HIGHER FIXED POINTS OF TOPOLOGICAL

HOCHSCHILD HOMOLOGY OF THE INTEGERS

John Rognes

June 9th 1999

The spectral sequences.

Let p be an odd prime. The Tate spectral sequence for C_{p^{n}} acting on T HH(ℤ) with 𝔽_{p} coefficients has E^{2} term

Ê_{∗∗}^{2}(C_{p^{n}}) = Ĥ^{−∗}(C_{p^{n}}; π_{∗}(T HH(ℤ); 𝔽_{p}))
= E(u_{n}) ⊗ P (t, t^{−1}) ⊗ E(e) ⊗ P (f)

and converges to π_{∗}(ℍ̂(C_{p^{n}}, T HH(ℤ)); 𝔽_{p}). The bidegrees are: deg(u_{n}) = (−1, 0), deg(t) = (−2, 0), deg(e) = (0, 2p − 1) and deg(f) = (0, 2p).

Restricting to the second quadrant, we have the homotopy fixed point spectral sequence for C_{p^{n}} acting on T HH(ℤ) with 𝔽_{p} coefficients:

E_{∗∗}^{2}(C_{p^{n}}) = H^{−∗}(C_{p^{n}}; π_{∗}(T HH(ℤ); 𝔽_{p}))
= E(u_{n}) ⊗ P (t) ⊗ E(e) ⊗ P (f) .

It converges to π_{∗}(T HH(ℤ)^{hC_{p^{n}}}; 𝔽_{p}). Both spectral sequences are algebra spectral sequences.

The first case.

For n = 1, there is a nonzero differential

(1)  d^{2p}(t^{1−p}) = te

(up to a unit in 𝔽_{p}, which we hereafter omit to mention). Also e and tf are infinite cycles and d^{2p}(u_{1}) = 0, so

Ê_{∗∗}^{2p+1}(C_{p}) = E(u_{1}) ⊗ P (t^{p}, t^{−p}) ⊗ E(e) ⊗ P (tf) .

There is also a nonzero differential

(2)  d^{2p+1}(u_{1}t^{−p}) = tf .

Thus

Ê_{∗∗}^{2p+2}(C_{p}) = P (t^{p}, t^{−p}) ⊗ E(e)

which is also the E^{∞} term.

Typeset by AMS-TEX

It follows that Γ̂_{1} : T HH(ℤ) → ℍ̂(C_{p}, T HH(ℤ)) induces an isomorphism on homotopy with 𝔽_{p} coefficients in all nonnegative dimensions. Hence the maps Γ_{n}: T HH(ℤ)^{C_{p^{n}}} → T HH(ℤ)^{hC_{p^{n}}} and Γ̂_{n}: T HH(ℤ)^{C_{p^{n−1}}} → ℍ̂(C_{p^{n}}, T HH(ℤ)) induce isomorphisms on homotopy with 𝔽_{p} coefficients in all nonnegative dimensions.

Restricting to the second quadrant, the first differential d^{2p}(t) = t^{p+1}e in the homotopy fixed point spectral sequence yields

E_{∗∗}^{2p+1}(C_{p}) = E(u_{1}) ⊗ E(e) ⊗ P (tf) ⊗ P (f^{p}, f^{−p})
⊕ E(u_{1}) ⊗ P_{p}(tf){ef^{j} | v_{p}(j) = 0}

in nonnegative total degrees. Here P_{h}(x) = P (x)/(x^{h} = 0) is the truncated polynomial algebra of height h, and v_{p}(j) is the p-adic valuation of j. The second differential d^{2p+1}(u_{1}) = t^{p+1}f yields

E_{∗∗}^{2p+2}(C_{p}) = E(e) ⊗ P_{p+1}(tf) ⊗ P (f^{p}, f^{−p})
⊕ E(u_{1}) ⊗ P_{p}(tf){ef^{j} | v_{p}(j) = 0}

in nonnegative total degrees, which is also the E^{∞} term.

