(C\) as above, the induced map \(M_c \to N_{Cc}\) is a morphism of profinite sets, i.e. continuous.

The reason the two definitions of continuity are equivalent is that a continuous injection from a compact space to a Hausdorff space must be a topological embedding.

A morphism \(f \colon M \to N\) is strict if \(x \in M_c\) implies \(f(x) \in N_c\) (in other words, if we can take \(C = 1\) in the boundedness condition above).

We will also consider the analogue of an \(r\)-normed \(\mathbb Z[T^{-1}]\)-module in the pseudo-normed setting.

Let \(r'\) be a positive real number. A CH-filtered pseudo-normed group \(M\) has an \(r'\)-action of \(T^{-1}\) if it comes endowed with a distinguished morphism of CH-filtered pseudo-normed groups \(T^{-1} \colon M \to M\) that is bounded by \(r'^{-1}\): if \(x \in M_c\) then \(T^{-1}x \in M_{c/r'}\).

A morphism of CH-filtered pseudo-normed groups with \(r'\)-action of \(T^{-1}\) is a morphism \(f \colon M \to N\) of CH-filtered pseudo-normed groups that commutes with the action of \(T^{-1}\).