Čech cohomology of infinite projective spaces, flag manifolds, and related spaces

Abstract

We compute the Čech cohomology ring of a countable product of infinite projective spaces, and that of an infinite flag manifold. The method of our first result in fact computes the cohomology ring of a countably infinite product of paracompact Hausdorff spaces, under some mild assumptions.

1 Introduction

Let $\mathbb {C}\mathbb{P} ^\infty $ be infinite-dimensional complex projective space, topologized as a rising union of finite-dimensional projective spaces, $\mathbb {C}\mathbb{P} ^\infty =\bigcup \mathbb {C}\mathbb{P} ^k$ , or equivalently, as the quotient space $\mathbb {C}\mathbb{P} ^\infty = (\mathbb {C}^\infty \setminus 0)/\mathbb {C}^*$ , where $\mathbb {C}^\infty $ has again the colimit topology of the spaces $\mathbb {C}^{k}$ . The latter description makes $\mathbb {C}\mathbb{P} ^\infty $ a model for the classifying space of $\mathbb {C}^*$ , and we have the well-known calculation

$$\begin{align*}H^*\mathbb{C}\mathbb{P}^\infty = H^*B\mathbb{C}^* = H^*_{\mathbb{C}^*}(\mathrm{pt}) = \mathbb{Z}[t], \end{align*}$$

a polynomial ring in one generator of degree $2$ , often taken to be the Chern class of the tautological line bundle. Here, cohomology is taken with integer coefficients, and one may use any standard cohomology theory: the space $\mathbb {C}\mathbb{P} ^{\infty }$ is locally contractible, paracompact, and Hausdorff, so singular, Čech, Alexander–Spanier, and sheaf cohomology all agree (see, e.g., [Sp66, Corollary 6.9.5, Exercise 6.D3] and [Br97, Section I.7 and Chapter III]).

Likewise, the cohomology ring of a finite product $(\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ is well known to be a polynomial ring in n generators of degree $2$ :

$$\begin{align*}H^*(\mathbb{C}\mathbb{P}^\infty)^{\times n} = H^*_{(\mathbb{C}^*)^n}(\mathrm{pt}) = \mathbb{Z}[t_1,\ldots,t_n]. \end{align*}$$

Again, this calculation is valid in any standard cohomology theory.

The first aim of this note is to compute the analogous ring for the countably infinite product of infinite projective spaces. The principal bundle $(\mathbb {C}^\infty \setminus 0)^{\times \mathbb {N}} \to (\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ has contractible total space, so $(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ is a model for the classifying space of the infinite-dimensional torus $T = (\mathbb {C}^*)^{\times \mathbb {N}}$ , with the caveat that this bundle is not locally trivial in the product topology. (It becomes locally trivial in the finer box topology, but we will not use this.) Equivariant cohomology with respect to T comes up in several contexts (see, e.g., [An24, LLS21, LS12]) and so does the cohomology of its classifying space.

Using Čech cohomology ¹ with integer coefficients, we will show that

(1.1) $$ \begin{align} H^* (\mathbb{C}\mathbb{P}^\infty)^{\times \mathbb{N}} = H^*_T(\mathrm{pt}) = \mathbb{Z}[t_1,t_2,\ldots], \end{align} $$

a polynomial ring in countably many variables of degree $2$ , where $t_i$ may be regarded as the Chern class of the tautological line bundle pulled back along the projection to the ith factor. In fact, the same holds with $\mathbb {Q}$ coefficients: see the corollary at the end of this introduction. The calculation uses a construction of $(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ as an inverse limit of spaces $(\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ .

This time, the choice of cohomology theory matters. The rational version of the asserted isomorphism (1.1) says that $H^2\big ((\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}; \mathbb {Q}\big ) = \bigoplus _{i\geq 1} \mathbb {Q} t_i$ is a vector space of countably infinite rank, so it is not the dual of any vector space (e.g., by the Erdős–Kaplansky theorem). The universal coefficient theorem therefore implies that it cannot be the singular cohomology of any space.