This computes π_{∗}(T HH(ℤ)^{C_{p}}; 𝔽_{p}), giving

dim_{𝔽_{p}}(π_{k}(T HH(ℤ)^{C_{p}}; 𝔽_{p})) = \{ \begin{matrix}1 & k ≢ 0, −1 mod 2p^{2} \\ 2 & k ≡ 0, −1 mod 2p^{2} . \end{matrix}

The second case.

Now consider the Tate spectral sequence for n = 2. By naturality with respect to the Frobenius and Verschiebung maps, its E^{2p+1} term agrees with that for n = 1, replacing u_{1} by u_{2}, and furthermore d^{2p+1}(u_{2}) = 0. Thus:

Ê_{∗∗}^{2p+2}(C_{p^{2}}) = E(u_{2}) ⊗ P (t^{p}, t^{−p}) ⊗ E(e) ⊗ P (tf) .

All classes of the form t^{i}ef^{j} with j ≥ 0 are in the image of the Tate spectral sequence for S^{1} acting on T HH(ℤ) with ℤ_{p} coefficients, under the natural map induced by restriction over C_{p^{n}} ⊂ S^{1} and the coefficient homomorphism ℤ_{p} → 𝔽_{p}. For bidegree reasons all classes above the horizontal axis in the latter spectral sequence are infinite cycles. Thus so are the image classes t^{i}ef^{j} in Ê_{∗∗}^{2}(C_{p^{n}}) for all n ≥ 1.

We will show that there is a nonzero differential of the form

d^{r}(t^{p−p^{2}}) = t^{i} · e · f^{j}

with r ≥ 2p + 2. The class t^{i}ef^{j} lies in Ê_{∗∗}^{2p+2}(C_{p^{2}}) when i ≡ j mod p. It is in the total degree of d^{r}(t^{p−p^{2}}) when i + p^{2} = 2p + pj. Of these classes, the two with least j are t^{2p−p^{2}} · e and t^{2p} · e · f^{p} = t^{p} · e · (tf)^{p}.

HIGHER FIXED POINTS OF T HH(ℤ) 3

The first cannot be hit by a d^{r}-differential with r ≥ 2p+2, hence survives to E^{∞}. It is in total degree 2p^{2} − 2p − 1 ≢ 0, −1 mod 2p^{2}, so there is only one permanent cycle in this total degree. Hence the second infinite cycle must be a boundary. We claim the differential hitting it is

(3)  d^{2p^{2}+2p}(t^{p−p^{2}}) = t^{p} · e · (tf)^{p} .

For the classes in total degree 2p^{2} −2p are t^{pi} ·(tf)^{j} ·t^{p−p^{2}} or t^{pi} ·u_{2}e·(tf)^{j−1} ·t^{p−p^{2}} with pi = (p − 1)j. The only possible sources for a differential are t^{p−p^{2}}, (tf)^{p} and u_{2}e · (tf)^{p−1}. Since r ≥ 2p + 2 the latter two are excluded. This proves the claimed formula (3).

We will also show that there is a nonzero differential of the form

d^{r}(u_{2}e · t^{−p^{2}}) = t^{i} · e · f^{j}

with r ≥ 2p + 2. The class t^{i} · e · f^{j} is in the total degree of d^{r}(u_{2}e · t^{−p^{2}}) when i + p^{2} = 1 + pj. Of these classes with i ≡ j mod p, the two with the least j are t^{−p^{2}+p+1} · e · f = (t^{p})^{1−p} · e · (tf) and e · (tf)^{p+1}.

The first cannot be hit by a differential of length r ≥ 2p + 2, hence survives to E^{∞}. It is in total degree 2p^{2} + 2p − 3 ≢ 0, −1 mod 2p^{2}, so there is only one permanent cycle in this total degree. Hence the second infinite cycle must be a boundary. We claim the differential hitting it is

d^{2p^{2}+2p+1}(u_{2}e · t^{−p^{2}}) = e · (tf)^{p+1} .

