2.1 Coefficient systems

The modules V and $V^\vee$ for $\mathrm{GL}(V)$ both fit into the setting of [Reference van der KallenvdK80, § 5]. We recall this setting, in modernised language, as follows. There is a category $\mathsf{VIC}(k)$ whose objects are the finite-dimensional k-modules, and whose morphisms from V to W are given by a linear injection $f : V \to W$ along with a choice of subspace $U \leq W$ complementary to f(V). A coefficient system will for us be defined to be a functor

$$F : \mathsf{VIC}(k) \longrightarrow k\text{-Mod}.$$

Unravelling definitions, this provides a ‘strongly central coefficient system’ in the sense of [Reference van der KallenvdK80, § 5.2]. Here are two examples. The first is $I(V) := V$ , with its functoriality given on a morphism $(f,U) : V \to W$ by $I(f,U) = f : V \to W$ . The second is $I^\vee(V) := V^\vee$ , given on a morphism $(f,U) : V \to W$ by

$$I^\vee(f,U) : V^\vee \overset{f^\vee}{\underset{\sim}\longleftarrow} f(V)^\vee \overset{inc}\longrightarrow f(V)^\vee \oplus U^\vee = W^\vee.$$

From these, for finite sets S and T we can form coefficient systems $I^{\otimes S} \otimes (I^\vee)^{\otimes T}$ , whose values at V are of course $V^{\otimes S} \otimes (V^\vee)^{\otimes T}$ . (Here the tensor product is, of course, taken over k.) Under the shifting operation $\Sigma$ on coefficient systems [Reference van der KallenvdK80, § 5.3] we have a decomposition

(2.1) \begin{equation}\Sigma (I^{\otimes S} \otimes (I^\vee)^{\otimes T}) \cong \bigoplus_{\substack{A \subseteq S \\ B \subseteq T}} I^{\otimes A} \otimes (I^\vee)^{\otimes B},\end{equation}

from which it quickly follows that $I^{\otimes S} \otimes (I^\vee)^{\otimes T}$ has degree $|S|+|T|$ in the sense of [Reference van der KallenvdK80, § 5.5].

The evaluation maps yield commutative squares

giving a morphism of coefficient systems $ev : I \otimes I^\vee \to k$ , the target being the constant coefficient system. We can therefore form the coefficient systems

$$I^{[S,T]} := {\operatorname*{Ker}}\bigg(\!I^{\otimes S}\otimes (I^\vee)^{\otimes T} \to \bigoplus_{\substack{s \in S \\ t\in T}} I^{\otimes S-s} \otimes (I^\vee)^{\otimes T-t}\!\bigg).$$

The decomposition (2.1) restricts to a decomposition $\Sigma(I^{[S,T]}) \cong \bigoplus_{{A \subseteq S, B \subseteq T}} I^{[A,B]}$ , so $I^{[S,T]}$ also has degree $|S|+|T|$ .

Example 2.1. The coefficient system $I^{[1,1]}$ associates to V the vector space

$$sl(V) := \operatorname*{Ker}(ev_V : V \otimes V^\vee \longrightarrow k).$$

When $V = k^n$ we will sometimes denote this by $sl_n(k)$ instead.

We write $V^{[S,T]}$ for the value of the functor $I^{[S,T]}$ at V, given of course by the kernel ${\operatorname*{Ker}}(V^{\otimes S} \otimes (V^\vee)^{\otimes T} \to \bigoplus_{\substack{s \in S, t \in T}} V^{\otimes S-s} \otimes (V^\vee)^{\otimes T-t})$ .

2.2 Stability

By [Reference van der KallenvdK80, Theorem 5.6] (using that we can take $sdim=0$ as k is a field [Reference van der KallenvdK80, § 2.2]) it follows that for fixed finite sets S and T the maps

\begin{align*}H_i(\mathrm{SL}(V) ; V^{\otimes S} \otimes (V^\vee)^{\otimes T}) &\longrightarrow H_i(\mathrm{SL}(V \oplus k) ; (V\oplus k)^{\otimes S} \otimes ((V\oplus k)^\vee)^{\otimes T}),\\H_i(\mathrm{SL}(V) ; V^{[S,T]}) &\longrightarrow H_i(\mathrm{SL}(V \oplus k) ; (V \oplus k)^{[S,T]})\end{align*}

are isomorphisms as long as $2i \leq \dim(V)-2-(|S|+|T|)$ . We will not keep track of explicit stability ranges, and will just use the fact that this range of degrees diverges with $\dim(V)$ for S and T fixed.

We can obtain a similar result on cohomology, by dualising. Recalling that in the introduction we defined

$$V_{[S,T]} ={\operatorname*{Coker}}\bigg(\bigoplus_{\substack{s \in S \\ t \in T}} V^{\otimes S-s} \otimes (V^\vee)^{\otimes T-t} \to V^{\otimes S}\otimes (V^\vee)^{\otimes T}\!\bigg),$$

there is an isomorphism $V^{[T,S]} \overset{\sim}\to V_{[S,T]}^\vee$ induced by the (swapped) evaluation map $V^{\otimes S} \otimes (V^\vee)^{\otimes T} \overset{\sim}\to \mathrm{Hom}_k(V^{\otimes T} \otimes (V^\vee)^{\otimes S}, k)$ . By the universal coefficient theorem the maps

\begin{align*}H^i(\mathrm{SL}(V \oplus k) ; (V\oplus k)^{\otimes T} \otimes ((V\oplus k)^\vee)^{\otimes S}) \longrightarrow & H^i(\mathrm{SL}(V) ; V^{\otimes T} \otimes (V^\vee)^{\otimes S}), \\H^i(\mathrm{SL}(V \oplus k) ; (V \oplus k)_{[T,S]}) \longrightarrow & H^i(\mathrm{SL}(V) ; V_{[T,S]})\end{align*}

are also isomorphisms in a stable range of degrees. Note that these maps are induced by the inclusions $\mathrm{SL}(V) \to \mathrm{SL}(V \oplus k)$ and the maps $(V\oplus k)^{\otimes T} \otimes ((V\oplus k)^\vee)^{\otimes S} \to V^{\otimes T} \otimes (V^\vee)^{\otimes S}$ dual to the maps considered on homology (i.e., induced by projection $V \oplus k \to V$ on the first tensor factors and restriction $(V \oplus k)^\vee \to V^\vee$ on the second).

2.3 SL versus GL

The extension

(2.2) \begin{equation}1 \longrightarrow \mathrm{SL}(V) \longrightarrow \mathrm{GL}(V) \overset{\det}\longrightarrow k^\times \longrightarrow 1\end{equation}

induces a $k^\times$ -action on $H^*(\mathrm{SL}(V) ; V_{[S,T]})$ .

Proof. Choose a decomposition $V=W \oplus k$ . The cohomological stability result above applies to show that the natural map

$$H^*(\mathrm{SL}(W \oplus k) ; (W \oplus k)_{[S,T]}) \longrightarrow H^*(\mathrm{SL}(W) ; W_{[S,T]})$$

is an isomorphism in a stable range of degrees. For the $k^\times$ -action on the source given by conjugation with $\mathrm{diag}(1_W, k^\times)$ , and the trivial $k^\times$ -action on the target, this map is equivariant, and so in the range in which this map is an isomorphism the $k^\times$ -action on the domain is trivial.

Corollary 2.3. The natural map

$$H^*(\mathrm{GL}(V) ; V_{[S,T]}) \longrightarrow H^*(\mathrm{SL}(V) ; V_{[S,T]})$$

is an isomorphism in a stable range of degrees.

Proof. As the finite field k has characteristic p, its units $k^\times$ form a finite group whose order is invertible in k. It follows by transfer that $H^s(k^\times ; M)=0$ for $s\gt 0$ and any $k[k^\times]$ -module M. The spectral sequence for the extension (2.2) takes the form

$$E_2^{s,t} = H^s(k^\times ; H^t(\mathrm{SL}(V) ; V_{[S,T]})) \Longrightarrow H^{s+t}(\mathrm{GL}(V) ; V_{[S,T]})$$

and is therefore supported along the line $s=0$ so collapses to give an isomorphism $H^*(\mathrm{GL}(V) ; V_{[S,T]}) \overset{\sim}\to H^0(k^\times ; H^*(\mathrm{SL}(V) ; V_{[S,T]}))$ . However, as the $k^\times$ -action is trivial in a stable range of degrees, the latter is $H^*(\mathrm{SL}(V) ; V_{[S,T]})$ in this range.

3.1 Ext and products

We will be interested in various cohomology groups of the form $H^*(\mathrm{GL}(V) ; U \otimes W^\vee)$ where V is a finite-dimensional k-vector space and U and W are $k[\mathrm{GL}(V)]$ -modules which are finite-dimensional as k-modules (typically U and W will be constructed functorially from V). It will be convenient to translate between such cohomology groups and Ext-groups over $k[\mathrm{GL}(V)]$ .

For $k[\mathrm{GL}(V)]$ -modules A and B we write $\mathrm{Ext}^i_{\mathrm{GL}(V)}(A,B) := \mathrm{Ext}^i_{k[\mathrm{GL}(V)]}(A,B)$ , which we consider as the abelian group of morphisms in the derived category of $k[\mathrm{GL}(V)]$ -modules from A to the i-fold shift B[i] (e.g., cf. [Reference WeibelWei94, § 10.7]). The cocommutative coalgebra structure on the Hopf algebra $k[\mathrm{GL}(V)]$ defines a symmetric monoidal structure $- \otimes_k -$ on $k[\mathrm{GL}(V)]$ -modules, which is exact in each variable and so descends to a symmetric monoidal structure on the derived category of $k[\mathrm{GL}(V)]$ -modules (which preserves exact triangles in each variable).

Some of this structure can be spelled out as follows. If A, B and C are $k[\mathrm{GL}(V)]$ -modules, then composition in the derived category yields the Yoneda product [Reference BensonBen98, § 2.6]

$$-\circ- : \mathrm{Ext}_{\mathrm{GL}(V)}^i(B, C) \otimes \mathrm{Ext}_{\mathrm{GL}(V)}^j(A, B) \longrightarrow \mathrm{Ext}_{\mathrm{GL}(V)}^{i+j}(A, C).$$

If D is a further $k[\mathrm{GL}(V)]$ -module then the symmetric monoidal structure $- \otimes_k -$ induces the cup product [Reference BensonBen98, § 3.2]

$$-\otimes- : \mathrm{Ext}_{\mathrm{GL}(V)}^i(A, B) \otimes \mathrm{Ext}_{\mathrm{GL}(V)}^j(C, D) \longrightarrow \mathrm{Ext}_{\mathrm{GL}(V)}^{i+j}(A \otimes_k C, B \otimes_k D).$$

It is associative and graded commutative. Its compatibility with the Yoneda product is encoded by the fact that $- \otimes_k -$ defines a bifunctor on the derived category of $k[\mathrm{GL}(V)]$ -modules.

As $H^*(\mathrm{GL}(V) ; U \otimes W^\vee) = \mathrm{Ext}^*_{\mathrm{GL}(V)}(k, U \otimes W^\vee)$ we can form the map

$$\mathrm{Ext}^*_{\mathrm{GL}(V)}(k, U \otimes W^\vee)\overset{- \otimes \mathrm{Id}_W}\longrightarrow\mathrm{Ext}^*_{\mathrm{GL}(V)}(W, U \otimes W^\vee \otimes W)\overset{\,(U \otimes ev) \circ -}\longrightarrow \mathrm{Ext}^*_{\mathrm{GL}(V)}(W, U)$$

using the evaluation $ev : W^\vee \otimes W \to k$ , and this map is an isomorphism (with inverse given by composing $- \otimes\mathrm{Id}_{W^\vee}$ with $- \circ coev$ using $coev : k \to W\otimes W^\vee$ ).

3.2 Defining cohomology classes

As an instance of the discussion above, for a finite-dimensional k-vector space V we have constructed an isomorphism

$$H^*(\mathrm{GL}(V) ; V \otimes V^\vee) = \mathrm{Ext}_{\mathrm{GL}(V)}^*(k, V \otimes V^\vee) \cong \mathrm{Ext}_{\mathrm{GL}(V)}^*(V, V),$$

and the latter has an associative k-algebra structure by the Yoneda product. We define

$$\mathrm{Ext}_{\mathrm{GL}}^*(I, I) := \lim_{V \in \mathsf{VIC}(k)^{\mathrm{op}}} \mathrm{Ext}_{\mathrm{GL}(V)}^*(V,V),$$

which again has an associative k-algebra structure. The limit here is taken over the opposite of the (essentially small) category $\mathsf{VIC}(k)$ from § 2.1: a morphism $(f, U) : V \to W$ in this category gives an isomorphism $f \oplus {inc} : V \oplus U \overset{\sim}\to W$ and so induces a map

$$\mathrm{Ext}_{\mathrm{GL}(W)}^*(W,W) \overset{\text{restriction}}\longrightarrow \mathrm{Ext}_{\mathrm{GL}(V)}^*(V \oplus U,V \oplus U) \overset{\text{projection}}\longrightarrow \mathrm{Ext}_{\mathrm{GL}(V)}^*(V,V)$$

of associative k-algebras. In fact any automorphism $(f,0) : V \to V$ in $\mathsf{VIC}(k)$ acts on $\mathrm{Ext}_{\mathrm{GL}(V)}^*(V,V)$ as the identity, as an instance of the fact that inner automorphisms act trivially on group cohomology [Reference BrownBro94, III (8.3)], so the limit is the same if we just take it over the standard split inclusions $0 \to k \to k^2 \to k^3 \to \cdots$ . In each cohomological degree this limit is attained at a finite stage, by the stability results of § 2.2.

Theorem 3.1. There are classes $x^{[i]} \in \mathrm{Ext}_{\mathrm{GL}}^{2i}(I, I)$ such that the map

$$\Gamma_{k}[x] = k\{x^{[0]}, x^{[1]}, x^{[2]}, \ldots\} \longrightarrow \mathrm{Ext}_{\mathrm{GL}}^{*}(I, I)$$

is an isomorphism of k-algebras from the free divided power algebra on $x = x^{[1]}$ .