Nevertheless, the calculation of $H^*(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ gives the “expected” answer, in the following sense. Let $X_{\leq n} = (\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ , and for $n\geq n'$ , consider the projection ${X_{\leq n} \to X_{\leq n'}}$ onto the first $n'$ factors. So the $X_{\leq n}$ form an inverse system, with limit

$$\begin{align*}X = \varprojlim_n X_{\leq n} = (\mathbb{C}\mathbb{P}^\infty)^{\times \mathbb{N}}. \end{align*}$$

Reversing the arrows, there is a natural homomorphism of cohomology rings

(1.2) $$ \begin{align} \varinjlim H^*X_{\leq n} \to H^*(\varprojlim X_{\leq n}). \end{align} $$

If this map were known to be an isomorphism, we would obtain

$$\begin{align*}H^*X = \varinjlim_n H^* X_{\leq n} = \varinjlim_n \mathbb{Z}[t_1,\ldots,t_n] = \mathbb{Z}[t_1,t_2,\ldots], \end{align*}$$

as asserted by (1.1).

When the spaces involved are compact, the continuity property for Čech–Alexander–Spanier cohomology (reviewed below) implies that the homomorphism (1.2) is an isomorphism. For inverse limits of non-compact spaces, however, simple counterexamples show that (1.2) can fail to be an isomorphism in any reasonable cohomology theory. ² In our setting, the spaces $X_{\leq n} = (\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ are not compact, so further argument is required.

The reasoning sketched above has led several authors (including the present first author) to assert the calculation (1.1) without sufficient justification, so it seems worth providing details. In particular, this note fills a gap left in [AF24, Section A.8].

We will establish the isomorphism (1.1) in a more general context. From now on, unless otherwise indicated, we use Čech cohomology with coefficients in a countable principal ideal domain R. (Typical examples are $\mathbb {Z}$ , $\mathbb {Q}$ , finite fields, and subrings of $\mathbb {Q}$ .) Recall that a CW complex is said to be of finite type if it has only finitely many cells in each dimension.

Theorem A Let $X_1$ , $X_{2}$ , …be a countable family of CW complexes of finite type. Write X for their product, and set $X_{\le n}=X_{1}\times \dots \times X_{n}$ . The natural homomorphism

$$\begin{align*}\varinjlim_n H^*(X_{\leq n}) \xrightarrow{\sim} H^*X, \end{align*}$$

induced by the canonical projections $X\to X_{\le n}$ , is an isomorphism of graded R-algebras.

The space $X_{\leq n}$ is again a CW complex since each $X_{k}$ has at most countably many cells [Ha02, Theorem A.6]. And since CW complexes are locally contractible and paracompact Hausdorff (see, e.g., [LW69] or [Ha02, Appendix]) Čech cohomology agrees with singular cohomology for these spaces. (This fact also shows that the countable product X is not a CW complex in general.)

The calculation of the cohomology of $(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ is an immediate consequence.

Corollary For Čech cohomology with coefficients in any countable principal ideal domain R, we have

$$\begin{align*}H^*\Bigl( (\mathbb{C}\mathbb{P}^\infty)^{\times \mathbb{N}} \Bigr) \cong \varinjlim_n R[t_1,t_2,\ldots,t_n] = R[t_1,t_2,\ldots]. \end{align*}$$

In particular, this holds with $R=\mathbb {Z}$ or $R=\mathbb {Q}$ .

Using this calculation together with an explicit homotopy equivalence, we will deduce a computation of the cohomology ring of the infinite flag manifold (Theorem B in Section 3).

2 Proof of Theorem A

We first collect the required ingredients. Recall that R indicates a countable PID, and coefficients for cohomology are assumed to be taken in R, when unspecified.

We use the general tautness property of Čech cohomology: this says that for any closed subspace Y of a paracompact space Z and coefficients in any R-module M, the natural homomorphism

$$\begin{align*}\varinjlim H^k(U;{M}) \to H^k(Y;{M}) \end{align*}$$

is an isomorphism, where the direct limit is over all neighborhoods U of Y in Z (see [Sp66, Theorem 6.6.2]).

There is also the above-mentioned continuity property: for an inverse system of compact Hausdorff spaces $Z_n$ , with limit Z, the homomorphism

$$\begin{align*}\varinjlim H^k(Z_n;{M}) \to H^k(Z;{M}) \end{align*}$$

is an isomorphism, again for any R-module M [Sp66, Theorem 6.6.6].

We make use of the following general Künneth formula for Čech cohomology.

Proposition 2.1 Assume that P is locally contractible and that $P\times Y$ is paracompact Hausdorff. Then, for any $k\ge 0,$ there is a natural isomorphism

$$\begin{align*}H^k( P \times Y) \cong \bigoplus_{i=0}^k H^i(P; H^{k-i}(Y)), \end{align*}$$

functorial in P and Y.