For the only possible differentials of length r ≥ 2p+2 hitting e·(tf)^{p+1} are t^{−p^{2}} ·tf, with r = 2p^{2} + 2p and u_{2}e · t^{−p^{2}} with r = 2p^{2} + 2p + 1. But d^{2p^{2}+2p}(t^{−p^{2}} · tf) = 0 because t^{−p} survives to the E^{2p^{2}+2p} term and tf is an infinite cycle. This proves the claim.

Canceling the e’s, we deduce the differential

(4)  d^{2p^{2}+2p+1}(u_{2}t^{−p^{2}}) = (tf)^{p+1} .

Hence t^{p} and t^{−p} survive to the E^{2p^{2}+2p} term, while u_{2} survives to the E^{2p^{2}+2p+1} term. Thus there are no earlier differentials; the first odd differential in Ê_{∗∗}^{∗}(C_{p^{2}}) is of length 2p^{2} + 2p + 1. We compute:

Ê_{∗∗}^{2p^{2}+2p+1}(C_{p^{2}}) = E(u_{2}) ⊗ P (t^{p^{2}}, t^{−p^{2}}) ⊗ E(e) ⊗ P (tf)
⊕ E(u_{2}) ⊗ P_{p}(tf){t^{i}e | v_{p}(i) = 1}

and

Ê_{∗∗}^{2p^{2}+2p+2}(C_{p^{2}}) = P (t^{p^{2}}, t^{−p^{2}}) ⊗ E(e) ⊗ P_{p+1}(tf)
⊕ E(u_{2}) ⊗ P_{p}(tf){t^{i}e | v_{p}(i) = 1}

which is also the E^{∞} term.

Now restrict to the second quadrant, considering the homotopy fixed point spectral sequence E_{∗∗}^{∗}(C_{p^{2}}). The first differential d^{2p}(t) = t^{p+1}e yields

E_{∗∗}^{2p+1}(C_{p^{2}}) = E(u_{2}) ⊗ E(e) ⊗ P (tf) ⊗ P (f^{p}, f^{−p})
⊕ E(u_{2}) ⊗ P_{p}(tf){ef^{j} | v_{p}(j) = 0}

in nonnegative total degrees. The second differential d^{2p^{2}+2p}(t^{p}) = t^{p^{2}+p} · e · (tf)^{p} yields

E_{∗∗}^{2p^{2}+2p+1}(C_{p^{2}}) = E(u_{2}) ⊗ E(e) ⊗ P (tf) ⊗ P (f^{p^{2}}, f^{−p^{2}})
⊕ E(u_{2}) ⊗ P_{p}(tf){ef^{j} | v_{p}(j) = 0}
⊕ E(u_{2}) ⊗ P_{p^{2}+p}(tf){ef^{j} | v_{p}(j) = 1}

in nonnegative total degrees. The third differential d^{2p^{2}+2p+1}(u_{2}) = t^{p^{2}}(tf)^{p+1} yields

E_{∗∗}^{2p^{2}+2p+2}(C_{p^{2}}) = E(e) ⊗ P_{p^{2}+p+1}(tf) ⊗ P (f^{p^{2}}, f^{−p^{2}})
⊕ E(u_{2}) ⊗ P_{p}(tf){ef^{j} | v_{p}(j) = 0}
⊕ E(u_{2}) ⊗ P_{p^{2}+p}(tf){ef^{j} | v_{p}(j) = 1}

in nonnegative total degrees, which is also the E^{∞} term.

This computes π_{∗}(T HH(ℤ)^{C_{p^{2}}} ; 𝔽_{p}), giving

dim_{𝔽_{p}}(π_{k}(T HH(ℤ)^{C_{p^{2}}} ; 𝔽_{p})) = \{ \begin{matrix}2 & k ≢ 0, −1 mod 2p^{3} \\ 3 & k ≡ 0, −1 mod 2p^{3} . \end{matrix}

The general case.