More generally, for finite sets S and T we define

$$\mathrm{Ext}_{\mathrm{GL}}^*(I^{\otimes S}, I^{\otimes T}) := \lim_{V \in \mathsf{VIC}(k)^{\mathrm{op}}} \mathrm{Ext}_{\mathrm{GL}(V)}^*(V^{\otimes S},V^{\otimes T})$$

analogously to the above: we wish to determine these groups. The Yoneda product and cup product extend to these, by their naturality. For a function $\ell : S \to \mathbb{N}$ , we can take cup products of the classes in Theorem 3.1 to obtain cohomology classes

$$\kappa(\ell) := \bigotimes_{s \in S} x^{[\ell(s)]} \in \mathrm{Ext}_{\mathrm{GL}}^{\sum_{s \in S} 2\ell(s)}(I^{\otimes S}, I^{\otimes S}).$$

Given in addition a bijection $f : T \to S$ , the morphisms $V^{\otimes f} : V^{\otimes T} \to V^{\otimes S}$ assemble to an element $[f] \in \mathrm{Ext}_{\mathrm{GL}}^{0}(I^{\otimes T}, I^{\otimes S})$ , and we can form the Yoneda product

$$\kappa(\ell, f) := \kappa(\ell) \circ [f] \in \mathrm{Ext}_{\mathrm{GL}}^{\sum_{s \in S} 2\ell(s)}(I^{\otimes T}, I^{\otimes S}).$$

Writing $\mathrm{Bij}(T,S)$ for the set of bijections from T to S, and $k\{\mathrm{Bij}(T,S)\}$ for the free k-module on this set, this construction defines a map

(3.1) \begin{equation}\Psi_{S,T}: \Gamma_{k}[x]^{\otimes S} \otimes k\{\mathrm{Bij}(T,S)\} \longrightarrow \mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes T}, I^{\otimes S}).\end{equation}

We will often write $\Gamma_{k}[x]^{\otimes S} = \Gamma_{k}[x_s \,|\, s \in S]$ .

By the stability discussion in § 2.2 the map

$$\mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes T}, I^{\otimes S}) \longrightarrow \mathrm{Ext}_{\mathrm{GL}(V)}^{*}(V^{\otimes T}, V^{\otimes S}) \cong H^*(\mathrm{GL}(V) ; V^{\otimes S} \otimes (V^\vee)^{\otimes T})$$

is an isomorphism in a stable range of degrees, so Theorem 3.2 determines the latter cohomology groups in a stable range.

3.3 Proof of Theorems 3.1 and 3.2

The proofs of these theorems are, at least implicitly, available in the literature on functor homology; we explain how to extract them, taking the paper of Franjou et al. [Reference Franjou, Friedlander, Scorichenko and SuslinFFS⁺99] as our main reference.

Let $\mathsf{F}$ denote the category of functors from finite-dimensional k-modules to k-modules, with $\mathrm{Hom}_{\mathsf{F}}(F,G)$ the k-module of natural transformations from F to G. This is an abelian category with a set of projective generators, so one may do homological algebra in this category. In particular, one may form $\mathrm{Ext}_{\mathsf{F}}^*(-,-)$ as the derived functors of $\mathrm{Hom}_{\mathsf{F}}(-,-)$ .

For each $V \in \mathsf{VIC}(k)$ there is a functor

$$F \mapsto F(V) : \mathsf{F} \longrightarrow k[\mathrm{GL}(V)]\text{-}\mathsf{mod}$$

which is exact, so induces a map $\mathrm{Ext}_{\mathsf{F}}^*(F,G) \to \mathrm{Ext}_{\mathrm{GL}(V)}^{*}(F(V), G(V))$ for each V. Taking the limit over $\mathsf{VIC}(k)^{\mathrm{op}}$ , these assemble into a map

$$\mathrm{Ext}_{\mathsf{F}}^*(F,G) \longrightarrow \mathrm{Ext}_{\mathrm{GL}}^{*}(F, G),$$

which by [Reference Franjou, Friedlander, Scorichenko and SuslinFFS⁺99, Theorem A.1] or [Reference BetleyBet99] is an isomorphism.

Taking F and G to be the ‘identity’ functor I, combining Théorème 7.3 and § 11 of [Reference Franjou, Lannes and SchwartzFLS94] identifies $\mathrm{Ext}_{\mathsf{F}}(I,I)$ , with its Yoneda product, with the divided power algebra $\Gamma_{k}[x] = k\{x^{[0]}, x^{[1]}, x^{[2]}, \ldots\}$ . Combined with the previous paragraph, this gives Theorem 3.1.

Consider the extension

(3.2) \begin{equation}1 \longrightarrow \mathrm{SL}_n(\mathbb{Z}/p^2, p) \longrightarrow \mathrm{SL}_n(\mathbb{Z}/p^2) \longrightarrow \mathrm{SL}_n(\mathbb{Z}/p) \longrightarrow 1,\end{equation}

where the right-hand map is reduction modulo p, so the left-hand group consists of matrices over $\mathbb{Z}/p^2$ of determinant 1 which reduce to the identity mod p. Elements of $\mathrm{SL}_n(\mathbb{Z}/p^2, p)$ can be uniquely written in the form $I + p A$ for A an $n \times n$ matrix with entries in $\mathbb{Z}/p$ (using $p \cdot - : \mathbb{Z}/p \overset{\sim}\to p\mathbb{Z}/p^2$ ), and the identity $\det(I + p A) = 1 + p \cdot \mathrm{tr}(A) \in \mathbb{Z}/p^2$ shows that $I+pA$ has determinant 1 if and only if A has trace 0. This shows that the function $I + p A \mapsto A : \mathrm{SL}_n(\mathbb{Z}/p^2, p) \to sl_n(\mathbb{Z}/p)$ is a bijection, and furthermore $(I+pA)(I+pB) = I+p(A+B)$ , so this function is an isomorphism.

Under this identification, the class of the abelian extension (3.2) is an element $e_n \in H^2(\mathrm{SL}_n(\mathbb{Z}/p) ; sl_n(\mathbb{Z}/p))$ . Writing $V_n = (\mathbb{Z}/p)^n$ , the exact sequence $0 \to sl_n(\mathbb{Z}/p) \to V_n \otimes V_n^\vee \overset{ev}\to \mathbb{Z}/p \to 0$ gives

and the outer terms vanish (for n large enough) by the theorem of Quillen [Reference QuillenQui72] so the middle map is an isomorphism. We define $x'_n \in H^2(\mathrm{SL}_n(\mathbb{Z}/p) ; V_n \otimes V_n^\vee)$ to be the image of minus the class $e_n$ . One may check that these classes are compatible under stabilisation so give an $x' \in \mathrm{Ext}^2_{\mathrm{GL}}(I, I)$ . This class is not zero. If it were, then $e_n$ would vanish for n large enough so the extensions (3.2) would be split, but they are not: see [Reference SahSah77, Proposition 0.3]. Thus, $x' = u \cdot x$ for some unit u in $\mathbb{Z}/p$ , so we can let $(x')^{[i]} := u^i \cdot x^{[i]}$ so that $\mathrm{Ext}^*_{\mathrm{GL}}(I, I) = \Gamma_{\mathbb{F}_p}[x']$ . In Theorem 3.1 for $k=\mathbb{F}_p$ we take this x’ for x. (A similar normalisation can be made for any finite field k, substituting the length 2 Witt vectors $W_2(k)$ for $\mathbb{Z}/p^2$ in the above discussion.)

If $|S| \neq |T|$ then Pirashvili’s cancellation lemma [Reference Betley and PirashviliBP94, Theorem A.1] shows that $\mathrm{Ext}_{\mathsf{F}}^*(I^{\otimes T}, I^{\otimes S}) \cong \mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes T}, I^{\otimes S})$ vanishes. If $|S|=|T|$ then [Reference Franjou, Friedlander, Scorichenko and SuslinFFS⁺99, Corollary 1.8], using Pirashvili’s cancellation lemma to neglect most terms, gives an isomorphism

$$\mathrm{Ext}_{\mathsf{F}}^*(I,I)^{\otimes S} \otimes k\{\mathrm{Bij}(T,S)\} \cong \mathrm{Ext}_{\mathsf{F}}^*(I^{\otimes T}, I^{\otimes S}).$$

Combined with the previous two paragraphs, and after checking that the maps which induce this isomorphism agree with those that we have described above, this gives Theorem 3.2.

4.1 Functoriality on the upward walled Brauer category

The upward walled Brauer category $\mathsf{uwBr}$ is the category with objects given by pairs (S,T) of finite sets, and with morphisms $\mathsf{uwBr}((S,T), (U,V))$ given by a pair of injections $f : S \to U$ , $g: T \to V$ and well as a bijection $m : U \setminus f(S) \to V \setminus g(T)$ . We visualise such morphisms as in the following diagram, where the composition is given by gluing such one-dimensional cobordisms.

The construction $(S, T) \mapsto \mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes T}, I^{\otimes S})$ defines a functor from $\mathsf{uwBr}$ to graded k-modules: a morphism (f,g,m) acts as

$$\mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes T}, I^{\otimes S}) \overset{(f,g)}\longrightarrow \mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes g(T)}, I^{\otimes f(S)}) \overset{m}\longrightarrow \mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes V}, I^{\otimes U}),$$

where the first map is induced by the bijections $f : S \to f(S)$ and $g : T \to g(T)$ , and the second is given by inserting $1 \in \mathrm{Ext}_{\mathrm{GL}}^{0}(I, I)$ along each of the pairs $\{(u, m(u))\}_{u \in U \setminus f(S)}$ .

Similarly, the construction $(S,T) \mapsto \Gamma_k[x]^{\otimes S} \otimes k\{\mathrm{Bij}(T,S)\}$ defines a functor from $\mathsf{uwBr}$ to graded k-modules: a morphism (f,g,m) acts by sending the element $(\bigotimes_{s \in S} x_s^{[\ell(s)]}) \otimes \phi$ to $(\bigotimes_{u \in U} x_u^{[\ell'(u)]}) \otimes \phi'$ where $\ell'(u)$ is $\ell(s)$ if $u=f(s)$ and is 0 otherwise, and the bijection $\phi' : V \to U$ is equal to $f\circ\phi\circ g^{-1}$ on $g(T) \subset V$ and is equal to $m^{-1}$ on $V \setminus g(T)$ .

As $x^{[0]} = 1 \in \mathrm{Ext}_{\mathrm{GL}}^{0}(I, I)$ , it follows from these descriptions that the map (3.1) we have described is a natural transformation

$$\Psi : \Gamma_k[x]^{\otimes -} \otimes k\{\mathrm{Bij}(\bullet,-)\} \Longrightarrow \mathrm{Ext}_{\mathrm{GL}}^{*}(I^{\otimes \bullet}, I^{\otimes -}) $$

of functors from $\mathsf{uwBr}$ to graded k-modules. Theorem 3.2 then says that it is in fact a natural isomorphism of such functors.

4.2 Functoriality on the full walled Brauer category

The discussion in this section is not needed for the proof of Theorem C, but will be used in the proof of Theorem A.

For $\delta \in k$ the walled Brauer category $\mathsf{wBr}_\delta$ is the k-linear category with objects given by pairs (S,T) of finite sets, and with morphisms $\mathsf{wBr}_\delta((S,T), (U,V))$ given by the k-vector space with basis given by tuples (f, g, m, n) where $f : S' \to U'$ is a bijection from a subset $S' \subset S$ to a subset $U' \subset U$ , $g : T' \to V'$ is a bijection from a subset $T' \subset T$ to a subset $V' \subset V$ , and $m : S \setminus S' \to T \setminus T'$ and $n : U \setminus U' \to V \setminus V'$ are bijections. We depict such morphisms as in the following diagram, where as shown the composition is given by gluing such one-dimensional cobordisms, and replacing any circles that are formed by the scalar $\delta \in k$ .

For V a finite-dimensional vector space, the functor

$$(S,T) \longmapsto H^*(\mathrm{GL}(V) ; V^{\otimes S} \otimes (V^\vee)^{\otimes T})$$

from $\mathsf{uwBr}$ to graded k-modules extends to a functor on $\mathsf{wBr}_\delta$ with $\delta := \dim(V)$ . This extension is induced by the evaluation maps $ev: V \otimes V^\vee \to k$ . Concretely, the morphism

$$(inc : S \setminus s \to S, inc : T \setminus t \to T, \emptyset \overset{\sim}\to\emptyset, \{s\} \overset{\sim}\to \{t\}) : (S,T) \longrightarrow (S \setminus s, T \setminus t)$$

gives the map induced on cohomology by the morphism of coefficient systems

$$\varepsilon_{s,t} : V^{\otimes S } \otimes (V^\vee)^{\otimes T} \longrightarrow V^{\otimes S \setminus s} \otimes (V^\vee)^{\otimes T \setminus t}$$

which evaluates the sth and tth terms. Using that the composition $k \overset{coev}\to V \otimes V^\vee \overset{ev}\to k$ is multiplication by $\dim(V)=\delta$ , it is elementary to verify that there is a unique extension to a functor on $\mathsf{wBr}_\delta$ given in this way on such morphisms.

Composing the maps $\Psi_{S,T}$ from (3.1) with evaluation at V gives maps

$$\psi_{S,T} : \Gamma_{k}[x]^{\otimes S} \otimes k\{\mathrm{Bij}(T,S)\} \longrightarrow H^*(\mathrm{GL}(V) ; V^{\otimes S} \otimes (V^\vee)^{\otimes T}),$$

which are isomorphisms in a stable range by Theorem 3.2, and which are natural transformations of functors on $\mathsf{uwBr}$ . We wish to explain how the $\mathsf{wBr}_\delta$ -functoriality of the target translates to the source.

Define a map

$$\delta_{s,t} : \Gamma_{k}[x]^{\otimes S} \otimes k\{\mathrm{Bij}(T, S)\} \longrightarrow \Gamma_{k}[x]^{\otimes S \setminus s } \otimes k\{\mathrm{Bij}(T \setminus t, S \setminus s)\}$$

by the formula

$$\bigg(\bigotimes_{u \in S} x_u^{[\ell(u)]}\bigg) \otimes \sigma \longmapsto \begin{cases}\dim(V) (\bigotimes_{u \in S \setminus s} x^{[\ell(u)]}_u ) \otimes\sigma\vert_{T \setminus t}, & \sigma(t)=s \text{ and } \ell(s)=0,\\0, & \sigma(t)=s \text{ and } \ell(s) \gt 0,\\\binom{\ell(s)+ \ell(\sigma(t))}{\ell(s)} (\bigotimes_{u \in S\setminus s} x_u^{[\ell'(u)]} )\otimes \sigma', & \sigma(t) \neq s,\end{cases}$$

where $\ell'$ is given by $\ell'(\sigma(t)) = \ell(s)+ \ell(\sigma(t))$ and by the restriction of $\ell$ on all other elements of $S \setminus s$ , and $\sigma'$ is given by $\sigma'(\sigma^{-1}(s)) = \sigma(t)$ , and by the restriction of $\sigma$ on all other elements of $T \setminus t$ .