When P is compact, a proof via the spectral sequence of a double complex may be found in Bredon’s book. (Use the short exact sequence of [Br97, Theorem IV.7.6] together with the universal coefficient theorem for sheaf cohomology [Br97, Theorem II.15.3].) The version stated above is due to Bartik, and uses a natural isomorphism between kth Čech cohomology and homotopy classes of maps into the Eilenberg–MacLane space $K({R},k)$ [Ba68, Theorem C(b)].

In preparation for the proof of our main result, we state some lemmas. Writing $X_n^{(k)}$ for the $(k+2)$ -skeleton of $X_{n}$ , we have an isomorphism

(2.1) $$ \begin{align} H^i(X_n;M) \to H^i(X_n^{(k)};M) \end{align} $$

for $i\le k+1$ and $M=R$ , hence also for $i\le k$ and any M by the universal coefficient theorem for singular cohomology.

In addition to the spaces X and $X_{\le n}$ , we introduce

$$ \begin{align*} X_{>n} = \prod_{m>n} X_m, \end{align*} $$

and we define $X^{(k)}$ , $X_{\leq n}^{(k)}$ , and $X_{>n}^{(k)}$ as the analogous products involving the spaces $X_{m}^{(k)}$ . Note that the latter three products are all compact. Moreover, $X_{\leq n}^{(k)}$ contains the $(k+2)$ -skeleton of $X_{\leq n}$ , so in fact the $(k+2)$ -skeleta of $X_{\leq n}^{(k)}$ and of $X_{\leq n}$ are the same, which yields an isomorphism of the form (2.1) for $X_{\leq n}$ in place of $X_{n}$ .

CW complexes are not only paracompact, but in fact hereditarily paracompact in the sense that every subspace is again paracompact (see [LW69, Corollary II.4.4]). Because we can express the countable product X as the inverse limit

$$ \begin{align*} X = \varprojlim_{n} X_{\le n}, \end{align*} $$

this implies that X is again paracompact [DG86, Lemma 7], and so are all $X_{>n}$ .

Lemma 2.2 The map

$$ \begin{align*} H^{i}(X) \to H^{i}(X_{\le n}^{(k)}\times X_{>n}) \end{align*} $$

is an isomorphism for any $n\ge 0$ and any $i\le k$ .

Proof By Proposition 2.1, this reduces to the fact that the map

$$ \begin{align*} H^{*}\bigl(X_{\le n};H^{*}(X_{>n})\bigr) \to H^{*}\bigl(X_{\le n}^{(k)};H^{*}(X_{>n})\bigr) \end{align*} $$

is an isomorphism in total degree $i\le k$ , which is a special case of what has been discussed above.

Lemma 2.3 For any $n\ge 0$ and any $i\le k,$ there is an isomorphism

$$ \begin{align*} \varinjlim_{V} H^{i}(V\times X_{>n}) = H^{i}(X^{(k)}_{\le n}\times X_{>n}), \end{align*} $$

where V ranges over the open neighborhoods of $X^{(k)}_{\le n}$ in $X_{\le n}$ .

Proof Note that $V\times X_{>n}$ is the inverse limit of the spaces $V\times X_{n+1}\times \dots \times X_{m}$ with $m\ge n+1$ . The latter are subspaces of CW complexes and therefore hereditarily paracompact, which ensures that $V\times X_{>n}$ itself is paracompact.

We may therefore appeal to Proposition 2.1, which together with tautness gives isomorphisms

$$ \begin{align*} \varinjlim_{V} H^{i}(V\times X_{>n}) &= \varinjlim_{V} \bigoplus_{j} H^{j}\bigl(V;H^{i-j}(X_{>n})\bigr) \\ &= \bigoplus_{j} H^{j}\bigl(X^{(k)}_{\le n};H^{i-j}(X_{>n})\bigr) = H^{i}(X^{(k)}_{\le n}\times X_{>n}).\\[-45pt] \end{align*} $$