Inductively, suppose that in the Tate spectral sequence Ê_{∗∗}^{∗}(C_{p^{n}}) there are differentials

(5)  d^{r}(t^{p^{k}−p^{k+1}}) = t^{p^{k}} · e · (tf)^{p^{k}+···+p}

with r = 2(p^{k+1} + ⋯ + p) for all 0 ≤ k < n, and a differential

(6)  d^{r}(u_{n}t^{−p^{n}}) = (tf)^{p^{n−1}+···+1}

with r = 2(p^{n} + ⋯ + p) + 1. Also suppose that t^{p^{n}} survives to the E^{r} term, with r = 2(p^{n} + ⋯ + 1). In these formulas, each expression p^{a} + ⋯ + p^{b} means the partial geometric series ∑_{a≥i≥b} p^{i}.

The inductive hypothesis completely determines the differential structure in the algebra spectral sequence Ê_{∗∗}^{∗}(C_{p^{n}}). The intermediate terms are, for 0 < k ≤ n:

Ê_{∗∗}^{r}(C_{p^{n}}) = E(u_{n}) ⊗ P (t^{p^{k}}, t^{−p^{k}}) ⊗ E(e) ⊗ P (tf)

⊕ ⨁_{m=v_{p}(i)<k} E(u_{n}) ⊗ P_{p^{m}+···+p}(tf){t^{i}e | v_{p}(i) = m}

with r = 2(p^{k} + ⋯ + p) + 1. The final (odd) differential leaves the E^{r} term with r = 2(p^{n} + ⋯ + 1), which equals the E^{∞} term since all remaining classes are close to the horizontal axis:

Ê_{∗∗}^{∞}(C_{p^{n}}) = P (t^{p^{n}}, t^{−p^{n}}) ⊗ E(e) ⊗ P_{p^{n−1}+···+1}(tf)

⊕ ⨁_{m=v_{p}(i)<n} E(u_{n}) ⊗ P_{p^{m}+···+p}(tf){t^{i}e | v_{p}(i) = m} .

By restriction to the second quadrant, the homotopy fixed point spectral sequence E_{∗∗}^{∗}(C_{p^{n}}) has differentials

(7)  d^{r}(t^{p^{k}}) = t^{p^{k+1}+p^{k}} · e · (tf)^{p^{k}+···+p}

with r = 2(p^{k+1} + ⋯ + p) for all 0 ≤ k < n, and a differential

(8)  d^{r}(u_{n}) = (tf)^{p^{n}+···+1}

with r = 2(p^{n} + ⋯ + p) + 1.

This completely determines the differential structure of the spectral sequence E_{∗∗}^{∗}(C_{p^{n}}). The intermediate terms are

E_{∗∗}^{r}(C_{p^{n}}) = E(u_{n}) ⊗ E(e) ⊗ P (tf) ⊗ P (f^{p^{k}}, f^{−p^{k}})

⊕ ⨁_{m=v_{p}(j)<k} E(u_{n}) ⊗ P_{p^{m+1}+···+p}(tf){ef^{j} | v_{p}(j) = m}

in nonnegative total degrees, for 0 < k ≤ n with r = 2(p^{k} + ⋯ + p) + 1. The final (odd) differential leaves the E^{r} term with r = 2(p^{n} + ⋯ + 1), which again equals the E^{∞} term for degree reasons:

E_{∗∗}^{∞}(C_{p^{n}}) = E(e) ⊗ P_{p^{n}+···+1}(tf) ⊗ P (f^{p^{n}}, f^{−p^{n}})

⊕ ⨁_{m=v_{p}(j)<n} E(u_{n}) ⊗ P_{p^{m+1}+···+p}(tf){ef^{j} | v_{p}(j) = m}

in nonnegative total degrees.