Lemma 4.1. The square

commutes in a stable range of degrees.

If $\sigma(t) = s' \neq s$ , then we can write $t' := \sigma^{-1}(s) \neq t$ and again by taking tensor products we reduce to the case $S=\{s,s'\}$ and $T=\{t,t'\}$ . Then

\begin{equation*}\psi_{S,T}(x_s^{[\ell(s)]} x_{s'}^{[\ell(s')]} \otimes \sigma) \in H^*(\mathrm{GL}(V) ; V^{\otimes S } \otimes (V^\vee)^{\otimes T})\end{equation*}

is the cup product of the classes

\begin{align*}\psi_{\{s\}, \{t'\}}(x_s^{[\ell(s)]}) &\in H^*(\mathrm{GL}(V) ; V^{\otimes \{s\}} \otimes (V^\vee)^{\otimes\{t'\}}) \cong \mathrm{Ext}^*_{\mathrm{GL}(V)}(V, V),\\\psi_{\{s'\}, \{t\}}(x_{s'}^{[\ell(s')]}) &\in H^*(\mathrm{GL}(V) ; V^{\otimes \{s'\}} \otimes (V^\vee)^{\otimes\{t\}}) \cong \mathrm{Ext}^*_{\mathrm{GL}(V)}(V, V),\end{align*}

and applying $\varepsilon_{s,t}$ corresponds to evaluating the Yoneda product. By the divided power algebra structure described in Theorem 3.1, the result is

$$\binom{\ell(s)+ \ell(s')}{\ell(s)} x^{[\ell(s)+ \ell(s')]},$$

which agrees with $\psi_{\{s'\}, \{t'\}} \delta_{s,t}$ applied to $x_s^{[\ell(s)]} x_{s'}^{[\ell(s')]} \otimes \sigma$ .

4.3 Graphical interpretation

Rather than the formulas given above, we can interpret the functoriality of $\Gamma_{k}[x]^{\otimes -} \otimes k\{\mathrm{Bij}(\bullet,-)\}$ on the walled Brauer category by interpreting elements of $\Gamma_{k}[x]^{\otimes S} \otimes k\{\mathrm{Bij}(T,S)\}$ as given graphically in the diagram below. That is, an element of $\mathsf{wBr}_\delta((\emptyset, \emptyset),(S,T))$ with each strand labelled by an $x^{[i]}$ .

Then the functoriality is given by concatenating with an element of the walled Brauer category, multiplying labels which now lie on the same strand together using the divided power multiplication, then setting any closed components labelled by $x^{[i]}$ with $i\gt 0$ equal to zero, and setting any closed components labelled by $x^{[0]}$ equal to $\dim(V)$ .

6.1 Some semisimplicity

As long as $p \gt n,m$ the algebra $k[\Sigma_n \times \Sigma_m]$ is semisimple, and for partitions $\lambda \vdash n$ and $\mu \vdash m$ the external tensor products of Specht modules $S^\lambda \otimes S^\mu$ form a complete set of simple modules. Now $\Sigma_n \times \Sigma_m$ acts on $V_{[n,m]}$ , and so defining

$$S_{\lambda, \mu}(V) := \mathrm{Hom}_{k[\Sigma_n \times \Sigma_m]} ( S^\lambda \otimes S^\mu, V_{[n,m]}),$$

the evaluation map

$$\bigoplus_{\lambda, \mu} S^\lambda \otimes S^\mu \otimes S_{\lambda, \mu}(V) \longrightarrow V_{[n,m]}$$

is an isomorphism.

As a final ingredient we should like to know that the quotient map

(6.1) \begin{equation}q: V^{\otimes n} \otimes (V^\vee)^{\otimes m} \longrightarrow V_{[n,m]}\end{equation}

is split as a map of $\mathrm{GL}(V)$ -modules. Unfortunately this is not generally true: in the exact sequence

(6.2) \begin{equation}0 \longrightarrow k \overset{coev}\longrightarrow V \otimes V^\vee \overset{q}\longrightarrow V_{[1,1]} \longrightarrow 0\end{equation}

we have $ev : H_0(GL(V) ; V \otimes V^\vee) \overset{\sim}\to k$ , and $ev \circ coev : k \to k$ is multiplication by $\dim(V)$ ; thus, there is an exact sequence

$$\cdots \longrightarrow H_1(\mathrm{GL}(V) ; V_{[1,1]}) \overset{\partial}\longrightarrow k \overset{-\cdot\dim(V)}\longrightarrow k \overset{q_*}\longrightarrow H_0(\mathrm{GL}(V) ; V_{[1,1]}) \longrightarrow 0$$

and so if $\dim(V) \equiv 0 \mod p$ then $\partial$ is non-trivial and so (6.2) cannot be $\mathrm{GL}(V)$ -equivariantly split. However, we have the following partial result, which will suffice.

Let $I \subset \mathcal{B}_{n,m}(\delta)$ denote the right sub- $\mathcal{B}_{n,m}(\delta)$ -module spanned, as a k-module, by those walled Brauer diagrams which contain at least one matched pair, that is, contain an arc which crosses the wall. This is a right sub- $\mathcal{B}_{n,m}(\delta)$ -module as such diagrams must contain a wall-crossing arc at the left-hand end, and this is clearly preserved by right-multiplication. The exact sequence $0 \to I \to \mathcal{B}_{n,m}(\delta) \to \mathcal{B}_{n,m}(\delta)/I \to 0$ yields an exact sequence

and the definition of I identifies the quotient map with (6.1). Thus, in order to show that (6.1) is $\mathrm{GL}(V)$ -equivariantly split, it suffices to show that $\mathcal{B}_{n,m}(\delta) \to \mathcal{B}_{n,m}(\delta)/I$ is split as a map of right $\mathcal{B}_{n,m}(\delta)$ -modules.

For the latter it suffices that the algebra $\mathcal{B}_{n,m}(\delta)$ be semisimple, and for this we apply a result of Andersen, Stroppel and Tubbenhauer [Reference Andersen, Stroppel and TubbenhauerAST17]. In the notation of that paper (Conventions 3.11 of that paper), as $\dim(V) \equiv ({p+1})/{2} \mod p$ (and p is odd) we have $\delta_p = ({p+1})/{2}$ , so $\delta_p \neq 0$ and hence [Reference Andersen, Stroppel and TubbenhauerAST17, Theorem 6.1(1)] applies and says that $\mathcal{B}_{n,m}(\delta)$ is semisimple providing $m+n \leq \min\{\delta_p+1, p-\delta_p+1\} = ({p+1})/{2}$ . (Note that our assumption $\dim(V) \equiv ({p+1})/{2} \mod p$ was made to maximise the value of $m+n$ for which their result applies.)

6.2 Proof of Theorem C

To prove Theorem C, using Corollary 2.3, it suffices to identify

(6.3) \begin{equation}H^i(\mathrm{GL}(V) ; V_{[n,m]}) \cong [V_{[n,m]} \otimes X^i]^{\mathrm{GL}(V)}\end{equation}

as $k[\Sigma_n \times \Sigma_m]$ -modules, for $i \lt 2p$ and $n+m \leq ({p+1})/{2}$ and all large enough $\dim(V)$ . Applying the exact functor $\mathrm{Hom}_{k[\Sigma_n \times \Sigma_m]}(S^\lambda \otimes S^\mu, -)$ then gives the claimed result.

By the results of § 2 the two sides stabilise with $\dim(V)$ , so it suffices to establish this identity for all large enough $\dim(V)$ with $\dim(V) \equiv ({p+1})/{2} \mod p$ . In this case by Proposition 6.1 and the assumption $n+m \leq ({p+1})/{2}$ the quotient map in the sequence

$$\bigoplus_{i=1}^n \bigoplus_{j=1}^m V^{\otimes n-1} \otimes (V^\vee)^{\otimes m-1} \to V^{\otimes n} \otimes (V^\vee)^{\otimes m} \longrightarrow V_{[n,m]} \longrightarrow 0$$

is split as $\mathrm{GL}(V)$ -modules and hence this sequence remains exact after applying $H^i(\mathrm{GL}(V) ; -)$ or $[- \otimes X^i]^{\mathrm{GL}(V)}$ . Combining this with the natural isomorphism $H^i(\mathrm{GL}(V) ; -) \cong [- \otimes X^i]^{\mathrm{GL}(V)}$ (for $i \lt 2p$ ) of functors on the upward Brauer category, given by Corollary 5.7, yields the isomorphism (6.3) as required.

PART II. Stable cohomology of congruence subgroups

For the rest of the paper all cohomology will be taken with $\mathbb{F}_p$ -coefficients, which we will often omit from the notation.

7.1 Completed cohomology

Following Calegari and Emerton [Reference Calegari and EmertonCE16] we consider the system of congruence subgroups defined as the kernels

(7.1) \begin{equation}1 \longrightarrow \mathrm{SL}_n(\mathbb{Z}, p^r) \longrightarrow \mathrm{SL}_n(\mathbb{Z}) \longrightarrow \mathrm{SL}_n(\mathbb{Z}/p^r) \longrightarrow 1\end{equation}

as a pro-group, $\{\mathrm{SL}_n(\mathbb{Z}, p^r)\}_r$ , and let the associated completed ( $\mathbb{F}_p$ -)cohomologyFootnote ²

$$\widetilde{H}^i(\mathrm{SL}_n(\mathbb{Z})) := \operatorname*{colim}_{r \to \infty} H^i(\mathrm{SL}_n(\mathbb{Z}, p^r) )$$

be the cohomology of this pro-group. It of course depends on the (implicit) prime p.

We also consider the pro-(finite group) $\mathrm{SL}_n(\mathbb{Z}_p) := \{\mathrm{SL}_n(\mathbb{Z}/p^r)\}_r$ , whose cohomology we call the continuous cohomology of $\mathrm{SL}_n(\mathbb{Z}_p)$ . Taking the limit of this pro-(finite group) gives the usual group $\mathrm{SL}_n(\mathbb{Z}_p)$ with the topology induced from the p-adic topology on $\mathbb{Z}_p$ , and the pro-group plays for us the role of this topological group: working with the pro-object allows us to form the ‘classifying pro-space’ $\{B\mathrm{SL}_n(\mathbb{Z}/p^r)\}_r$ , which should be thought of as an implementation of the classifying space of the group $\mathrm{SL}_n(\mathbb{Z}_p)$ which takes into account the topology.

By the main theorem of [Reference Calegari and EmertonCE16] the completed cohomology groups enjoy homological stability with respect to n, and (hence) the outer action of the pro-(finite group) $ \{\mathrm{SL}_n(\mathbb{Z}/p^r)\}_r$ on these groups is trivial in the stable range. This also follows from slightly more recent work of Iwasa [Reference IwasaIwa20, Theorem 1.3]. We write

$$\widetilde{H}^i(\mathrm{SL}(\mathbb{Z})) := \lim_{n \to \infty} \widetilde{H}^i(\mathrm{SL}_n(\mathbb{Z}))$$

for the stable completed cohomology. We may identify it with the cohomology of the pro-space

$$\operatorname*{hocolim}_{n \to \infty}{\kern.1pt} \{B\mathrm{SL}_n(\mathbb{Z}, p^r)\}_r,$$

the homotopy colimit formed in pro-spaces. Generally speaking, (homotopy) limits and colimits in pro-spaces are not formed objectwise, and so this should not be confused with the pro-space $\{\operatorname*{hocolim}_{n \to \infty} B\mathrm{SL}_n(\mathbb{Z}, p^r)\}_r$ , which will play no role. For background on the homotopy theory of pro-spaces see [Reference Artin and MazurAM69, § 2 and Appendices] and [Reference IsaksenIsa01].

Taking colimits of the Leray–Hochschild–Serre spectral sequences of the extensions (7.1) gives a spectral sequence

$$E_2^{s,t} = H^s_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) ;\widetilde{H}^t(\mathrm{SL}_n(\mathbb{Z}))) \Longrightarrow H^{s+t}(\mathrm{SL}_n(\mathbb{Z})),$$

and in the stable range the coefficient system is untwisted. Here the continuous cohomology refers to

$$H^s_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) ; M) := \operatorname*{colim}_{r \to \infty} H^s(\mathrm{SL}_n(\mathbb{Z}/p^r) ; M)$$

and is defined for $\mathrm{SL}_n(\mathbb{Z}_p)$ -modules M for which the action factors through $\mathrm{SL}_n(\mathbb{Z}_p) \to \mathrm{SL}_n(\mathbb{Z}/p^m)$ for some m (which allows us to form the colimit).

More generally, considering the extension of pro-groups

$$1 \longrightarrow \{\mathrm{SL}_n(\mathbb{Z}, p^{k+r})\}_r \longrightarrow \mathrm{SL}_n(\mathbb{Z}, p^k) \longrightarrow \{\mathrm{SL}_n(\mathbb{Z}/p^{r+k}, p^k)\}_r \longrightarrow 1$$

gives a spectral sequence

(7.2) \begin{equation}E_2^{s,t} = H^s_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p^k) ; \widetilde{H}^t(\mathrm{SL}_n(\mathbb{Z}) )) \Longrightarrow H^{s+t}(\mathrm{SL}_n(\mathbb{Z}, p^k)),\end{equation}

again untwisted in a stable range, whose $E_2$ -term is quite accessible for $k \gt 0$ as then $\mathrm{SL}_n(\mathbb{Z}_p, p^k)$ is a p-adic analytic group (see § 7.6).