Lemma 2.4 For any $i\le k,$ the inclusion $\iota \colon X^{(k)}\hookrightarrow X$ induces an isomorphism

$$ \begin{align*} \iota^{*}\colon H^{i}(X) = H^{i}(X^{(k)}). \end{align*} $$

Proof Consider the open neighborhoods of $X^{(k)} \subseteq X$ that are of the form

$$ \begin{align*} U = V \times X_{>n}, \end{align*} $$

where V is an open neighborhood of $X_{\leq n}^{(k)}$ in $X_{\leq n}$ , for some n. The collection $\mathcal {U}$ of all such U is cofinal in the directed set of all open neighborhoods of $X^{(k)}$ , by compactness of $X^{(k)}$ and the definition of the product topology. Let $\mathcal {U}_n\subseteq \mathcal {U}$ be the subset of all U that can be written in the above form for a given n, so $\mathcal {U}_n \subset \mathcal {U}_{n'}$ for $n\leq n'$ . In this notation, Lemma 2.3 reads

$$ \begin{align*} \varinjlim_{U\in \mathcal{U}_n} H^i(U) = H^{i}(X^{(k)}_{\le n}\times X_{>n}) \end{align*} $$

for all $i\leq k$ .

Using this identity together with Lemma 2.2 and the tautness property for $X^{(k)} \subseteq X$ gives

$$ \begin{align*} H^i(X^{(k)}) &\cong \varinjlim_{U\in \mathcal{U}} H^i(U) = \varinjlim_n \varinjlim_{U\in \mathcal{U}_n} H^i(U) \\ &\cong \varinjlim_n H^i(X_{\leq n}^{(k)} \times X_{>n}) = \varinjlim_n H^i(X) = H^{i}(X) \end{align*} $$

for all $i\leq k$ , which completes the proof.

Now, we will show that the natural homomorphism $\alpha \colon \varinjlim _n H^kX_{\leq n} \to H^kX$ is an isomorphism for any fixed degree k, proving Theorem A.

We have a commuting square

(2.2)

Since $X^{(k)} = \varprojlim _n X_{\leq n}^{(k)}$ is a limit of compact spaces, the homomorphism $\beta $ is an isomorphism by the continuity property. The homomorphism j is a colimit of isomorphisms $H^k(X_{\leq n}) \to H^k(X_{\leq n}^{(k)})$ , so it is an isomorphism itself. Finally, the map $\iota ^{*}$ is an isomorphism by Lemma 2.4, and we conclude that so is $\alpha $ .

3 Flag manifolds

Infinite-dimensional flag manifolds are closely related to products of projective spaces. Let $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ be the space parametrizing chains of subspaces $V_1 \subset \cdots \subset V_n \subset \mathbb {C}^\infty $ with $\dim V_{i}=i$ , topologized as a union

$$\begin{align*}{Fl} (1,\ldots,n;\mathbb{C}^\infty) = \bigcup_m {Fl} (1,\ldots,n;\mathbb{C}^m). \end{align*}$$

A standard calculation shows that

$$\begin{align*}H^* {Fl} (1,\ldots,n;\mathbb{C}^\infty) = \mathbb{Z}[t_1,\ldots,t_n] \end{align*}$$

is again a polynomial ring in n variables of degree $2$ (see, e.g., [AF24, Section 2.5]). Since the space $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ is locally contractible, paracompact, and Hausdorff, any usual cohomology theory will do: in particular, singular and Čech cohomology agree.

The infinite flag manifold $ {Fl} (\mathbb {C}^\infty )$ parametrizes chains of subspaces $V_1 \subset V_2 \subset \cdots \subset \mathbb {C}^\infty $ , with $\dim V_i = i$ , and is topologized as an inverse limit of partial flag manifolds:

$$\begin{align*}{Fl} (\mathbb{C}^\infty) = \varprojlim_n {Fl} (1,\ldots,n;\mathbb{C}^\infty), \end{align*}$$

where the maps in the inverse system are projections onto the first part of a chain. As with $(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ , this space is no longer locally contractible, so the choice of cohomology theory matters.

Theorem B Using Čech cohomology with coefficients in a countable principal ideal domain R, we have

$$\begin{align*}H^* {Fl} (\mathbb{C}^\infty) = H^*(\mathbb{C}\mathbb{P}^\infty)^{\times\mathbb{N}} = R[t_1,t_2,\ldots], \end{align*}$$

a polynomial ring in countably many variables of degree $2$ .

With $\mathbb {Q}$ coefficients, it seems feasible to prove this directly by following the pattern of the previous argument, using a version of the Leray–Hirsch theorem in place of the Künneth formula, but we have not pursued this line of reasoning. Instead, we will deduce Theorem B from the calculation of $H^*(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ by constructing a homotopy equivalence between $ {Fl} (\mathbb {C}^\infty )$ and $(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ . Here, we will be using the fact that Čech cohomology satisfies the homotopy axiom [Sp66, Theorem 6.5.6].

