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Notes on condensed mathematics

Kiran S. Kedlaya

Section 18 Norms on analytic rings

In this section, we introduce the concept of a norm on an analytic ring, set up in a way that will make it compatible with the theory of analytic stacks.

Reference.

This section is based on [8], Lectures 19 and 20.

Subsection 18.1 Norms and Gelfand–Berkovich spectra

The concepts of a submultiplicative seminorm on a commutative ring, and the resulting concept of a commutative Banach ring, were introduced in Definition 11.1.5. While this is ultimately not the model for the concept of a norm on an analytic ring, it will be helpful to recall some related points.

Example 18.1.1.

If \(R\) carries a submultiplicative seminorm \(|\bullet|\text{,}\) then for any \(\rho \geq 0\text{,}\) we can define a submultiplicative seminorm on \(R[T]\) by the formula
\begin{equation*} \left| \sum_i x_i T^i \right|_\rho = \sum_i |x|_i \rho^i. \end{equation*}
If the seminorm on \(R\) is nonarchimedean, we can replace the summation with the supremum in this definition to recover the \(\rho\)-Gauss seminorm. The terminology arises from the fact that if the original seminorm on \(R\) is both nonarchimedean and multiplicative, then adapting the proof of Gauss’s lemma on integer polynomials shows that the \(\rho\)-Gauss seminorm is again multiplicative.

Remark 18.1.2.

For \(R\) a commutative Banach ring, the set of units in \(R\) is an open subset. More precisely, for any \(x \in R^\times\text{,}\) the open ball \(\{y \in R\colon |x-y| \lt |x^{-1}|^{-1}\}\) consists entirely of units: for \(y\) in the ball, \(|1-y/x| \leq |x-y| |x^{-1}| \lt 1\) and so the series \(x^{-1} \sum_i (1-y/x)^i\) converges to an inverse of \(x(1-y/x) = y\text{.}\)

Remark 18.1.3.

For \(R\) a commutative Banach ring, for any maximal ideal \(\frakm\) of \(R\text{,}\) define the quotient seminorm on \(F := R/\frakm\) taking \(x\) to the infimum of \(|y|\) over all \(y \in R\) lifting \(x\text{.}\) We verify that this is in fact a submultiplicative norm with respect to which \(F\) is again complete.
Note first that if \(y \in 1 + \frakm\text{,}\) then \(|y| \geq 1\) as otherwise Remark 18.1.2 would imply that \(1-y \in \frakm\) is a unit, a contradiction (as the existence of a maximal ideal forces \(R \neq 0\)). By the same token, for any \(y \in R \setminus \frakm\text{,}\) we can find \(z \in R \setminus \frakm\) with \(yz \equiv 1 \pmod{\frakm}\text{,}\) and then \(|yz| \geq 1\) and so \(|y| \geq |z|^{-1}\text{;}\) since we can choose \(z\) depending only on the image of \(y\) in \(R\text{,}\) this shows that the quotient seminorm is a norm.
Finally, note that if \((x_n)\) is a null sequence in \(F\text{,}\) then for any fixed \(\epsilon \gt 0\text{,}\) we can lift each \(x_n\) to \(y_n \in R\) so that \(|y_n|\) is at most \(1+\epsilon\) times the quotient seminorm of \(x_n\text{.}\) Consequently, the \(y_n\) form a null sequence in \(R\text{,}\) which is thus summable to some limit \(y\text{;}\) hence the \(x_n\) also form a sequence summable to the image \(x\) of \(y\) in \(F\text{.}\)

Remark 18.1.4.

Let \(R\) be a field equipped with a submultiplicative seminorm \(|\bullet|\) (e.g., the field \(F\) of Remark 18.1.3). Then for any \(x \in R^\times\text{,}\) \(|x| |x^{-1}| \geq |1| = 1\text{;}\) consequently, \(|\bullet|\) is actually a norm.
In general the norm need not be multiplicative. However, in this case multiplicativity can be deduced from the formally weaker condition that \(|x| |x^{-1}| = 1\) for all \(x \in R^\times\text{:}\) this condition implies that for \(x \in R^\times, y \in R\text{,}\)
\begin{equation*} |xy| \leq |x| |y| \leq |x| |x^{-1}| |xy| = |xy|\text{;} \end{equation*}
hence \(|xy| = |x| |y|\) when \(x\) is nonzero, and obviously also when \(x =0\text{.}\)
It is a different and subtler question to ask whether or not the topology on \(R\) can also be defined by some multiplicative norm. See [18] for discussion of this issue in the nonarchimedean case.

Definition 18.1.5.