Counting classes, one obtains:

dim_{𝔽_{p}}(π_{k}(T HH(ℤ)^{C_{p^{n}}}; 𝔽_{p})) = \{ \begin{matrix}n & k ≢ 0, −1 mod 2p^{n+1} \\ n + 1 & k ≡ 0, −1 mod 2p^{n+1} . \end{matrix}

Since Γ̂_{n+1} induces an isomorphism on homotopy with 𝔽_{p} coefficients in nonnegative dimensions, the same formula computes dim_{𝔽_{p}}(π_{k}(ℍ̂(C_{p^{n+1}}, T HH(ℤ)); 𝔽_{p})). We thus know the number of permanent cycles in the next Tate spectral sequence Ê_{∗∗}^{∗}(C_{p^{n+1}}).

By naturality with respect to the Frobenius and Verschiebung maps, the E^{r} terms of Ê_{∗∗}^{∗}(C_{p^{n+1}}) are isomorphic to those of Ê_{∗∗}^{∗}(C_{p^{n}}) for r ≤ 2(p^{n} +⋯+p)+1, replacing u_{n} by u_{n+1}, and d^{r} = 0 for r = 2(p^{n} + ⋯ + p) + 1. Thus

Ê_{∗∗}^{r}(C_{p^{n+1}}) = E(u_{n+1}) ⊗ P (t^{p^{n}}, t^{−p^{n}}) ⊗ E(e) ⊗ P (tf)

⊕ ⨁_{m=v_{p}(i)<n} E(u_{n+1}) ⊗ P_{p^{m}+···+p}(tf){t^{i}e | v_{p}(i) = m}

for r = 2(p^{n} + ⋯ + 1).

We will show that there is a differential

(9)  d^{r}(t^{p^{n}−p^{n+1}}) = t^{p^{n}} · e · (tf)^{p^{n}+···+p}

with r = 2(p^{n+1} + ⋯ + p). The target is in total degree 2p^{n+1} − 2p^{n} − 1 ≢ 0, −1 mod 2p^{n+1}, where there are a total of n permanent cycles. These can be seen to be of the form t^{i} · e · f^{j} with

i = p^{n} − p^{n+1} + p^{k+1} + ⋯ + p
j = p^{k} + ⋯ + p ,

for 0 ≤ k < n. Thus the next infinite cycle t^{p^{n}} · e · (tf)^{p^{n}+···+p}, which is the case k = n of the formulas above, must be a boundary. Checking bidegrees, the only possible source of the differential hitting it is t^{p^{n}−p^{n+1}}. This establishes the claimed differential (9).

We also claim there is a differential

d^{r}(u_{n+1}e · t^{−p^{n+1}}) = e · (tf)^{p^{n}+···+1}

with r = 2(p^{n+1} + ⋯ + p) + 1, which upon canceling the permanent cycle e yields the desired formula

(10)  d^{r}(u_{n+1} · t^{−p^{n+1}}) = (tf)^{p^{n}+···+1} .

The target class e·(tf)^{p^{n}+···+1} is in total degree 2p^{n+1}+2p−3 ≢ 0, −1 mod 2p^{n+1}, so there are precisely n permanent cycles in this total degree. These are the classes t^{i} · e · f^{j} with

i = −p^{n+1} + p^{k+1} + ⋯ + 1
j = p^{k} + ⋯ + 1 ,

for 0 ≤ k < n. Thus the next infinite cycle e · (tf)^{p^{n}+···+1} is a boundary, and for bidegree reasons the only possible differential hitting it is given by the claimed formula (10).

These counting arguments also show that t^{p^{n+1}} survives past the last odd differential, thus establishing the inductive hypothesis for n + 1. This completes the determination of the differentials in the Tate- and homotopy fixed point spectral sequences for the groups C_{p^{n}} acting on T HH(ℤ) with coefficient in 𝔽_{p}.