7.2 Relation to algebraic K-theory

We rephrase [Reference CalegariCal15, § 2.3]. Let $\mathrm{K}(-)$ denote the algebraic K-theory spectrum, so that there are acyclic maps $B\mathrm{GL}(-) \to \Omega_0^\infty \mathrm{K}(-)$ . Write $\mathrm{K}^{\mathrm{cts}}(\mathbb{Z}_p) :=\operatorname*{holim}\limits_{r \to \infty}\mathrm{K}(\mathbb{Z}/p^r)$ for the continuous K-theory of $\mathbb{Z}_p$ , so there are induced maps of pro-spectra

$$\mathrm{K}(\mathbb{Z}) \longrightarrow \mathrm{K}(\mathbb{Z}_p) \longrightarrow \mathrm{K}^{\mathrm{cts}}(\mathbb{Z}_p) \longrightarrow \{\mathrm{K}(\mathbb{Z}/p^r)\}_r;$$

here and below we implicitly identify objects with constant pro-objects. The middle map is comparing the K-theory of $\mathbb{Z}_p$ considered as a discrete ring with its continuous K-theory. Denote the 1-connected coverFootnote ³ of the algebraic K-theory spectrum by $\mathrm{SK}(-) = \tau_{\gt 1}\mathrm{K}(-)$ , and write

$$\kappa : \mathrm{SK}(\mathbb{Z}) \longrightarrow \mathrm{SK}(\mathbb{Z}_p)$$

for the map induced by $\mathbb{Z} \to \mathbb{Z}_p$ , which we call the completion map. Using this, we form the diagram of pro-spaces

whose columns are fibration sequences. (Fibres, and more generally finite limits, of pro-spaces may be computed objectwise; see [Reference Artin and MazurAM69, Appendix (4.1)].) All columns but the first are fibrations of pro-(infinite loop spaces), so the coefficient systems given by the cohomology of the fibres are trivial. The same property holds for the first column by the main theorem of [Reference Calegari and EmertonCE16].

For the left-hand map of the bottom row, the natural map of pro-spaces

$$\operatorname*{hocolim}\limits_{n \to \infty} \{B\mathrm{SL}_n(\mathbb{Z}/ p^r)\}_r \longrightarrow \{ \operatorname*{hocolim}\limits_{n \to \infty} B\mathrm{SL}_n(\mathbb{Z}/ p^r)\}_r \simeq \{ B\mathrm{SL}(\mathbb{Z}/ p^r)\}_r$$

is a cohomology isomorphism, because for each r the sequence $B\mathrm{SL}_n(\mathbb{Z}/ p^r)$ enjoys homological stability with respect to n, with a stability range that is independent of r (this follows from [Reference van der KallenvdK80, Theorem 4.11]). Furthermore, the maps $B\mathrm{SL}(\mathbb{Z}/ p^r) \to \Omega^\infty \mathrm{SK}(\mathbb{Z}/p^r)$ are cohomology isomorphisms, as in the first paragraph of this proof.

For the middle map of the bottom row, the argument of [Reference CalegariCal15, Sublemma 2.18] shows that for each i the dimension of $H_i(\Omega^\infty \mathrm{SK}(\mathbb{Z}/p^r) )$ is finite and bounded independently of r. By [Reference GoerssGoe96, Theorem B], using that the homology of infinite loop spaces are abelian Hopf algebras, it follows that the map

$$H_*(\Omega^\infty \mathrm{SK}^{\mathrm{cts}}(\mathbb{Z}_p)) \longrightarrow \lim_{r \to \infty} H_*(\Omega^\infty \mathrm{SK}(\mathbb{Z}/p^r) )$$

is an isomorphism (in principle the limit is taken in the category of $\mathbb{F}_p$ -coalgebras, but the uniform boundedness means that the limit in $\mathbb{F}_p$ -modules has finite type, and so inherits a coalgebra structure and agrees with the limit in $\mathbb{F}_p$ -coalgebras). Dualising shows that this map is a $\mathbb{F}_p$ -cohomology isomorphism.

The right-hand map of the bottom row is a $\mathbb{F}_p$ -cohomology isomorphism by [Reference Hesselholt and MadsenHM97, Theorem C (iii)] applied with $k=\mathbb{F}_p$ and $A = W(k) = \mathbb{Z}_p$ . That theorem is formulated as an equivalence of p-adically complete spectra, but this yields an equivalence of their p-adically complete infinite loop spaces, and therefore an $\mathbb{F}_p$ -(co)homology equivalence between their infinite loop spaces.

For each of the maps in the top row of the diagram, we use that for each column the action of the fundamental group of the base on the cohomology of the fibre is trivial, then apply the Zeeman comparison theorem to each of the maps of Serre spectral sequences between the columns.

As $\widetilde{H}^*(\mathrm{SL}(\mathbb{Z}))$ is the cohomology of the pro-space $\operatorname*{hocolim}_{n \to \infty}\{B\mathrm{SL}_n(\mathbb{Z}, p^r)\}_r$ , the top row of the diagram provides an isomorphism

(7.3) \begin{equation}\widetilde{H}^*(\mathrm{SL}(\mathbb{Z})) \cong H^*(\mathrm{hofib}(\Omega^\infty\kappa)),\end{equation}

relating the stable completed cohomology to the cohomology of the fibre of the completion map in K-theory. In the following two sections we will explain how the latter may be determined, at least in degrees $* \lt 2p-2$ , using deep results in algebraic K-theory. We denote by $\mathrm{K}(-;\mathbb{Z}_p) := \operatorname*{holim}_{m \to \infty} \mathrm{K}(-)/p^m$ the p-adic completion of the algebraic K-theory spectrum $\mathrm{K}(-)$ , which we shall use to formulate things.

7.3 The completion map

In this section we will describe the structure of the completion map

$$\kappa : \mathrm{SK}(\mathbb{Z} ; \mathbb{Z}_p) \longrightarrow \mathrm{SK}(\mathbb{Z}_p ; \mathbb{Z}_p)$$

in stable homotopy-theoretic terms, for p an odd prime. We first recall some well-known spectra and their homotopy groups: a source for this is [Reference AdamsAda95, pp.204–206] and [Reference AdamsAda62, § 5]. We will make use of the connective real and complex topological K-theory spectra ko and ku, and their periodic versions KO and KU, all of which we implicitly complete at p. We will also make use of the (connective, p-complete) image-of-J spectrum j, defined by the fibre sequence

$$j \longrightarrow ku \overset{\psi^r-1}\longrightarrow \tau_{\gt 0} ku$$

for r an integer which topologically generates $\mathbb{Z}_p^\times$ and $\psi^r$ the corresponding Adams operation.

The homotopy groups of these spectra are as follows. We have $\pi_*(ku) = \mathbb{Z}_p[u]$ for $u \in \pi_2(ku)$ the Bott element, and $\pi_*(ko) = \mathbb{Z}_p[u^2]$ considered as a subring of $\pi_*(ku)$ via complexification $c : ko \to ku$ . The Adams operation $\psi^r$ acts on the Bott class $u \in \pi_2(ku)$ by multiplication by r, so the map $\psi^r-1 : ku \to \tau_{\gt 0}ku$ acts on homotopy groups as $u^{k} \mapsto(r^{k}-1) \cdot u^{k}$ . As $r \in \mathbb{Z}_p^\times$ is a topological generator, $(r^{k}-1)$ is a unit in $\mathbb{Z}_p$ if $p-1 \nmid k$ and is $p \cdot i$ times a unit if $k= (p-1) \cdot i$ , giving

$$\pi_*(j) = \begin{cases}\mathbb{Z}_p, & *=0,\\\mathbb{Z}_p/p \cdot i, & * = i(2p-2)-1,\\0 & \text{otherwise}.\end{cases}$$

7.3.1 K-theory of $\mathbb{Z}_p$ .

By the main theorem of [Reference Bökstedt and MadsenBM95] there is an equivalence

(7.4) \begin{equation}\mathrm{K}(\mathbb{Z}_p ; \mathbb{Z}_p) \simeq j \oplus \Sigma j \oplus \Sigma^3 ku.\end{equation}

At the level of homotopy groups this gives

$$SK_*(\mathbb{Z}_p ; \mathbb{Z}_p) = \begin{cases}\mathbb{Z}/p, & *=2p-3,\\0 & \text{otherwise}\end{cases} \oplus \begin{cases}\mathbb{Z}_p, & *=3,5,7,9,\ldots,\\0 & \text{otherwise}\end{cases}\quad\text{for } * \lt 2p-2.$$

Note that the contributions of $\pi_0(j)=\mathbb{Z}_p$ and $\pi_1(\Sigma j) = \mathbb{Z}_p$ to (7.4) are removed by passing to the 1-connected cover $\mathrm{SK}(\mathbb{Z}_p;\mathbb{Z}_p)$ .

7.3.2 K-theory of $\mathbb{Z}$ and the completion map.

The structure of $K_*(\mathbb{Z};\mathbb{Z}_p)$ (at irregular primes) is far more complicated. Following Weibel’s survey [Reference WeibelWei05], we have

$$SK_*(\mathbb{Z} ; \mathbb{Z}_p) \cong \begin{cases}\text{order } |\mathbb{Z}_p/\mathrm{Num}(B_{2k}/4k)|, & *=4k-2,\\0, & *=4k-1,\\?, & *=4k,\\\mathbb{Z}_p, & * = 4k+1\\\end{cases}\quad\text{for } * \lt 2(2p-3)$$

where $B_{2k}$ denotes the 2kth Bernoulli number, and $?$ is unknown but finite. If the Vandiver conjecture holds for p, then in degrees $\equiv 0 \mod 4$ these groups vanish and in degrees $\equiv 2\mod 4$ they are cyclic. The Vandiver conjecture has been checked for $p \leq 2^{31}$ (cf. [Reference Hart, Harvey and OngHHO17]).

This can be extracted from [Reference WeibelWei05] as follows. In degrees $\equiv 1,3 \mod 4$ it follows from Theorem A and the description of $w_i(\mathbb{Q})$ in Lemma 27; in degrees $\equiv 2 \mod 4$ it follows from Corollary 95; in degrees $\equiv 0 \mod 4$ it follows from Theorem 6; the claim involving the Vandiver conjecture follows from Corollary 107.

We wish to show that the truncations

$$\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}; \mathbb{Z}_p) = \tau_{\leq 2p-3} \tau_{\geq 2} \mathrm{K}(\mathbb{Z}; \mathbb{Z}_p) = \tau_{\leq 2p-3}\mathrm{SK}(\mathbb{Z}; \mathbb{Z}_p)$$

and $\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}_p; \mathbb{Z}_p)$ are coproducts of Eilenberg–MacLane spectra, and thereby understand to some extent the map induced by $\kappa$ between these truncations by understanding its effect on homotopy groups. We will use the following lemma to control stable homotopy classes of maps between Eilenberg–MacLane spectra. For $\mathbb{Z}$ -modules A and B, the universal coefficient theorem identifies [HA, HB] with $\mathrm{Hom}_{\mathbb{Z}}(A,B)$ , and $[HA, \Sigma HB]$ with $\mathrm{Ext}^1_{\mathbb{Z}}(A, B)$ , functorially in A and B.

Lemma 7.2. If A and B are finitely generated $\mathbb{Z}_p$ -modules, then $[HA, \Sigma^i HB]=0$ for $1 \lt i \lt 2p-2$ .

Proof. Recall that $[H\mathbb{Z}/p, \Sigma^*H\mathbb{Z}/p]$ is by definition the $\mathbb{Z}/p$ -Steenrod algebra $\mathcal{A}_p$ , and (for p odd) this is generated under composition by the Bockstein map $\beta : H\mathbb{Z}/p \to \Sigma H\mathbb{Z}/p$ as well as operations $\mathcal{P}^n : H\mathbb{Z}/p \to \Sigma^{2n(p-1)} H\mathbb{Z}/p$ for $n \geq 1$ , subject to certain relations which need not concern us. Recall that there is a cofibre sequence $H\mathbb{Z}_p \overset{p}\to H\mathbb{Z}_p \overset{\rho}\to H\mathbb{Z}/p \overset{\partial}\to \Sigma H\mathbb{Z}_p$ and the Bockstein is defined to be $\beta := (\Sigma\rho) \circ \partial$ . There is a commutative diagram

where the bottom row is exact and is given by applying $[-, \Sigma^* H\mathbb{Z}/p]$ to this cofibre sequence, and the vertical map is a (degree-shifted) copy of the horizontal map $- \circ \rho$ . The bottom row shows that $- \circ \rho$ is surjective, so it follows that the image of $- \circ \partial$ in $\mathcal{A}_p = {[H\mathbb{Z}/p, \Sigma^* H\mathbb{Z}/p]}$ is precisely the left $\mathcal{A}_p$ -submodule generated by $\beta$ , and therefore identifies $[H\mathbb{Z}_p, \Sigma^* H\mathbb{Z}/p] \cong \mathcal{A}_p / \mathcal{A}_p \beta$ . By the description of the Steenrod algebra this vanishes in degrees $0 \lt * \lt 2p-2$ .

Any finitely generated $\mathbb{Z}_p$ -module is a finite sum of $\mathbb{Z}_p$ and $\mathbb{Z}/p^r$ . Applying $[-, \Sigma^*H\mathbb{Z}/p]$ to the cofibre sequence $H\mathbb{Z}_p \overset{p^r}\to H\mathbb{Z}_p \to H\mathbb{Z}/p^r \to \Sigma H\mathbb{Z}_p$ gives a short exact sequence

$$0 \longleftarrow {[H\mathbb{Z}_p, \Sigma^* H\mathbb{Z}/p]} \longleftarrow {[H\mathbb{Z}/p^r, \Sigma^* H\mathbb{Z}/p]} \longleftarrow {[ H\mathbb{Z}_p, \Sigma^{*-1} H\mathbb{Z}/p]} \longleftarrow 0 $$

so using $[H\mathbb{Z}_p, \Sigma^* H\mathbb{Z}/p] \cong\mathcal{A}_p / \mathcal{A}_p \beta$ , which vanishes in degrees $0 \lt * \lt 2p-2$ , we see that $[H\mathbb{Z}/p^r, \Sigma^* H\mathbb{Z}/p]=0$ for $1 \lt * \lt 2p-2$ . This shows that the claim holds for $B=\mathbb{Z}/p$ and for all finitely generated $\mathbb{Z}_p$ -modules A. Applying $[HA, \Sigma^*-]$ to the cofibre sequences $H\mathbb{Z}/p^{r-1} \to H\mathbb{Z}/p^r \to H\mathbb{Z}/p \to \Sigma H\mathbb{Z}/p^{r-1}$ , then shows, by induction on r, that the claim holds for $B=\mathbb{Z}/p^r$ and for all finitely generated $\mathbb{Z}_p$ -modules A.