Let \(R\) be a ring equipped with a submultiplicative norm \(|\bullet|\text{.}\) The Gelfand–Berkovich spectrum of \(R\) is the set \(\calM(R)\) of multiplicative seminorms \(\alpha\) on \(R\) which are dominated by \(|\bullet|\) (meaning that \(\alpha(x) \leq |x|\) for all \(x \in R\)). We upgrade this to a topological space by viewing this as a subspace of \(\prod_{x \in R} \RR_{\geq 0}\text{,}\) or equivalently \(\prod_{x \in R} [0, |x|]\text{;}\) from the latter point of view we see that \(\calM(R)\) is compact Hausdorff.
Note that \(\calM(R)\) is unchanged by replacing \(R\) with the completion with respect to the seminorm.

Proof.

We give the argument from [3], Theorem 1.2.1 (filling in some details). Using Remark 18.1.3, by quotienting by some maximal ideal we may reduce to the case where \(R\) is a field, which ensures that every submultiplicative seminorm on \(R\) is actually a norm (Remark 18.1.4). By Zorn’s lemma, the space of submultiplicative norms on \(R\) dominated by \(|\bullet|\) contains a minimal element \(\alpha\) with respect to domination. It will suffice to check that any such element is in fact multiplicative. By Remark 18.1.4, it suffices to check that \(\alpha(x)^{-1} = \alpha(x^{-1})\) for all \(x \in R^\times\text{.}\)
Suppose the contrary; since \(\alpha(x) \alpha(x^{-1}) \geq \alpha(1) = 1\) by submultiplicativity, we must have \(\alpha(x^{-1}) \gt \alpha(x)^{-1}\text{.}\) Set \(\rho := \alpha(x^{-1})^{-1}\text{;}\) by the same token, for any positive integer \(i\text{,}\) \(\alpha(x^{-i}) \alpha(x^{-1})^i \geq 1\) and so \(\alpha(x^{-i}) \geq \rho^{-i}\text{.}\) In particular, the sum \(\sum_{i=0}^\infty \alpha(x^{-i}) \rho^i\) diverges. Now let \(S\) be the completion of \(R[T]\) for the seminorm defined in Example 18.1.1 with respect to \(\alpha\) and \(\rho\text{,}\) viewed as a subring of \(R \llbracket T \rrbracket\text{;}\) then \((1-x^{-1}T)^{-1} \notin S\text{,}\) so \(1-x^{-1} T\) is not a unit in \(S\text{.}\) Hence the map \(R \to S/(x-T)\) is injective and we can pull back the quotient seminorm from \(S/(x-T)\) to \(R\text{;}\) this gives a submultiplicative norm \(\beta\) on \(R\) such that \(\beta(y) \leq \alpha(y)\) for all \(y \in R\) (by lifting \(y\) to itself) but \(\beta(x) \lt \alpha(x)\) (by lifting \(x\) to \(T\)), a contradiction.

Proof.

Let \(I\) be the closure of the ideal \((x)\text{.}\) By hypothesis, \(\calM(R/I)\) is empty, so Proposition 18.1.6 that \(I\) is the unit ideal. By Remark 18.1.2, this means that \((x)\) contains a unit, and so \(x\) is itself a unit.

Example 18.1.8.

Take \(R = \ZZ\) equipped with the archimedean absolute value \(|\bullet|_\infty\text{.}\) Using Ostrowski’s theorem, we may classify the points \(\alpha \in \calM(\ZZ)\) as follows.
  • The trivial norm: \(\alpha(x) = 1\) for all nonzero \(x \in \ZZ\text{.}\)
  • For \(c \in (0,1]\text{,}\) the scaled archimedean norm \(\alpha(x) = |x|_\infty^c\text{.}\) (We cannot take \(c \gt 1\) as then the triangle inequality would fail; see Remark 18.1.9.)
  • For \(p\) prime and \(c \in (0, +\infty)\text{,}\) the scaled \(p\)-adic norm \(\alpha|x| = |x|_p^c\text{.}\)
  • For \(p\) prime, the seminorm pulled back from the trivial norm on \(\FF_p\text{.}\)
We can get some partial information about the topology by noting that on one hand the map \([0,1] \to \calM(\ZZ)\) taking \(c\) to \(|x|_\infty^c\) (meaning the trivial norm for \(c=0\)) is continuous; and on the other hand, for each prime \(p\) the map \([0, \infty] \to \calM(\ZZ)\) taking \(c\) to \(|x|_p^c\) (meaning the trivial norm for \(c=0\) and the pullback of the trivial norm on \(\FF_p\) for \(c = \infty\)) is also continuous.
To describe the topology completely, note that on one hand, any nontrivial \(\alpha \in \calM(\ZZ)\) lies on exactly one of the segments described above and this forms a compact neighborhood of \(\alpha\text{.}\) On the other hand, any neighborhood of the trivial norm contains the entire segments corresponding to all but many primes \(p\text{.}\) An efficient description of the resulting space is that it is homeomorphic to the union of the segment from \([0,0]\) to \([0,1]\) (corresponding to the segment ending at \(|\bullet|_\infty\)) with, for each prime \(p\text{,}\) the segment from \([0,0]\) to \([1/p, 1/p^2]\text{.}\)

Remark 18.1.9.

