The non-nil-invariance of TP

arXiv:1712.03187v1 [math.AT] 8 Dec 2017

Ryo Horiuchi

1 Introduction
In [7], Hesselholt defined a spectrum TP(X), called periodic topological cyclic
homology, for a scheme X using topological Hochshild homology and the
Tate construction, which is a topological analogue of the Connes-Tsygan
periodic cyclic homology HP defined by Hochschild homology and the Tate
construction. In [6, Theorem II.5.1], Goodwillie proved that for R an algebra
over a field of characteristic 0 and I a nilpotent ideal of R, the quotient map
R R/I induces an isomorphim on HP. In this article, we show that the
analogous result for TP does not hold, that is, there is an algebra of positive
characteristic and a nilpotent ideal such that the quotient map does not
induce an isomorphism on TP, even rationally. More precisely, we prove the
following result.

Theorem 1.1. Let p be a prime number and k 2 a natural number. Then

the canonical map TP (Fp [x]/(xk ))[1/p] TP (Fp )[1/p] is not an isomor-
phism.

In [7], Hesselholt gives a cohomological interpretation of the Hasse-Weil

zeta function of a scheme smooth and proper over a finite field using TP
inspired by [4] and [3]. In [1] and [2], it is proved that TP satisfies the
Kunneth formula for stable -categories smooth and proper over a perfect
field of positive characteristic. Therefore, the new cohomology theory TP is
considered to be an important cohomology theory for p-adic geometry and
non-commutative geometry. Our result concerns a fundamental property
of this theory. In Theorem 3.3, we evaluate the TP-group of Fp [x]/(xk )
completely.

1
2 Periodic topological cyclic homology
Periodic topological cyclic homology TP is proposed in [7]. In this section, we
recall some notions from there. We let T denote the circle group throughout
this article.
Let E be a free T-CW-complex whose underlying space is contractible.
Then we consider the following cofibration sequence of pointed T-spaces

E+ S 0 E,

here E+ is the pointed space E {} and S 0 = {0, }, and the left map
sends to the base point S 0 and all other points to 0 S 0 .
Let X be a T-spectrum. Smashing the internal hom [E+ , X] with the
above diagram and taking homotopy fixed points of a subgroup C T, we
have the following sequence called Tate cofibration sequence

(E+ S [E+ , X])C ([E+ , X])C (E S [E+ , X])C .

We write this sequence as

(
H (C, X), if C ( T
(E+ [E+ , X])C =
H (C, X) if C = T,

([E+ , X])C = H (C, X)

(E [E+ , X])C = H (C, X)

Let X be a scheme. The topological periodic cyclic homology of X is the

spectrum given by
TP(X) = H (T, THH(X)),
where THH is topological Hochschild homology defined in [5] and [BM2]. In
the present paper, we will only consider affine schemes. For a commutative
ring R, there is a conditionally convergent spectral sequence [12, 4],
2
Ei,j = S{t, t1 } THHj (R) TPi+j (R),

where deg(t) = (2, 0).

2
3 Truncated polynomial algebras
Our main result is the following
Theorem 3.1. Let p be a prime number and k 2 a natural number. Then
the canonical map TP (Fp [x]/(xk ))[1/p] TP (Fp )[1/p] is not an isomor-
phism.
Before proving our main result, we recall from [11] and [8] some calcula-
tions concerning THH(Fp [x]/(xk )) .
We give the pointed finite set k ={0, 1, x, . . . , xk1 } with the base point
0 the pointed commutative monoid structure, where 1 is the unit, 0 1 =
0 xi = 0, xi xj = xi+j , xk = 0. We denote the cyclic bar construction of k
by Ncy (k ). More precisely, the set of l-simplicies is

Ncy
l (k ) = k k ,

where there are l + 1 smash factors and the structure maps are given by
di (x0 xl ) = x0 xi xi+1 xl , 0 i < l,
dl (x0 xl ) = xl x0 x1 xk1 ,
si (x0 xl ) = x0 xi 1 xi+1 xl , 0 i l,
tl (x0 xl ) = xl x0 x1 xl1 .
We let Ncy (k ) denote the geometric realization of Ncy (k ).
In [10, Theorem 7.1], it is proved that there is a natural equivalence of
cyclotomic spectra
THH(Fp [x]/(xk ))) THH(Fp ) Ncy (k ). (a)
For each positive integer i, we also have the cyclic subset
Ncy cy
(k , i) N (k )

generated by the (i1)-simplex x x (i factors), and denote the geometric

realization by Ncy (k , i). We also have the cyclic subset Ncy
(k , 0) generated
by the 0-simplex 1 with the geometric realization Ncy (k , 0). Thus we obtain
the following wedge decomposition
_
Ncy (k , i) = Ncy (k ).
i0

We consider the complex T-representation, where d = (i 1)/k is the

integer part of (i 1)/k for i 1,

3
 d = C(1) C(2) C(d),
where C(i) = C with the T action;

T C(i) C(i)

defined by (z, w) 7 z i w. Then we have the following by [11, theorem B], for
i 1 such that i
/ kN, there is an equivalence

Ncy (k , i) S d (T/Ci )+ ,

where Ci is the i-th cyclic group.