Finally, writing $H\mathbb{Z}_p = \operatorname*{holim}\limits_r H\mathbb{Z}/p^r$ and applying $[HA, \Sigma^*-]$ to it gives a Milnor sequence

$$0 \longrightarrow \lim_{r \to \infty}{}^1 [HA, \Sigma^{i-1} H\mathbb{Z}/p^r] \longrightarrow [HA, \Sigma^i H\mathbb{Z}_p] \longrightarrow \lim_{r \to \infty} [HA, \Sigma^i H\mathbb{Z}/p^r] \longrightarrow 0.$$

The outer terms vanish for $i \gt 2$ . For $i=2$ the right-hand term vanishes, and the left-hand term is $\lim{_{r \to \infty}^1} \mathrm{Ext}^1_{\mathbb{Z}}(A, \mathbb{Z}/p^r)$ . As $\mathrm{Ext}^2_{\mathbb{Z}}(-,-)$ vanishes identically, $\mathrm{Ext}^1_{\mathbb{Z}}(A, -)$ preserves epimorphisms and so $\{\mathrm{Ext}^1_{\mathbb{Z}}(A, \mathbb{Z}/p^r)\}_r$ is an inverse system of epimorphisms and hence has vanishing $\lim^1$ by the Mittag–Leffler condition.

We use this lemma to show that $\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}; \mathbb{Z}_p)$ and $\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}_p; \mathbb{Z}_p)$ are coproducts of Eilenberg–MacLane spectra, by showing that their Postnikov towers must split: assuming a splitting $\tau_{[2, i]} \mathrm{K}(\mathbb{Z}; \mathbb{Z}_p) \simeq \bigoplus_{j=2}^i \Sigma^j HK_j(\mathbb{Z};\mathbb{Z}_p)$ has been chosen, there is a pullback

but the lower map is nullhomotopic as long as $i \lt 2p-2$ by the lemma. However, this splitting is not completely canonical: when $i=4k$ the nullhomotopy of the lower map may not be unique (though it is if the Vandiver conjecture holds). The analogous discussion goes through for $\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}_p; \mathbb{Z}_p)$ , though in this case the splitting is canonical.

Using the lemma again, it follows that the map $\kappa$ on truncations is determined by its components

\begin{align*}\kappa_{4k-2} &: \Sigma^{4k-2} HK_{4k-2}(\mathbb{Z};\mathbb{Z}_p) \longrightarrow \Sigma^{4k-1}HK_{4k-1}(\mathbb{Z}_p;\mathbb{Z}_p),\\\kappa_{4k} &: \Sigma^{4k} HK_{4k}(\mathbb{Z};\mathbb{Z}_p) \longrightarrow \Sigma^{4k+1}HK_{4k+1}(\mathbb{Z}_p;\mathbb{Z}_p),\\\kappa_{4k+1} &: \Sigma^{4k+1} HK_{4k+1}(\mathbb{Z};\mathbb{Z}_p)\longrightarrow \Sigma^{4k+1} HK_{4k+1}(\mathbb{Z}_p;\mathbb{Z}_p).\end{align*}

7.3.3 The completion map at regular odd primes.

If p is a regular odd prime then the completion map may be completely described at the level of spectra, improving upon the description in the previous section. Our reference for the following is [Reference RognesRog03, § 3]. It follows from the (affirmed) Quillen–Lichtenbaum conjecture that there is an equivalence $\mathrm{K}(\mathbb{Z}; \mathbb{Z}_p) \simeq j \oplus \Sigma^5 ko$ ; similarly, (7.4) gives an equivalence $\mathrm{K}(\mathbb{Z}_p ; \mathbb{Z}_p) \simeq j \oplus \Sigma j \oplus \Sigma^3 ku$ .

Lemma 7.3 (Rognes). Under these equivalences the completion map is the identity on the j-summand and on the $\Sigma^5 ko$ -summand is the map $\chi: \Sigma^5 ko \to\Sigma^3 ku$ induced by the suspension of the complexification map $\Sigma c : \Sigma ko \to \Sigma ku$ by taking 1-connected covers.

Proof sketch. Following [Reference RognesRog03, § 3], we develop the diagram

where the right-hand square is as indicated by [Reference RognesRog03, Proposition 3.1], the horizontal maps in the middle square are equivalences by the (affirmed) Quillen–Lichtenbaum conjecture in the two cases, and the left-hand square is cartesian by the localisation sequence in K-theory. More precisely, the map of horizontal fibres in the left-hand square is identified with $\mathrm{K}(\mathbb{Z}/p ; \mathbb{Z}_p) \overset{\sim}\to \mathrm{K}(\mathbb{Z}_p/p ; \mathbb{Z}_p) \simeq H\mathbb{Z}_p$ . In each row this copy of $H\mathbb{Z}_p$ cancels against the lowest homotopy group of $\Sigma ko$ or $\Sigma ku$ (by considering the left-hand square on $\pi_1$ ), giving the claimed description of the completion map in terms of the lift of $\Sigma c : \Sigma ko \to \Sigma ku$ to 1-connected covers (see first line of the proof of [Reference RognesRog03, Theorem 3.8]).

Taking the suspension of the Wood cofibre sequence $\Sigma KO \overset{\eta}\to KO \overset{c}\to KU$ and then taking 1-connective covers gives a cofibre sequence $\Sigma^2 ko \to \Sigma^5 ko \overset{\chi}\to \Sigma^3 ku$ . Taking 1-connected covers of the maps in the lemma, we see that $\kappa$ is identified with $\mathrm{Id}_{\tau_{\gt 1} j} \oplus \chi : \tau_{\gt 1}j \oplus \Sigma^5 ko \to \tau_{\gt 1}j \oplus \tau_{\gt 1} \Sigma j \oplus \Sigma^3 ku$ , giving an equivalence

\begin{equation*}\mathrm{hofib}(\kappa) \simeq \tau_{\gt 0} j \oplus \Sigma^2 ko.\end{equation*}

7.4 Transgressive fibrations

The discussion so far gives an analysis of the stable homotopy types of $\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}; \mathbb{Z}_p)$ and $\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}_p; \mathbb{Z}_p)$ , and of the map between them: we now wish to take their associated infinite loop spaces in order to describe the behaviour of the Serre spectral sequence associated to the fibration

$$\Omega^\infty\kappa : \Omega^\infty \mathrm{SK}(\mathbb{Z}; \mathbb{Z}_p) \longrightarrow \Omega^\infty \mathrm{SK}(\mathbb{Z}_p; \mathbb{Z}_p)$$

in degrees $* \lt 2p-2$ .

Let us say that a fibration $\pi: E \to B$ with 0-connected fibre F is transgressive (with $\mathbb{F}_p$ -coefficients) if $\pi_1(B)$ acts trivially on $H^*(F;\mathbb{F}_p)$ , and if $H^*(F;\mathbb{F}_p)$ is freely generated as a graded-commutative $\mathbb{F}_p$ -algebra by a set of classes which are transgressive in the Serre spectral sequence for $\pi$ . We say $\pi$ is transgressive in degrees $*\lt N$ if the above two conditions hold in this range of cohomological degrees. The class of such fibrations is closed under forming pullbacks, and (by the Künneth theorem) is closed under forming products of fibrations. The following lemma gives a class of examples.

1. (i) $\pi: E \to B$ a principal K(A,n)-fibration, for A a finitely generated $\mathbb{Z}_p$ -module;

2. (ii) $\alpha : K(A, n) \to K(B, n+1)$ for A and B finitely generated $\mathbb{Z}_p$ -modules, and $\alpha \in H^{n+1}(K(A,n) ; B) \cong \mathrm{Ext}^1_{\mathbb{Z}}(A, B) \cong [HA, \Sigma HB]$ ;

3. (iii) $N\cdot \mathrm{Id} + \alpha : K(\mathbb{Z}_p, n+1)\times K(A, n) \to K(\mathbb{Z}_p, n+1)$ for $N \in \mathbb{Z}_p$ , A a finitely generated $\mathbb{Z}_p$ -module, and $\alpha \in H^{n+1}(K(A,n) ; \mathbb{Z}_p) \cong \mathrm{Ext}^1_{\mathbb{Z}}(A, \mathbb{Z}_p) \cong [HA, \Sigma H\mathbb{Z}_p]$ .

For (ii), this fibration arises from delooping the extension $0 \to B \to A' \to A \to 0$ classified by $\alpha$ , so is a principal K(A’, n)-fibration and so is transgressive by (i).

For (iii), if $N=0$ then this fibration is the product of the fibrations $K(\mathbb{Z}_p, n+1) \to *$ and $\alpha : K(A, n) \to K(\mathbb{Z}_p, n+1)$ . The first is clearly transgressive, and the second is too by (ii). If $N \neq 0$ then the homotopy groups of the fibre F of this fibration sit in an exact sequence

$$ \cdots 0 \longrightarrow \pi_{n+1}(F) \longrightarrow \mathbb{Z}_p \overset{N}\longrightarrow \mathbb{Z}_p \longrightarrow \pi_n(F) \longrightarrow A \longrightarrow 0 \longrightarrow \pi_{n-1}(F) \longrightarrow 0 \cdots$$

so $F \simeq K(A', n)$ for the extension $0 \to \mathbb{Z}_p/N \to A' \to A \to 0$ determined by $\alpha$ and reduction modulo N. Furthermore, the fibration is principal as both maps Id and $\alpha$ deloop, so (i) applies.

Corollary 7.5. The fibration

$$\Omega^\infty\tau_{[2, 2p-3]}\kappa : \Omega^\infty\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}; \mathbb{Z}_p) \longrightarrow \Omega^\infty\tau_{[2, 2p-3]} \mathrm{K}(\mathbb{Z}_p; \mathbb{Z}_p)$$

is transgressive in degrees $* \lt 2p-2$ .

Proof. By the discussion in § 7.3 it suffices to show that each of the fibrations

\begin{align*} K(K_{4k-2}(\mathbb{Z};\mathbb{Z}_p), 4k-2) &\xrightarrow{\Omega^\infty \kappa_{4k-2} } K(K_{4k-1}(\mathbb{Z}_p;\mathbb{Z}_p), 4k-1),\\ K(K_{4k}(\mathbb{Z};\mathbb{Z}_p), 4k) \times K(K_{4k+1}(\mathbb{Z};\mathbb{Z}_p), 4k+1) &\xrightarrow{\Omega^\infty (\kappa_{4k} + \kappa_{4k+1})} K(K_{4k+1}(\mathbb{Z}_p;\mathbb{Z}_p), 4k+1)\end{align*}

is transgressive. Using that $K_{4k- 1}(\mathbb{Z}_p;\mathbb{Z}_p)$ is $\mathbb{Z}_p$ or $\mathbb{Z}/p \oplus \mathbb{Z}_p$ , Lemma 7.4(ii) applies to the first; using that $K_{4k+ 1}(\mathbb{Z}_p;\mathbb{Z}_p) \cong \mathbb{Z}_p$ , Lemma 7.4(iii) applies to the second.

Using Lemma 7.1 and the discussion surrounding it, this translates into the following statement.

Corollary 7.6. For all large enough n, in degrees $* \lt 2p-2$ the group $\mathrm{SL}_n(\mathbb{Z}_p)$ acts trivially on $\widetilde{H}^*(\mathrm{SL}_n(\mathbb{Z}))$ , and the latter is freely generated as a graded-commutative $\mathbb{F}_p$ -algebra by elements $\{x_\alpha\}_{\alpha \in I}$ which are transgressive in the spectral sequence

\begin{equation*}E_2^{s,t} = H^s_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) ; \widetilde{H}^t(\mathrm{SL}_n(\mathbb{Z}) )) \Longrightarrow H^{s+t}(\mathrm{SL}_n(\mathbb{Z})),\end{equation*}

given by (7.2) with $k=0$ .

7.4.1 The spectral sequence at regular odd primes.

When p is a regular odd prime, the discussion in § 7.3.3 gives an explicit description of this spectral sequence. Limiting our interest to the slightly smaller range of degrees $* \lt 2p-3$ (as we will later limit ourselves to $* \lt p-1$ anyway) and using that $\tau_{[1,2p-4]}j = 0$ by the description of the homotopy groups of j at the beginning of § 7.3, we see that in this range of degrees the right-hand column of the large diagram in § 7.2 may be identified with the fibre sequence

\begin{equation*}\Omega^{\infty} \Sigma^2 ko \longrightarrow \Omega^\infty \Sigma^5 ko \longrightarrow \Omega^\infty \Sigma^3 ku.\end{equation*}

The cohomology of these spaces is well-known, and can be extracted from [Reference Dyer and LashofDL61], for example. To do so one should express these spaces in terms of their usual names from real and complex Bott periodicity: the fibre is $SO/U$ , the base is the universal cover of U (i.e., SU), and the total space is the universal cover of $U/Sp$ . In degrees $* \lt 2p-3$ we obtain isomorphisms

The first row of isomorphisms calculates the completed cohomology $\widetilde{H}^*(\mathrm{SL}(\mathbb{Z}))$ in degrees $* \lt 2p-3$ , and so justifies the deduction of Corollary B from Theorem A. The remaining isomorphisms show, following [Reference Dyer and LashofDL61], that in this range the Serre spectral sequence for the right-hand column in the large diagram in § 7.2 has the form

$$\Lambda^*_{\mathbb{F}_p}[ y_3, y_5, y_7, \ldots] \otimes \mathbb{F}_p[x_2, x_6, x_{10}, \ldots] \Longrightarrow \Lambda^*_{\mathbb{F}_p}[ y_5, y_9, y_{13}, \ldots]$$

with generating differentials $d_{4k+3}(x_{4k+2}) = y_{4k+3}$ , for an appropriate choice of generators $x_i$ and $y_i$ . Using the large diagram in § 7.2, this describes the spectral sequence (7.2) for $k=0$ , in degrees $* \lt 2p-3$ and for all large enough n.