In Example 18.1.8, the fact that the branch of \(\calM(\ZZ)\) containing the points \(\alpha(x) = |x|_\infty^c\) is truncated at \(c=1\) is arguably a bit artificial: for \(c \gt 1\text{,}\) the function \(|x|_\infty^c\) still satisfies a somewhat weaker form of the triangle inequality. More importantly, the balls with respect to this function still define the archimedean topology.

Definition 18.1.10.

For \(R\) a ring equipped with a submultiplicative seminorm \(|\bullet|_R\text{,}\) define the associated spectral seminorm by the formula
\begin{equation*} |x|_{\spect,R} = \lim_{n \to \infty} |x^n|_R^{1/n}\text{;} \end{equation*}
it is an elementary exercise to verify that the limit exists (and actually equals the infimum). By construction, \(|\bullet|_{\spect,R}\) is a power-multiplicative seminorm.
For \(R\) a commutative Banach ring, it is not always the case that the spectral seminorm defines the same topology as the original norm; for one, if \(R\) is not reduced, then any nilpotent element has spectral seminorm \(0\text{.}\) However, replacing the norm with its associated spectral seminorm does not affect \(\calM(R)\text{:}\) any multiplicative seminorm that is dominated by the original norm is also dominated by the spectral seminorm.

Proof.

In one direction, as pointed out in Definition 18.1.10, every \(\alpha \in \calM(R)\) is dominated by the spectral seminorm. In the other direction, if \(\alpha(x) \lt \rho\) for all \(\alpha \in \calM(R)\text{,}\) then let \(S\) be the completion of \(R[T]\) for the seminorm defined in Example 18.1.1 with respect to \(|\bullet|_R\) and \(\rho^{-1}\text{.}\) By Corollary 18.1.7, the element \(1-xT\) of \(S\) is invertible; this means that \(\sum_{n=0}^\infty |x^n| \rho^{-n}\) converges, implying \(|x|_{\spect,R} \leq \rho\text{.}\) (Compare [3], Theorem 1.3.1.)

Subsection 18.2 Norm functions

Remark 18.2.1.

In order to study norms on analytic rings in terms of analytic stacks, we need to adopt a shift in perspective. For \(R\) a ring, an element of \(R\) corresponds to a section of the projection \(\AAA^1_R \to \Spec R\text{;}\) while our conventional thinking about norms would suggest that we assign a single number to such a section, it is ultimately better to think of assigning a number to each individual point on the section, with the supremum over these numbers corresponding to the norm in the traditional sense. This forces the supremum to be power-multiplicative (compare Proposition 18.1.11); in the case where \(R\) is a Banach ring, this means that we do not recover the topology on \(R\) but we do recover the Gelfand spectrum (Definition 18.1.10).
Another way to say this is that we want to specify a way to pick out the closed ball of radius \(\rho\) for every \(\rho \in [0, \infty)\text{.}\) For any given section of the projection \(\AAA^1_R \to \Spec R\text{,}\) we would like to be able to detect whether any given point on the section indeed lies in this closed ball.
In order to do this, we need to make sense of writing down an “analytic” map from \(\AAA^1_R\) to the nonnegative real numbers. We give an ad hoc definition here, but it will be explained later in the language of analytic stacks.

Definition 18.2.2.