In general, suppose $X_n$ and $Y_n$ are two inverse systems, with respective limits ${X = \varprojlim X_n}$ and $Y=\varprojlim Y_n$ . If $f_n \colon X_n \to Y_n$ and $g_n \colon Y_n \to X_n$ are homotopy inverse maps of the systems with homotopies respecting the inverse systems, then the induced maps $f\colon X \to Y$ and $g\colon Y \to X$ are homotopy inverses.

To prove Theorem B, it suffices to construct homotopy equivalences between the partial flag manifold $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ and $(\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ , compatible with the projections.

Let $H_n = \operatorname {\mathrm {Hom}}(\mathbb {C}^n,\mathbb {C}^\infty )$ be the vector space of $\infty \times n$ matrices with finitely many non-zero entries. Define the subspaces $H^{\circ \circ }_n \subset H^\circ _n \subset H_n$ by

$$\begin{align*}H^{\circ\circ}_n = \{ \text{linear } f\colon \mathbb{C}^n \to \mathbb{C}^\infty \,|\, f(v) \neq 0 \text{ for all }v\neq 0\} \end{align*}$$

and

$$\begin{align*}H^\circ_n = \{ \text{linear } f\colon \mathbb{C}^n \to \mathbb{C}^\infty \,|\, f(\mathrm{e}_i) \neq 0 \text{ for }1\leq i\leq n\}, \end{align*}$$

where $(\mathrm {e}_{i})$ denotes the standard basis for $\mathbb {C}^{n}$ . So $H^{\circ \circ }_n$ is the space of full-rank matrices – that is, of linear embeddings of $\mathbb {C}^n$ in $\mathbb {C}^\infty $ – and $H^\circ _n$ is the space of matrices with all columns nonzero. These spaces form inverse systems, via maps $H_n \to H_{n'}$ for $n\geq n'$ , by projecting onto the first $n'$ columns.

The group $B_n$ of upper-triangular matrices in $GL_n$ acts freely on $H^{\circ \circ }_n$ by right multiplication, with quotient

$$\begin{align*}H^{\circ\circ}_n/B_n = {Fl} (1,\ldots,n;\mathbb{C}^\infty). \end{align*}$$

The diagonal torus $T_n = (\mathbb {C}^*)^n$ also acts, with quotient

$$\begin{align*}H^{\circ\circ}_n/T_n = {Fl} ^{\mathrm{split}}(1,\ldots,n;\mathbb{C}^\infty), \end{align*}$$

where $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty )$ is the “split” flag manifold, parametrizing n linearly independent lines $L_1$ , …, $L_n$ in $\mathbb {C}^\infty $ . We equip it with the quotient topology coming from the above identity. There is a natural projection $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty ) \to {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ , defined by $V_i = L_1\oplus \cdots \oplus L_i$ . Choose a Hermitian metric on $\mathbb {C}^\infty $ – say, by writing $\mathbb {C}^\infty $ as the union of $\mathbb {C}^m$ according to the standard bases, and equipping each $\mathbb {C}^m$ with the standard Hermitian metric. This defines a section $\sigma $ of the projection map, by splitting the chain $V_1 \subset \cdots \subset V_n$ orthogonally with respect to the metric. Using Gram–Schmidt orthonormalization, this makes $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ a deformation retract of $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty )$ . (To ensure that the deformation is compatible with the inverse system, one uses Gram–Schmidt to simultaneously move all the lines $L_i$ into orthogonal position.)

Let $ {Fl} (\mathbb {C}^\infty )$ be the inverse limit of the spaces $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ , and define $ {Fl} ^{\mathrm {split}}(\mathbb {C}^\infty )$ , H, $H^{\circ }$ , and $H^{\circ \circ }$ analogously. Note that $ {Fl} (\mathbb {C}^\infty )$ is the inverse limit of the system $H^{\circ \circ }_{n}/B_{n}$ , and $ {Fl} ^{\mathrm {split}}(\mathbb {C}^\infty )$ that of the system $H^{\circ \circ }_{n}/T_{n}$ .