7.5 Evaluating completed cohomology at irregular primes

To justify the examples from the introduction, we need to calculate $H^*(\mathrm{hofib}(\Omega^\infty \kappa))$ , and in degrees $* \leq 2p-3$ this can be formally obtained from $\pi_*(\mathrm{hofib}(\kappa))$ . The recent work of Blumberg and Mandell [Reference Blumberg and MandellBM20] expresses these homotopy groups in terms of étale cohomology of $\mathbb{Z}[{1}/{p}]$ , but we shall proceed in a more down-to-earth way.

We are only interested in degrees $* \lt p-1$ , and in this range the long exact sequence on homotopy groups for the map $\kappa$ has the form

so if p satisfies the Vandiver conjecture (e.g., if $p \leq 2^{31}$ ) so that $K_{4k}(\mathbb{Z};\mathbb{Z}_p)=0$ and $K_{4k-2}(\mathbb{Z};\mathbb{Z}_p) \cong \mathbb{Z}_p / \mathrm{Num}(B_{2k}/4k)$ , then $\pi_*(\mathrm{hofib}(\kappa))$ is determined by the maps $\kappa_{4k+1} : \mathbb{Z}_p \cong K_{4k+1}(\mathbb{Z};\mathbb{Z}_p) \to K_{4k+1}(\mathbb{Z}_p;\mathbb{Z}_p) \cong \mathbb{Z}_p$ and extensions $0 \to \mathbb{Z}_p \to \pi_{4k-2}(\mathrm{hofib}(\kappa)) \to \mathbb{Z}_p / \mathrm{Num}(B_{2k}/4k) \to 0$ . We do not know how to control these extensions, though [Reference Blumberg and MandellBM20] gives the expression $\pi_{4k-2}(\mathrm{hofib}(\kappa)) \cong H^1_{\mathrm{\unicode{x00E9}t}}(\mathbb{Z}[{1}/{p}] ; \mathbb{Z}/p^\infty(1-2k))^*$ for it which may be useful to some readers. Instead we shall restrict ourselves to work in the range of degrees where $\mathrm{Num}(B_{2k}/4k)$ are p-adic units, that is, where $K_*(\mathbb{Z};\mathbb{Z}_p)$ is torsion-free, in which case we just need to understand the maps $\kappa_{4k+1}$ . The map $\kappa_{4k+1}$ is given by multiplication by the value $L_p(1+2k, \omega^{-2k})$ of the p-adic L-function, up to a p-adic unit; see [Reference HesselholtHes18] for a concise discussion.

We may use this as follows. The $\mathbb{F}_p$ -cohomology of the fibre of

$$\Omega^\infty\kappa_{4k+1} : K(K_{4k+1}(\mathbb{Z};\mathbb{Z}_p), 4k+1) \longrightarrow K(K_{4k+1}(\mathbb{Z}_p;\mathbb{Z}_p), 4k+1)$$

in degrees $* \lt 4k+1 + 2p-2$ depends only on whether $L_p(1+2k, \omega^{-2k})$ is a p-adic unit: if so then the fibre has trivial $\mathbb{F}_p$ -cohomology, and if not then the fibre has $\mathbb{F}_p$ -cohomology $\mathbb{F}_p[y_{4k}] \otimes \Lambda^*_{\mathbb{F}_p}[y_{4k+1}]$ in this range.

By [Reference CohenCoh07, Proposition 11.3.12(1)] we have

$$2k \cdot L_p(1+2k, \omega^{-2k}) = \lim_{r \to \infty} B_{\phi(p^r)-2k},$$

where the latter denote Bernoulli numbers. Assuming that $4k+1 \lt p$ , so certainly $2k \lt p-1$ , then as $\phi(p^r)-2k \equiv p-1-2k \mod p-1$ we have the Kummer congruences $({B_{\phi(p^r)-2k}})/({\phi(p^r)-2k}) \equiv ({B_{p-1-2k}})/({p-1-2k}) \mod p$ and so

\begin{equation*}L_p(1+2k, \omega^{-2k})\ \dot{\equiv}\ B_{p-1-2k} \mod p,\end{equation*}

where $\dot{\equiv}$ denotes congruence up to a p-adic unit. Thus, we may determine whether or not $L_p(1+2k, \omega^{-2k})$ is a p-adic unit by calculating this residue class.

Let us now treat the three primes 37, 16,843 and 2,124,679 from the introduction. Certainly they all satisfy the Vandiver conjecture. For $p=37$ we find that $B_{p-1-2k} \not \equiv 0 \mod p$ for all $1+2k \lt p$ except for $1+2k=5$ . This Bernoulli number contributes to torsion in $K_{62}(\mathbb{Z} ; \mathbb{Z}_p)$ , so in degrees $* \lt 36$ there is no torsion in $K_*(\mathbb{Z}; \mathbb{Z}_p)$ . It follows that

$$H^*(\mathrm{hofib}(\Omega^\infty \kappa);\mathbb{F}_p) \cong \mathbb{F}_p[x_2, x_6, x_{10}, \ldots] \otimes \mathbb{F}_p[y_8] \otimes \Lambda^*_{\mathbb{F}_p}[y_9]$$

in degrees $* \lt 36$ . With Theorem A this justifies the first example in the introduction.

For $p=16{,}843$ we find that $B_{p-1-2k} \not \equiv 0 \mod p$ for all $1+2k \lt p$ except for $1+2k=3$ . This Bernoulli number contributes to torsion in $K_{33{,}678}(\mathbb{Z} ; \mathbb{Z}_p)$ , so in degrees $* \lt 16{,}842$ there is no torsion in $K_*(\mathbb{Z}; \mathbb{Z}_p)$ . This gives the second example in the introduction.

For $p=2{,}124{,}679$ we find that $B_{p-1-2k} \not \equiv 0 \mod p$ for all $1+2k \lt p$ except for $1+2k=3$ and $1+2k = 1{,}422{,}781$ . The first Bernoulli number contributes to torsion in $K_{4{,}249{,}350}(\mathbb{Z};\mathbb{Z}_p)$ , and the second contributes to torsion in $K_{1{,}403{,}794}(\mathbb{Z};\mathbb{Z}_p)$ , so in degrees $* \lt 1{,}403{,}794$ there is no torsion in $K_*(\mathbb{Z}; \mathbb{Z}_p)$ . This gives the third example in the introduction.

7.6 Cohomology of p-adic analytic groups

For $m \geq 1$ and p odd the groups $\mathrm{SL}_n(\mathbb{Z}_p, p^m)$ are uniform ( $=$ uniformly powerful) pro-p-groups [Reference Dixon, du Sautoy, Mann and SegalDdSM⁺99, Theorem 5.2] and hence their continuous $\mathbb{F}_p$ -cohomology may be described as the exterior algebra on their first $\mathbb{F}_p$ -cohomology (cf. [Reference LazardLaz65, V.2.2.7.2],Footnote ⁴ [Reference Symonds and WeigelSW00, Theorem 5.1.5]). On the other hand, the map

$$I + p^m A \mapsto A \mod p : \mathrm{SL}_n(\mathbb{Z}_p, p^m) \longrightarrow sl_n(\mathbb{F}_p)$$

induces an isomorphism $\mathrm{SL}_n(\mathbb{Z}_p,p^m)/\mathrm{SL}_n(\mathbb{Z}_p, p^{m+1}) \overset{\sim}\to sl_n(\mathbb{F}_p)$ and this is the maximal p-elementary abelian quotient for $n \geq 2$ by [Reference Dixon, du Sautoy, Mann and SegalDdSM⁺99, Lemma 5.1]. Thus, there is an identification $H^1(\mathrm{SL}_n(\mathbb{Z}_p, p^m);\mathbb{F}_p) \cong sl_n(\mathbb{F}_p)^\vee$ , from which we obtain the following proposition.

Proposition 7.7. The induced map

$$\Lambda_{\mathbb{F}_p}^*[sl_n(\mathbb{F}_p)^\vee] \longrightarrow H^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p^m);\mathbb{F}_p)$$

is an isomorphism.

8.1 Proof of Theorem 8.1 for $m\gt 1$

The proof in this case is essentially trivial. There is a factorisation

$$H^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) ) \longrightarrow H^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p) ) \longrightarrow H^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p^m) )$$

of the restriction map and by the discussion in § 7.6 we know the latter two cohomology rings: both are given by $\Lambda_{\mathbb{F}_p}^*[sl_n(\mathbb{F}_p)^\vee]$ . However, the composition

$$\mathrm{SL}_n(\mathbb{Z}_p, p^m) \longrightarrow \mathrm{SL}_n(\mathbb{Z}_p, p) \longrightarrow \mathrm{SL}_n(\mathbb{Z}_p, p)/\mathrm{SL}_n(\mathbb{Z}_p, p^2) = sl_n(\mathbb{F}_p)$$

is trivial for $m\gt 1$ , so the second map in the factorisation of the restriction map is trivial in degrees $*\gt 0$ , and so the restriction map is too.

8.2 Proof of Theorem 8.1 for $m=1$

The fibration of pro-spaces

(8.1) \begin{equation}\{B\mathrm{SL}_n(\mathbb{Z}/ p^{r+1}, p)\}_r \longrightarrow \{B\mathrm{SL}_n(\mathbb{Z}/ p^r)\}_r \longrightarrow B\mathrm{SL}_n(\mathbb{Z}/p)\end{equation}

yields a spectral sequence

$$^{I\!I\!I}\!E^{s,t}_2 = H^s(\mathrm{SL}_n(\mathbb{Z}/p) ; {H}^t_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p) )) \Longrightarrow H^{s+t}_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) ),$$

and the map of Theorem 8.1 factors as

$$H^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) ) \overset{\text{edge hom.}}\longrightarrow H^0(\mathrm{SL}_n(\mathbb{Z}/p) ; {H}^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p) )) \overset{\text{inc.}}\longrightarrow H^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p)),$$

so we must show that this edge homomorphism is trivial for $0 \lt * \lt p$ . The following proposition describes the $d_2$ -differential on this edge.

Proposition 8.3. In degrees $* \lt p$ and for all large enough n the differential

$$d_2 : {^{I\!I\!I}\!E}^{0,*}_2 \longrightarrow {^{I\!I\!I}\!E}^{2,*-1}_2 $$

is given by

(8.2) \begin{align}\begin{split}\Lambda^*_{\mathbb{F}_p}[c_3, c_5, c_7, \ldots] &\longrightarrow\mathbb{F}_p\{e_1, e_2, e_3, \ldots\} \otimes \Lambda^*_{\mathbb{F}_p}[c_3, c_5, c_7, \ldots],\\c_t &\longmapsto t \cdot e_{t-1} \otimes 1\end{split}\end{align}

and the Leibniz rule, where $c_t$ and $e_t$ have degree t.

We defer the (quite involved) proof of this proposition to § 9.

To finish the proof of Theorem 8.1 in the case $m=1$ , we claim that the map (8.2) is injective in degrees $0 \lt * \lt p$ . To see this, suppose that $x \in \Lambda^*_{\mathbb{F}_p}[c_3, c_5, c_7, \ldots]$ has $d_2(x)=0$ and $0\lt |x|\lt p$ . Then, for each $i \in \{3,5,7,\ldots\}$ with $i\lt p$ , write $x = A_i + c_i \cdot B_i$ with $A_i$ and $B_i$ not containing $c_i$ , so that

\begin{align*}0 = d_2(x) &= d_2(A_i+c_i \cdot B_i)\\&= d_2(A_i) + i \cdot e_{i-1} \otimes B_i - c_i \cdot d_2(B_i).\end{align*}

As $A_i$ and $B_i$ do not contain $c_i$ , $d_2(A_i)$ and $d_2(B_i)$ do not contain terms of the form $e_{i-1} \otimes ?$ and so cannot cancel with the middle term: thus $i \cdot B_i = 0$ , and as $i \lt p$ it follows that $B_i=0$ . Thus, $c_i$ does not occur in x, but this goes for all i, so $x=0$ . Now, by Proposition 8.3 this means that the differential $d_2 : {^{I\!I\!I}\!E}^{0,*}_2 \to {^{I\!I\!I}\!E}^{2,*-1}_2$ is injective for $0 \lt * \lt p$ , and hence that the edge homomorphism

$$H^{*}_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) ) \longrightarrow H^0(\mathrm{SL}(\mathbb{Z}/p) ; {H}^*_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p) ))$$

is trivial in degrees $0 \lt * \lt p$ , proving Theorem 8.1.

8.3 Resolving extensions

By Corollary 8.2 the lowest bidegree in which there could be a differential in the second spectral sequence is $d_p : {^{I\!I\!}}E_p^{0,p-1} \to {^{I\!I\!}}E_p^{p,0}$ . It follows that the associated graded of the Hochschild–Leray–Serre filtration on $H^{*}(\mathrm{SL}_n(\mathbb{Z}, p^m))$ satisfies

(8.3) \begin{equation}\mathrm{Gr}^\bullet H^{*}(\mathrm{SL}_n(\mathbb{Z}, p^m) ) \cong \Lambda^\bullet_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee] \otimes \widetilde{H}^{*-\bullet}(\mathrm{SL}_n(\mathbb{Z}) )\end{equation}

in degrees $* \lt p-1$ , as bigraded $\mathbb{F}_p$ -algebras and as $\mathrm{SL}(\mathbb{Z}/p^m)$ -representations. To prove Theorem A we must show that $H^{*}(\mathrm{SL}_n(\mathbb{Z}, p^m)) \cong \bigoplus_s \mathrm{Gr}^s H^{*}(\mathrm{SL}_n(\mathbb{Z}, p^m))$ , as graded $\mathbb{F}_p$ -algebras and $\mathrm{SL}_n(\mathbb{Z}/p^m)$ -representations, in degrees $* \lt p-1$ . Using the multiplicative structure, to do so it suffices to show that the quotient map

$$H^{*}(\mathrm{SL}_n(\mathbb{Z}, p^m)) \longrightarrow \mathrm{Gr}^0 H^{*}(\mathrm{SL}_n(\mathbb{Z}, p^m)) = \widetilde{H}^{*}(\mathrm{SL}_n(\mathbb{Z}))$$

is split as a map of $\mathbb{F}_p$ -algebras and $\mathrm{SL}_n(\mathbb{Z}/p^m)$ -representations, in degrees $* \lt p-1$ .