For \(R\) an analytic ring, a norm function on \(R\) consists of, for each \(\rho \in (0, \infty)\text{,}\) an idempotent covering \(\{\AnSpec R_{\leq \rho} \to \PP^1_R, \AnSpec R_{\geq \rho} \to \PP^1_R\}\) satisfying the following conditions.
  • For \(0 \lt \rho_1 \lt \rho_2 \lt \infty\text{,}\) \(\AnSpec R_{\leq \rho_1} \to \PP^1_R\) factors through \(R_{\leq \rho_2}\text{.}\)
  • For \(\rho \in (0, \infty)\text{,}\) \(R_{\leq \rho}\) is the colimit of \(R_{\leq \rho_1}\) over all \(\rho_1 \in (\rho, \infty)\text{.}\)
  • For all \(\rho \in (0, \infty)\text{,}\) \(\AnSpec R_{\leq \rho} \to \PP^1_R\) factors through \(\AnSpec R[T]\text{.}\)
  • For \(\rho \in (0, \infty)\text{,}\) the pullback of \(\AnSpec R_{\leq \rho} \to \PP^1_R\) along the inversion map on \(\PP^1_R\) is isomorphic to \(\AnSpec R_{\geq 1/\rho} \to \PP^1_R\text{.}\)
  • For \(\rho_1, \rho_2 \in (0, \infty)\text{,}\) the composition
    \begin{equation*} \AnSpec R_{\leq \rho_1} \times_R \AnSpec R_{\leq \rho_2} \to \AnSpec R[T] \times_R \AnSpec R[T] \stackrel{\mu}{\to} \AnSpec R[T]\text{,} \end{equation*}
    where \(\mu\) is the multiplication map, factors through \(\AnSpec R_{\leq \rho_1 \rho_2} \to \AnSpec R[T]\text{.}\)
  • For any \(\rho \gt 1\text{,}\) the map \(\AnSpec (P \otimes R) \to \PP^1_R\) identifying \([n]\) with \(T^n\) factors through \(\AnSpec R_{\leq \rho}\text{.}\)

Example 18.2.3.

For any analytic ring \(R\text{,}\) we get a norm function on \(R\) by taking
\begin{equation*} R_{\leq \rho} = \begin{cases} \AnSpec R[T] & \rho \geq 1 \\ \AnSpec R[T]_{(T)} & \rho \lt 1 \end{cases}\text{.} \end{equation*}

Remark 18.2.4.

In the language of analytic stacks, the data of a norm function will correspond to a morphism \(N\colon \PP^1_R \to [0, \infty]\) of analytic stacks subject to the following conditions.
  • For \(\infty_R \colon \AnSpec R \to \PP^1_R\text{,}\) \(N \circ \infty_R = \infty\text{.}\)
  • The map \(N\) commutes with inversion (on \(\PP^1_R\) and \([0,\infty]\)).
  • The map \(N\) commutes with multiplication (on \(\AAA^1_R\) and \([0, \infty)\)).
  • The restriction of \(N\) to \(\AnSpec (P \otimes R)\) factors through \([0,1]\text{.}\)
The analytic stack \([0,\infty]\) will admit a covering by \([0,\rho]\) and \([\rho,\infty]\) for any \(\rho \in (0, \infty)\text{;}\) this covering will pull back to the covering specified in Definition 18.2.2.

Example 18.2.5.

Let \(A\) be a Tate analytic ring and fix a unit \(q \in A^{\circ \circ}\) and a constant \(c \in (0,1)\text{.}\) For \(\rho \in (0, \infty)\text{,}\) let \(A_{\leq \rho}\) be the colimit of \(A[T]_{\liquid T^m/q^n}\) over all pairs of integers \((m,n)\) with \(m \gt 0\) and \(c^{n/m} \gt \rho\text{.}\) We then obtain a norm function on \(A\text{.}\)
When \(A\) is the initial Tate analytic ring (Definition 10.6.1), it can be shown that this norm function is uniquely determined by the property that \(q\) maps identically to \(c\text{.}\) However, in general, each topologically nilpotent unit of \(A\) map into \([0,1)\) but not necessarily to a constant.

Example 18.2.6.

Let \(A\) be an arbitrary analytic ring and let \(B\) be the initial Tate analytic ring (Definition 10.6.1). Then applying Example 18.2.5 over \(A \otimes B\) also gives a norm function on \(A\text{.}\)

Remark 18.2.7.

In Definition 18.2.2, the last condition corresponds to requiring that any topologically nilpotent ring element has norm in \([0,1]\text{.}\)

Subsection 18.3 Application to the Tate elliptic curve

One key application of this construction is to construct the “universal” Tate elliptic curve.

Definition 18.3.1.

Let \(A\) be the initial Tate analytic ring constructed in Definition 10.6.2. Fix the norm function given by Example 18.2.5 for some \(c \in (0,1)\text{.}\) We can then define \(\GG_{m,A}^{\an}\) as the colimit of \(\AnSpec R_{\leq \rho}\) as \(\rho \to \infty\text{.}\)
The analytic version of the Tate elliptic curve will be the “free quotient” \(\GG_{m,A}^{\an}/q^{\ZZ}\text{.}\) One can then prove an “algebraization” statement exhibiting this as the analytification of some elliptic curve over \(A\text{,}\) which for instance one can interpret as a map from \(\AnSpec A\) to the algebraic moduli stack \(M_{1,1}\) of elliptic curves over \(\ZZ\text{.}\)
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