The sections $\sigma $ described above are compatible with the inverse system maps: for $n\geq n'$ , the diagram

commutes. Taking inverse limits, it follows that the projection $ {Fl} ^{\mathrm {split}}(\mathbb {C}^\infty ) = H^{\circ \circ }/T \to {Fl} (\mathbb {C}^\infty ) = H^{\circ \circ }/B$ is a homotopy equivalence.

On the other hand, $T_n$ also acts on $H^\circ _n$ , with quotient

$$\begin{align*}H^\circ_n/T_n = (\mathbb{C}\mathbb{P}^\infty)^{\times n}. \end{align*}$$

For the inverse limit, we get $H^\circ /T=(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ . The projections making $H^{\circ \circ }_n$ and $H^\circ _n$ inverse systems induce the projections of flag manifolds and products of projective spaces.

Now, it suffices to construct a homotopy equivalence between $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty )$ and $(\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ , compatible with projections. The following lemma completes the proof of Theorem B.

Lemma The inclusions $\alpha \colon H^{\circ \circ }_n \hookrightarrow H^\circ _n$ are $T_n$ -equivariant homotopy equivalences, via maps which respect the inverse systems.

In fact, we will prove the lemma directly on the limit spaces $H^{\circ \circ }$ and $H^\circ $ , in such a way that the constructions evidently restrict compatibly to $H^{\circ \circ }_n$ and $H^{\circ }_n$ for finite n.

Proof We construct a map $\beta \colon H^\circ \hookrightarrow H^{\circ \circ }$ as follows. The idea is to send $f\colon \mathbb {C}^\infty \to \mathbb {C}^\infty $ (with $f(\mathrm {e}_i)\neq 0$ for all i) to an injective map $\hat {f}\colon \mathbb {C}^\infty \to \mathbb {C}^\infty \otimes \mathbb {C}^\infty $ , and then compose with an isomorphism $\mu \colon \mathbb {C}^\infty \otimes \mathbb {C}^\infty \to \mathbb {C}^\infty $ to obtain $\beta (f)$ .

First, given any $f\in H = \operatorname {\mathrm {Hom}}(\mathbb {C}^\infty ,\mathbb {C}^\infty )$ , let $\hat {f}\colon \mathbb {C}^\infty \to \mathbb {C}^\infty \otimes \mathbb {C}^\infty $ be defined by

$$\begin{align*}\hat{f}(v) = \sum_i v_i f(\mathrm{e}_i)\otimes \mathrm{e}_i. \end{align*}$$

Since each $f(\mathrm {e}_i)$ is nonzero, the vectors $\{ f(\mathrm {e}_i) \otimes \mathrm {e}_i \}_{i=1,2,\ldots }$ are linearly independent. So $\hat {f}$ is injective, as claimed.

Next, let $\mu \colon \mathbb {N} \times \mathbb {N} \to \mathbb {N}$ be the bijection defined by enumerating pairs in the order indicated by reading matrix coordinates below:

$$\begin{align*}\begin{array}{ccccc} 1 & 3 & 6 & 10 & \cdots \\ 2 & 5 & 9 & \cdots \\ 4 & 8 & \cdots \\ 7 & \cdots \\ \vdots \end{array} \end{align*}$$

That is, the enumeration of $\mathbb {N}\times \mathbb {N}$ proceeds as

$$\begin{align*}(1,1), \, (2,1), \, (1,2), \, (3,1), \, (2,2), \, (1,3),\, (4,1), \, (3,2), \, (2,3), \, (1,4),\, \ldots \end{align*}$$

Reusing notation, let $\mu \colon \mathbb {C}^\infty \otimes \mathbb {C}^\infty \xrightarrow {\sim } \mathbb {C}^\infty $ be the isomorphism defined by $\mathrm {e}_i \otimes \mathrm {e}_j \mapsto \mathrm {e}_{\mu (i,j)}$ .

Given $f\in H^\circ $ and $v = (v_1,v_2,\ldots ) \in \mathbb {C}^\infty $ , define $\beta (f)$ by

$$\begin{align*}\beta(f)(v) = \mu\bigl(\hat{f}(v)\bigr) = \mu\big( v_1 f(\mathrm{e}_1)\otimes \mathrm{e}_1 + v_2 f(\mathrm{e}_2) \otimes \mathrm{e}_2 + \cdots \big). \end{align*}$$

In terms of matrices, $\beta $ sends

(3.1) $$ \begin{align} f= \left[ \begin{array}{cccc} f_{11} & f_{12} & f_{13} & \cdots \\ f_{21} & f_{22} & f_{23} & \cdots \\ f_{31} & f_{32} & f_{33} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array} \right] \qquad \text{to} \qquad \beta(f)= \left[ \begin{array}{cccc} f_{11} & 0 & 0 & \cdots \\ f_{21} & 0 & 0 & \cdots \\ 0 & f_{12} & 0 & \cdots \\ f_{31} & 0 & 0 & \cdots \\ 0 & f_{22} & 0 & \cdots \\ 0 & 0 & f_{13} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array} \right]. \end{align} $$

Note that each column has only finitely many nonzero entries. Since no column of f is zero, the columns of $\beta (f)$ are evidently linearly independent.