As $\widetilde{H}^{*}(\mathrm{SL}_n(\mathbb{Z}))$ is a free graded-commutative $\mathbb{F}_p$ -algebra on certain classes $\{x_\alpha\}_{\alpha \in I}$ , by part of Corollary 7.6, it suffices to show that for each of the free generators $x_\alpha \in \widetilde{H}^*(\mathrm{SL}_n(\mathbb{Z}) )$ of degree $|x_\alpha| \lt p-1$ there exists an $\mathrm{SL}_n(\mathbb{Z}/p^m)$ -invariant element $\bar{x}_{\alpha} \in H^{*}(\mathrm{SL}_n(\mathbb{Z}, p^m) )$ which restricts to

$$x_{\alpha} \in \frac{H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p^m))}{F^{|x_\alpha|-1} H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p^m))} = H^0_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p^m) ; \widetilde{H}^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}) )).$$

By pulling back along the natural map $\mathrm{SL}_n(\mathbb{Z}_p, p^m) \to \mathrm{SL}_n(\mathbb{Z}_p, p)$ , it suffices to produce such $\bar{x}_{\alpha}$ in the case $m=1$ .

To do so, consider the filtration of the $\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]$ -module $H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p) )$ , whose associated graded is

$$\mathrm{Gr}^t H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p) ) \cong \Lambda^t_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee] \otimes \widetilde{H}^{|x_\alpha|-t}(\mathrm{SL}_n(\mathbb{Z}) ),$$

where the second factor has trivial $\mathrm{SL}_n(\mathbb{Z}/p)$ -action. The spectral sequence of this filtered module takes the form

$$^{I{\kern-1pt}V{\kern-3pt}}E^{s,t}_1 = H^s(\mathrm{SL}_n(\mathbb{Z}/p) ; \mathrm{Gr}^t H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p) )) \Longrightarrow H^{s}(\mathrm{SL}_n(\mathbb{Z}/p);H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p) ))$$

$$d_r : {^{I{\kern-1pt}V{\kern-3pt}}E^{s,t}_r} \longrightarrow {^{I{\kern-1pt}V{\kern-3pt}}E^{s+1,t+r}_r}.$$

It follows that for $|x_\alpha| \lt p-1$ the class $x_{\alpha} \in H^0(\mathrm{SL}_n(\mathbb{Z}/p) ; \mathrm{Gr}^{0} H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p) )) = {^{I{\kern-1pt}V{\kern-3pt}}E^{0,0}_1}$ is a permanent cycle in this spectral sequence, so that we may find a $\bar{x}_{\alpha} \in H^{0}(\mathrm{SL}_n(\mathbb{Z}/p);H^{|x_\alpha|}(\mathrm{SL}_n(\mathbb{Z}, p) ))$ restricting to $x_\alpha$ , as required.

Firstly the argument of § 8.1 only used that $0 \lt *$ , giving (8.3) in degrees $* \lt 2p-2$ for $m\gt 1$ and all large enough n. The right-hand side of (8.3) is a free graded-commutative algebra in degrees $* \lt 2p-2$ , so the multiplicative extensions are trivial. However, triviality of the extensions as $\mathrm{SL}_n(\mathbb{Z}/p^m)$ -representations was obtained by comparison with the case $m=1$ , so we only know them for $* \lt p-1$ .

Secondly, in the case $m=1$ , Corollary 8.2 gives the identity (8.3) not only in degrees $* \lt p-1$ but also for $*=p-1$ and $\bullet \gt 0$ . Furthermore, it gives an exact sequence

$$0 \to \mathrm{Gr}^0 H^{p-1}(\mathrm{SL}_n(\mathbb{Z}, p)) \to {^{I\!I\!}}E_p^{0,p-1} = \widetilde{H}^{p-1}(\mathrm{SL}_n(\mathbb{Z}) ) \overset{d_p}\to {^{I\!I\!}}E_p^{p,0} = \Lambda^p_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee].$$

Combined with the same line of reasoning as above, this can be used to also obtain an injection

$$H^{p-1}(\mathrm{SL}_n(\mathbb{Z}, p)) \longrightarrow \bigoplus_{a+b=p-1}\Lambda^{a}_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee] \otimes \widetilde{H}^{b}(\mathrm{SL}_n(\mathbb{Z}) )$$

of $\mathrm{SL}_n(\mathbb{Z}/p)$ -representations, compatible with the evident multiplicative structures in degrees $* \leq p-1$ .

9.1 Tensor powers of the dual adjoint representation

Recall that we write sl(V) for the kernel of $ev : V \otimes V^\vee \to \mathbb{F}_p$ , so there is a short exact sequence

(9.1) \begin{equation}0 \longrightarrow \mathbb{F}_p \overset{coev}\longrightarrow V \otimes V^\vee \longrightarrow sl(V)^\vee \longrightarrow 0.\end{equation}

In §§ 3 and 4 we have constructed maps

$$\psi_{S,T} : \Gamma_{\mathbb{F}_p}[x]^{\otimes S} \otimes k\{\mathrm{Bij}(T,S)\} \overset{\Psi_{S,T}}\longrightarrow \mathrm{Ext}_{\mathrm{GL}}^*(I^{\otimes T}, I^{\otimes S}) \longrightarrow \mathrm{Ext}_{\mathrm{GL}(V)}^*(V^{\otimes T}, V^{\otimes S})$$

and shown that $\Psi_{S,T}$ is an isomorphism (in Theorem 3.2) and that the second map is an isomorphism in a stable range of degrees (by the stability discussion in § 2.2). Furthermore, the latter may be written as $H^*(\mathrm{GL}(V) ; V^{\otimes S} \otimes (V^\vee)^{\otimes T})$ , and, as discussed in § 2.3, we may replace $\mathrm{GL}(V)$ by $\mathrm{SL}(V)$ in a stable range.

Using this, we may form the composition

(9.2) \begin{align}\Gamma_{\mathbb{F}_p}[x]^{\otimes T} \otimes\mathbb{F}_p\{\Sigma^{\rm ad}_T\}&\overset{\psi_{T,T}}\longrightarrow H^*(\mathrm{SL}(V) ; (V \otimes V^\vee)^{\otimes T})\nonumber\\&\longrightarrow H^*(\mathrm{SL}(V) ; (sl(V)^\vee)^{\otimes T})\end{align}

for which we have the following analogue of Theorem 3.2.

Proof. By the results of § 2, we may increase the dimension of V if we wish, and in particular we may suppose that $\dim(V) \not\equiv 0 \mod p$ , so that (9.1) is $\mathrm{GL}(V)$ -equivariantly split by a unit times the map $ev : V \otimes V^\vee \to \mathbb{F}_p$ . Consider $\mathbb{F}_p \overset{coev}\to V \otimes V^\vee$ as a chain complex whose homology is $sl(V)^\vee$ supported in degree zero. The Tth tensor power of this chain complex has the form

$$\cdots \longrightarrow \bigoplus_{j \in T} (V \otimes V^\vee)^{\otimes T \setminus j} \longrightarrow (V \otimes V^\vee)^{\otimes T},$$

where the rightmost map is given by inserting coev on each of the T summands. By the Künneth theorem the homology of this complex is $(sl(V)^\vee)^{\otimes T}$ supported in degree zero, giving an exact sequence

$$\bigoplus_{j \in T} (V \otimes V^\vee)^{\otimes T\setminus j} \longrightarrow (V \otimes V^\vee)^{\otimes T} \longrightarrow (sl(V)^\vee)^{\otimes T} \longrightarrow 0.$$

We have arranged that the middle map is $\mathrm{GL}(V)$ -equivariantly split, so this sequence remains exact after applying $H^*(\mathrm{SL}(V) ; -)$ . In a stable range of degrees the map $\psi_{T,T}$ is an isomorphism by Theorem 3.2 and the discussion after it, as are the maps $\psi_{T \setminus j, T \setminus j}$ . Thus, after applying $H^*(\mathrm{SL}(V) ; -)$ we obtain an exact sequence

where the left-hand map is described by the functoriality on the upward walled Brauer category, as in § 4.1. Namely, on the jth summand it is induced by the map $(inc, inc, \mathrm{Id}_{\{j\}}) : (T \setminus j, T \setminus j) \to (T,T)$ in the upward walled Brauer category, which we defined to send $(\bigotimes_{s \in T \setminus j} x_s^{[\ell'(s)]}) \otimes \sigma'$ to $(\bigotimes_{t \in T} x_t^{[\ell(t)]}) \otimes \sigma$ , where $\sigma \in \Sigma^{\rm ad}_T$ fixes $j \in T$ and is given by $\sigma'$ on $T \setminus j$ , and $\ell$ sends j to 0 and agrees with $\ell'$ on $T \setminus j$ . The image of this map is spanned by the claimed elements.

Corollary 9.2. There are classes

\begin{align*}c_t \in H^0(\mathrm{SL}(V) ; \Lambda^t[sl(V)^\vee]) & \text{ for $t \geq 3$ odd},\\e_t \in H^2(\mathrm{SL}(V) ; \Lambda^t[sl(V)^\vee]) & \text{ for $t \geq 1$}\end{align*}

and isomorphisms

\begin{align*}H^0(\mathrm{SL}(V) ; \Lambda^*[sl(V)^\vee]) &\cong \Lambda^*_{\mathbb{F}_p}[c_3, c_5, c_7, \ldots],\\H^1(\mathrm{SL}(V) ; \Lambda^*[sl(V)^\vee]) &\cong 0,\\H^2(\mathrm{SL}(V) ; \Lambda^*[sl(V)^\vee]) &\cong \mathbb{F}_p\{e_1, e_2, e_3, \ldots\} \otimes \Lambda^*_{\mathbb{F}_p}[c_3, c_5, c_7, \ldots]\end{align*}

in degrees $* \lt p$ for $\dim(V)$ sufficiently large.

Proof. Recall that we write $\underline{t} := \{1,2,\ldots, t\}$ . Let $\tilde{c}_t \in H^0(\mathrm{SL}(V) ; \Lambda^t_{\mathbb{F}_p}[V\otimes V^\vee])$ denote the image of the class

$$1 \otimes (1,2,3,\ldots, t) \in \Gamma_{\mathbb{F}_p}[x]^{\otimes \underline{t}} \otimes \mathbb{F}_p\{\Sigma^{\rm ad}_{\underline{t}}\}$$

under the maps

(9.3) \begin{equation}\Gamma_{\mathbb{F}_p}[x]^{\otimes \underline{t}} \otimes \mathbb{F}_p\{\Sigma^{\rm ad}_{\underline{t}}\} \to H^*(\mathrm{SL}(V) ; (V \otimes V^\vee)^{\otimes \underline{t}}) \to H^*(\mathrm{SL}(V) ; \Lambda^t_{\mathbb{F}_p}[V \otimes V^\vee]),\end{equation}

and $c_t$ its further image in $H^*(\mathrm{SL}(V) ; \Lambda^t_{\mathbb{F}_p}[sl(V)^\vee])$ . The discussion above shows that $c_1=0$ . Writing $\mathbb{F}_p^-$ for the sign representation, for $t\lt p$ we have

$$H^0(\mathrm{SL}(V) ; \Lambda^t_{\mathbb{F}_p}[V \otimes V^\vee]) \cong \mathbb{F}_p\{\Sigma^{\rm ad}_{\underline{t}}\} \otimes_{\Sigma_{\underline{t}}} \mathbb{F}_p^-.$$

Recalling that we have assumed that p is odd, this is given by the way conjugacy classes split in the alternating group: if a conjugacy class contains an even cycle or two odd cycles of the same length, then it becomes trivial on applying $- \otimes_{\Sigma_{\underline{t}}} \mathbb{F}_p^-$ ; otherwise it contributes a one-dimensional space. This identifies this space with the degree t part of $\Lambda^*_{\mathbb{F}_p}[\tilde{c}_1, \tilde{c}_3, \tilde{c}_5, \ldots]$ , and hence identifies $H^0(\mathrm{SL}(V) ; \Lambda^*_{\mathbb{F}_p}[V \otimes V^\vee])$ with this graded-commutative algebra.

As $\Lambda^t_{\mathbb{F}_p}[sl(V)^\vee]$ is a summand of $(sl(V)^\vee)^{\otimes \underline{t}}$ for $t \lt p$ , using Lemma 9.1 we obtain the claimed formula for $H^0(\mathrm{SL}(V) ; \Lambda^*_{\mathbb{F}_p}[sl(V)^\vee])$ , and also for $H^1(\mathrm{SL}(V) ; \Lambda^*_{\mathbb{F}_p}[sl(V)^\vee])$ because $H^*(\mathrm{SL}(V) ; (V \otimes V^\vee)^{\otimes \underline{t}})$ is supported in even degrees in the stable range by Theorem 3.2.

Let $\tilde{e}_t \in H^2(\mathrm{SL}(V) ; \Lambda^t_{\mathbb{F}_p}[V \otimes V^\vee])$ denote the image of the class

$$x_1^{[1]} \otimes (1,2,3,\ldots, t) \in\Gamma_{\mathbb{F}_p}[x]^{\otimes \underline{t}} \otimes\mathbb{F}_p\{\Sigma^{\rm ad}_{\underline{t}}\}$$

under the maps (9.3) and $e_t$ its further image in $H^*(\mathrm{SL}(V) ; \Lambda^t_{\mathbb{F}_p}[sl(V)^\vee])$ . We have

$$H^2(\mathrm{SL}(V) ; \Lambda^t_{\mathbb{F}_p}[V \otimes V^\vee]) \cong (\mathbb{F}_p\{x_i^{[1]} \, | \, i \in \underline{t}\} \otimes \mathbb{F}_p\{\Sigma^{\rm ad}_{\underline{t}}\}) \otimes_{\Sigma_{\underline{t}}} \mathbb{F}_p^-.$$

We think of the first factor as being the space of permutations of $\underline{t}$ written as disjoint cycles, with one entry marked, on which $\Sigma_{\underline{t}}$ acts by conjugation. As above we find that to contribute the unmarked cycles must all be of different odd lengths, and the marked cycle may be of any length. As a module over $H^0(\mathrm{SL}(V) ; \Lambda^*_{\mathbb{F}_p}[V \otimes V^\vee])$ we therefore have that $H^2(\mathrm{SL}(V) ; \Lambda^*_{\mathbb{F}_p}[V \otimes V^\vee])$ is free on the basis of marked cycles (i.e., $\tilde{e}_1, \tilde{e}_2, \tilde{e}_3, \ldots$ ). Using as above that $\Lambda^t_{\mathbb{F}_p}[sl(V)^\vee]$ is a summand of $(sl(V)^\vee)^{\otimes \underline{t}}$ for $t \lt p$ , we find the same description for $H^2(\mathrm{SL}(V) ; \Lambda^*_{\mathbb{F}_p}[sl(V)^\vee])$ as a module over $H^0(\mathrm{SL}(V) ; \Lambda^*_{\mathbb{F}_p}[sl(V)^\vee])$ .