Let THH(Fp [x]/(xk ), (x)) denote the fiber of the canonical map

THH(Fp [x]/(xk )) THH(Fp ),

and we write

TP(Fp [x]/(xk ), (x)) = H (T, THH(Fp [x]/(xk ), (x)))

The triviality of TP(Fp [x]/(xk ), (x))[1/p] shall imply that TP is not nil-
invariant up to p-inverted. In order to obtain the triviality, we use the
following decomposition.

Lemma 3.2. There is a canonical equivalence

Y
TP(Fp [x]/(xk ), (x)) H (T, THH(Fp ) Ncy (k , i)).
i1

Proof. By (a) and the wedge decomposition, we have

_
H (T, THH(Fp [x]/(xk ), (x))) H (T, THH(Fp ) Ncy (k , i)),
i1

since H (T, ) preserves all homotopy colimits.

Since the connectivity of H (T, THH(Fp ) Ncy (k , i)) goes to as i goes
to , we have
_ Y
H (T, THH(Fp ) Ncy (k , i)) H (T, THH(Fp ) Ncy (k , i)).
i1 i1

4
Similarly, since H (T, ) preserves all homotopy limits, we have
Y
H (T, THH(Fp [x]/(xk ), (x))) H (T, THH(Fp ) Ncy (k , i)).
i1

Since TP(Fp [x]/(xk ), (x)) is the cofiber of

H (T, THH(Fp [x]/(xk ), (x))) H (T, THH(Fp [x]/(xk ), (x))),
we get the desired equivalence.
It is known that, for a T-spectrum X, there is a T-equivalence
X (T/Ci )+ [(T/Ci )+ , X],
see for example [10, 8.1]. Hence, we have
H (T, THH(Fp ) (T/Ci )+ ) = (E [E+ , THH(Fp ) (T/Ci )+ ])T
(E [E+ , [(T/Ci )+ , THH(Fp )]])T
(E [(T/Ci )+ , [E+ , THH(Fp )]])T
(E (T/Ci )+ [E+ , THH(Fp )]])T
([(T/Ci )+ , E [E+ , THH(Fp )]])T
(E S [E+ , THH(Fp )])Ci
= H (Ci , THH(Fp )).
Furthermore, by [10, 3.2], we have an equivalence of spectra
H (Ci , THH(Fp ) S d ) H (Cpvp (i) , THH(Fp ) S d ),
where vp is the p-adic valuation.
Hesselholt and Madsen have calculated the homotopy groups of the above
spectra [10, 9],
H (Cpn , THH(Fp ) S d )
= SZ/pn Z {t, t1 },

where is the divided Bott element. More precisely, H (Cpn , THH(Fp )

S d ) is a free module of rank 1 over Z/pn Z[t, t1 ] on a generator of degree 2d.
A preferred generator is specified in [9, Proposition 2.5]. Combining these,
we obtain for i / kN a canonical isomorphism
(
Z/pvp (i) Z, j d + 1 even
j H (T, THH(Fp ) Ncy (k , i))
=
0, j d + 1 odd,

5
and by definition d + 1 is always odd. They have similarly calculated that
for i kN, there is a canonical isomorphism
(
cy Z/pvp (k) Z, j odd
j H (T, THH(Fp ) N (k , i)) =
0, j even.

From this, we obtain:

Theorem 3.3. If j is an odd integer, then there is a canonical isomorphism

TPj (Fp [x]/(xk ), (x))

Y Y
= Z/pvp (k) Z Z/pvp (i) Z.
i1,ikN i1,ikN
/

If j is an even integer, then

TPj (Fp [x]/(xk ), (x)) = 0.

Therefore, we get our main result by this theorem. In addition, by [13,

Corollary 1.5] and [11], we get the following as well.

Corollary 3.4. Topological negative cyclic homology is not nil-invariant,

even rationally.

Acknowledgements
I would like to thank Lars Hesselholt for his tremendous help and suggesting
me this topic. I also thank Martin Speirs for reading the draft carefully and
daily conversation and the DNRF Niels Bohr Professorship of Lars Hesselholt
for the support.

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