Both $\alpha $ and $\beta $ are torus-equivariant, since they preserve the column-indices of entries. And they are evidently compatible with the inverse system – in fact, the above description applies verbatim to maps $\beta \colon H_n^\circ \hookrightarrow H_n^{\circ \circ }$ , defined analogously for all n.

Finally, we construct homotopies $\alpha \circ \beta \simeq \mathrm {id}_{H^\circ }$ and $\beta \circ \alpha \simeq \mathrm {id}_{H^{\circ \circ }}$ . Since $\alpha $ is just a natural inclusion, we suppress it from the notation. (So both $\alpha \circ \beta $ and $\beta \circ \alpha $ are given by the same formula that defines $\beta $ ; in matrices, this is (3.1).) We use the straight line homotopy

$$\begin{align*}(f,t) \mapsto t f + (1-t) \beta(f). \end{align*}$$

When $f \in H^\circ $ , so each column is nonzero, it is easy to see that the same is true of $t f + (1-t) \beta (f)$ for each $t\in [0,1]$ . (For $t=1$ , this is true by assumption; for $t=0,$ it holds because the entries of the columns of $\beta (f)$ are the same as those of f, but with $0$ ’s inserted. For $0<t<1$ , the first nonzero entry in column j of f appears strictly above the corresponding entry of $\beta (f)$ , so it remains nonzero in $t f + (1-t)\beta (f)$ – except if $j=1$ and the first nonzero entry is $f_{11}$ or $f_{21}$ , but those entries are constant in t for all $tf + (1-t)\beta (f)$ .) So we have shown $\alpha \circ \beta \simeq \mathrm {id}_{H^\circ }$ .

For the other composition, we factor the linear maps $\beta (f)$ and f as follows. From the definition, $\beta $ factors as $\beta (f) = \mu \circ \hat {f}$ . Let $\phi \colon \mathbb {C}^\infty \otimes \mathbb {C}^\infty \to \mathbb {C}^\infty $ be defined by $\phi (\mathrm {e}_i\otimes \mathrm {e}_j) = \mathrm {e}_i$ , so $f = \phi \circ \hat {f}$ . The two factorizations are summarized by the diagram

The homotopy between f and $\beta (f)$ factors as

$$\begin{align*}tf + (1-t)\beta(f) = (t\phi + (1-t)\mu)\circ\hat{f}. \end{align*}$$

To show $\beta \circ \alpha \simeq \mathrm {id}_{H^{\circ \circ }}$ , we must show that $tf + (1-t)\beta (f)$ is injective whenever f is; for this, it suffices to show that $t\phi + (1-t)\mu $ is injective for all $t\neq 1$ . We do this by direct computation.

Given $v = \sum _{i,j} a_{ij} \mathrm {e}_i\otimes \mathrm {e}_j$ , suppose $(t\phi + (1-t)\mu )(v)=0$ for some $t\neq 1$ . Expanding the coefficients, we obtain equations

$$ \begin{align*} a_{11} &= -t(a_{12} + a_{13} + \cdots ), \\ a_{21} &= -t(a_{22} + a_{23} + \cdots ), \end{align*} $$

and

$$ \begin{align*} a_{ij} &= \frac{t}{t-1}\sum_{\ell\geq1} a_{\mu(i,j),\ell} \end{align*} $$

for all other $(i,j)$ . Since $\mu (i,j)>i$ for all $(i,j)$ not equal to $(1,1)$ or $(2,1)$ , these linear equations express $a_{ij}$ as a combination of $a_{k,\ell }$ for pairs $(k,\ell )$ greater than $(i,j)$ in lexicographic order. Since all but finitely many coefficients $a_{k,\ell }$ are $0$ , this implies ${v=0}$ , completing the proof of the lemma.