9.2 Multiplicative structure on the $E_2$ -page of the Serre spectral sequence

We will need to use some details of the $d_2$ -differentials in the spectral sequence for a group extension

(9.4) \begin{equation}1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1,\end{equation}

with coefficients in a field $\mathbb{k}$ : it has the form

$$E_2^{s,t} = H^s(Q ; H^t(K ; \mathbb{k})) = \mathrm{Ext}^s_{\mathbb{k}[Q]}(\mathbb{k}, H^t(K;\mathbb{k})) \Longrightarrow H^{s+t}(G;\mathbb{k}).$$

1. (i) The differential $d_2 : E_2^{s, t} \to E_2^{s+2, t-1}$ is given by the Yoneda product with $d_2^t$ .

2. (ii) The element $-d_2^1 {\kern-1pt}\in{\kern-1pt}\mathrm{Ext}_{\mathbb{k}[Q]}^2(H^1(K;\mathbb{k}),\mathbb{k}){\kern-1pt}\cong{\kern-1pt} H^2(Q ; H_1(K;\mathbb{k}))$ classifies the extension obtained from (9.4) by pushout along $K \overset{\text{ab}}\to H_1(K;\mathbb{Z}) \to H_1(K;\mathbb{k})$ .

3. (iii) The square

commutes in the derived category of $\mathbb{k}[Q]$ -modules.

We were surprised to be unable to find this result in the literature, although the analogue of (i) and (ii) in homology has been developed in various works of Legrand (see, for example, [Leg83]), and is discussed for the change-of-rings spectral sequence by Suárez and Alvarez [Reference Suárez-AlvarezSuá07, Theorem 2.2.3].

Proof of Lemma 9.3. Let $E_*G \to \mathbb{k}$ be the standard free resolution of the trivial $\mathbb{k}[G]$ -module, which in homological degree p is $\mathbb{k}[G^p]$ , and consider the chain complex

$$C := \mathrm{Hom}_{\mathbb{k}[K]}(E_*G, \mathbb{k}) = [\mathrm{Hom}_{\mathbb{k}}(E_*G, \mathbb{k})]^K,$$

which has a residual action of $G/K=Q$ . The homology of this complex in degree $-i$ is $H^i(K;\mathbb{k})$ , with the Q-action induced by the outer Q-action on K coming from the extension (9.4).

Deconcatenation gives a map of $\mathbb{k}[G]$ -modules

\begin{align*}\Delta : E_*G &\longrightarrow E_*G \otimes_{\mathbb{k}} E_*G,\\(g_1, \ldots, g_p) &\longmapsto \sum_{i=0}^{p} (g_1, \ldots, g_i) \otimes (g_{i+1}, \ldots, g_p),\end{align*}

which yields a morphism $\phi \otimes \psi \mapsto (\phi \otimes \psi) \circ \Delta : C \otimes_{\mathbb{k}} C \to C$ of chain complexes over $\mathbb{k}[Q]$ . Precomposition with $E_*G \to \mathbb{k}$ gives a morphism $\mathbb{k} \to C$ of chain complexes over $\mathbb{k}[Q]$ . On homology this gives the cup-product on $H^*(K;\mathbb{k})$ , and its unit.

Using this structure, we consider C to be a unital associative ring object in the derived category $\mathsf{D}(\mathbb{k}[Q])$ of $\mathbb{k}[Q]$ , which we equip with the natural t-structure and corresponding truncation functors denoted by $\tau$ , as well as the symmetric monoidal structure given by $- \otimes_{\mathbb{k}} -$ (with the diagonal Q-action). The monoidal product of connective objects is connective, from which it follows that the monoidal product of an a-connective and a b-connective object is $(a+b)$ -connective: thus the multiplication and unit on C gives morphisms $\tau_{\geq -i} C \otimes_{\mathbb{k}} \tau_{\geq -j} C \to \tau_{\geq -i-j}C$ and $\mathbb{k} \to \tau_{\geq 0} C$ .

The distinguished triangles for adjacent connective covers of C yield its Whitehead tower, having the following form.

The horizontal compositions represent classes

\begin{equation*}\begin{aligned}d_2^t \in & \,\, \mathrm{Ext}^2_{\mathbb{k}[Q]}(H^t(K;\mathbb{k}), H^{t-1}(K;\mathbb{k})) \\&= [H^t(K;\mathbb{k}), H^{t-1}(K;\mathbb{k})[2]]_{\mathsf{D}(\mathbb{k}[Q])}\\& \cong [H^t(K;\mathbb{k})[-t-1], H^{t-1}(K;\mathbb{k})[1-t]]_{\mathsf{D}(\mathbb{k}[Q])}.\end{aligned}\end{equation*}

Applying $[\mathbb{k}, -]_{\mathsf{D}(\mathbb{k}[Q])}$ to this diagram yields an exact couple and so a spectral sequence: this is the (Lyndon–Hochschild–)Serre spectral sequence for the group extension (9.4), with

$$E_2^{s,t} := [\mathbb{k}[-s-t], H^t(K;\mathbb{k})[-t]]_{\mathsf{D}(\mathbb{k}[Q])} \cong \mathrm{Ext}^s_{\mathbb{k}[Q]}(\mathbb{k}, H^t(K;\mathbb{k}))$$

and converging to $[\mathbb{k}[-s-t], C]_{\mathsf{D}(\mathbb{k}[Q])} \cong H^{s+t}(G;\mathbb{k})$ . The $d_2$ -differentials are tautologically given by Yoneda product with the $d_2^t$ , giving (i).

For (ii), note that if M is a $\mathbb{k}[Q]$ -module then the filtered object $(\tau_{\geq \bullet} C) \otimes_{\mathbb{k}} M$ yields the (Lyndon–Hochschild–)Serre spectral sequence with coefficients in M,

$$E_2^{s,t} = H^s(Q; H^t(K;\mathbb{k})\otimes_{\mathbb{k}} M) \Longrightarrow H^{s+t}(G;M).$$

Thus, consider the map between spectral sequences with $M=H_1(K;\mathbb{k})$ induced by the pushout extension.

Writing $coev \in H^1(H_1(K;\mathbb{k}) ; \mathbb{k}) \otimes_{\mathbb{k}} H_1(K;\mathbb{k}) = \mathrm{Hom}_{\mathbb{Z}}(H_1(K;\mathbb{k}) , \mathbb{k}) \otimes_{\mathbb{k}} H_1(K;\mathbb{k})$ for the class dual to evaluation, its image under the differential

$$d_2 : H^0(Q ; H^1(H_1(K;\mathbb{k}) ; \mathbb{k}) \otimes H_1(K;\mathbb{k}) ) \longrightarrow H^2(Q ; H_1(K;\mathbb{k}) )$$

in the spectral sequence for the lower extension is minus the class classifying the lower extension [Reference Hochschild and SerreHS53, Theorem 4]. By naturality this class is the composition

$$\mathbb{k}[-2] \overset{coev}\longrightarrow H^1(K;\mathbb{k}) [-2] \otimes_{\mathbb{k}} H_1(K;\mathbb{k}) \overset{d_2^1 \otimes H_1(K;\mathbb{k}) }\longrightarrow \mathbb{k} \otimes_{\mathbb{k}} H_1(K;\mathbb{k}),$$

which corresponds to the class $d_2^1$ under the isomorphism $\mathrm{Ext}_{\mathbb{k}[Q]}^2(H^1(K;\mathbb{k}), \mathbb{k}) \cong \mathrm{Ext}_{\mathbb{k}[Q]}^2(\mathbb{k}, H_1(K;\mathbb{k})) = H^2(Q ; H_1(K;\mathbb{k}))$ .

For (iii) we appeal to the fact that $\mathsf{D}(\mathbb{k}[Q])$ has an enhancement to the unbounded derived $\infty$ -category as constructed in [Reference LurieLur17, § 1.3.5] by localising the category $\mathsf{Ch}(\mathbb{k}[Q])$ of chain complexes at the quasi-isomorphisms. This $\infty$ -category is stable [Reference LurieLur17, Proposition 1.3.5.9], and the t-structure we have been using refines to an $\infty$ -categorical t-structure [Reference LurieLur17, § 1.2.1]. Furthermore, $- \otimes_{\mathbb{k}} -$ on $\mathsf{Ch}(\mathbb{k}[Q])$ induces a symmetric monoidal structure, compatible with the t-structure in the sense that the monoidal product of connective objects is connective. Placing ourselves in this setting, we may rely on Hedenlund’s thesis [Reference HedenlundHed20, Part II]. Namely, C defines a (commutative) algebra object in the derived $\infty$ -category of $\mathbb{k}[Q]$ , and the Whitehead tower gives a multiplicative filtration of it, so by [Reference HedenlundHed20, Theorem II.1.21] we have a Leibniz rule in the form of the commutative diagram

in $\mathsf{D}(\mathbb{k}[Q])$ . There is a subtlety about signs. Shifting is implemented by $\mathbb{k}[1] \otimes_{\mathbb{k}} -$ , so the bottom cup-product map on the second summand actually requires commuting $\mathbb{k}[1]$ past $H^{t'}(K;\mathbb{k})[-t']$ before cupping, incurring a sign $(-1)^{t'}$ . When we shift up by $t'+t''$ in order to put this square in the form stated in the lemma, this can be expressed by writing the left-hand vertical map as $d_2^{t'}\otimes \mathrm{Id} + (-1)^{t'} \mathrm{Id} \otimes d_2^{t''}$ , and the bottom map as just cupping.

9.3 Proof of Proposition 8.3

Recall that we wish to understand the differentials $d_2 :{^{I\!I\!I}\!E}^{0,*}_2 \to {^{I\!I\!I}\!E}^{2,*-1}_2$ in the spectral sequence

$$^{I\!I\!I}\!E^{s,t}_2 = H^s(\mathrm{SL}_n(\mathbb{Z}/p) ;{H}^t_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p))) \Longrightarrow H^{s+t}_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p) )$$

associated to the fibration (8.1). In § 7.6 we explained how to identify $H^t_{\mathrm{cts}}(\mathrm{SL}_n(\mathbb{Z}_p, p) ) = \Lambda^t_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee]$ as an $\mathrm{SL}(\mathbb{Z}/p)$ -module, so by Corollary 9.2 we have calculated that

\begin{align*}{^{I\!I\!I}\!E}^{0,*}_2 &\cong \Lambda^*_{\mathbb{F}_p}[c_3, c_5, c_7, \ldots],\\{^{I\!I\!I}\!E}^{2,*}_2 &\cong \mathbb{F}_p\{e_1, e_2, e_3, \ldots\} \otimes \Lambda^*_{\mathbb{F}_p}[c_3, c_5, c_7, \ldots],\end{align*}

for $* \lt p$ and all large enough n. To evaluate the differential we apply the discussion in § 9.2. That discussion provides elements

$$d_2^t \in \mathrm{Ext}^2_{\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]}(\Lambda^t_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee], \Lambda^{t-1}_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee]),$$

such that $d_2 : H^s(\mathrm{SL}_n(\mathbb{Z}/p) ; \Lambda^t_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee]) \to H^{s+2}(\mathrm{SL}_n(\mathbb{Z}/p) ; \Lambda^{t-1}_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee])$ is given by Yoneda product with $d_2^t$ . For each $t \geq 3$ odd, we therefore wish to evaluate the composition

$$\mathbb{F}_p \overset{c_t}\longrightarrow \Lambda^t_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee] \overset{d_2^t}\longrightarrow \Lambda^{t-1}_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee][2]$$

as a morphism in the derived category of $\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]$ -modules, that is, an element of $\mathrm{Ext}^2_{\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]}(\mathbb{F}_p,\Lambda^{t-1}_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee]) =H^2(\mathrm{SL}_n(\mathbb{Z}/p);\Lambda^{t-1}_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee])$ . The Leibniz rule as described in Lemma 9.3(iii), applied t times, gives a commutative diagram

in the derived category of $\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]$ -modules. Writing $V_n = \mathbb{F}_p^n$ , defining

$$\tilde{d}_2^1 : V_n \otimes V_n^\vee \longrightarrow sl(V_n)^\vee \overset{d_2^1}\longrightarrow \mathbb{F}_p[2],$$

and recalling the classes

$$\tilde{c}_t \in H^0(\mathrm{SL}_n(\mathbb{F}_p) ; \Lambda^t_{\mathbb{F}_p}[V_n \otimes V_n^\vee]) = \mathrm{Ext}^0_{\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]}(\mathbb{F}_p, \Lambda^{t}_{\mathbb{F}_p}[sl_n(\mathbb{F}_p)^\vee])$$

defined in the proof of Corollary 9.2, the diagram

commutes in the derived category of $\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]$ -modules: the triangle commutes by the definition of $c_t$ in terms of $\tilde{c}_t$ , and the square commutes because it does so on each summand by definition of $\tilde{d}_2^1$ .

Lemma 9.6. The element

\begin{align*}\tilde{d}_2^1 &\in \mathrm{Ext}^2_{\mathbb{F}_p[\mathrm{SL}_n(\mathbb{Z}/p)]}(V_n \otimes V_n^\vee, \mathbb{F}_p)\\& \quad \cong H^2(\mathrm{SL}_n(\mathbb{Z}/p); \mathrm{Hom}(V_n \otimes V_n^\vee, \mathbb{F}_p))\\& \quad \cong H^2(\mathrm{SL}_n(\mathbb{Z}/p); V_n \otimes V_n^\vee) \cong \mathbb{F}_p\{x_1^{[1]}\} \otimes \mathbb{F}_p\{\Sigma^{\rm ad}_1\}\end{align*}

corresponds to $x^{[1]}_1 \otimes (1)$ .