C2(E)]e-
Since [L00 fill>00 , S1 A 4>C2 (E) ]e is non-equivariant, we can use Proposition 2.3.8 to state
[L00 ru1>00 , S1 A 4>C2 (E) ]e = [L00 fill>00 , S1 A Hk ]e
= H1 (fill>00 ; k).
By a similar argument, we have that
We know that H 1 (rui>1; k) = k and using the Universal Coefficient Theorem we have that
H 1(rui>00 ;k) = Hom(lF2,k) is k when the characteristic of k is 2, and O else. Thus, if k is
not characteristic two then there exists no x e H 1 (fill>00 ; k) that maps to the unit in k, so HF would
not be real oriented.
In the other direction, say k is characteristic two. We can use the cofibration sequence CJP1' ➔
CJP00' ➔ ClP0'0 /CJP1' to induce the following exact sequence:
By the computations above this gives an exact sequence:
Since k is characteristic two, we get
45
The C2-action on the subset CJP1 c CJP00 is closed, therefore (CJP00 /CJP1) c2 ==J RJ0P0 '/JRJP1 'which
is connected. Since JRJ0P0 '\JRJP1 'is connected then Ext!(Ho(fill' 00 \JRJP1;' Z), k) =E x1i:(Z, k) =0 .
Further, since JRJ0P0 '\JRJP1 'has no 1-cells then H l (JRJ0P0 '\JRJP1;' Z) = 0. So the Universal Coefficient
Theorem tells us that H1 ( ( CJP00 / CJP1) C2; k) = 0. Then the map k ~ k in this exact sequence is
injective, which makes it an isomorphism since k is finite by assumption. So the identity element
maps to the identity element, therefore HF is real oriented. □
Now that we know some examples of C2-determined C2-Mackey fields F which have an
Eilenberg-Mac Lane spectrum that is real oriented, we can use Corollary 3.5.2 to obtain the
following result.
Lemma 3.5.4. Let F be a C2-determined C2-Mackey field. If HF is real oriented, then HF* (MU~)
is a free HF* -module, that is,
where deg(bi) = ip.
Proof. Since HF is real oriented, we can use Corollary 3.5.2 to show that
MUIR. A HF== HF A /\ s0 [siP]
i~l
which gives an isomorphism of RO ( C2)-graded Green functors
for deg(bi) = ip. □
There is a classical standard argument which is a result of Cartan and Eilenberg's Theorem
X.6.1 in [CE99]. The argument is that fork a commutative ring, and A a commutative k-algebra
that is flat as a module over k, then
46
Using the homological algebra from [LM06] we can extend Cartan and Eilenberg's argument to
the equivariant setting. That is, if R~ is a commutative G-Green functor, and M * is a commutative
R* -algebra that is flat as a module over R*, then
M * □Ji* M~p ( ) ~ M * ( ) Tor*,* M*,M* = M* □!i* Tor;,-* R*,R*.
We will use this in our calculations.
From the discussion in the beginning of this section and Proposition 3.5.3 we know that for the
C2-Mackey field F where F(C2/C2) = lF2, and F(C2/e) = 0, HF has a trivial C2-action and is
real oriented.
Theorem 3.5.5. For F the C2-Mackey field where F ( C2/ C2) = lF2, and F ( C2/ e) = 0,
as an HF *-module. Here lbil = ip and lzil = 1 + ip.
Proof. Proposition 3.5.3 shows that HF has a trivial C2-action. In order to use the Bokstedt
spectral sequence, we need to show that HF* (MUR) is flat over HF*. The following isomorphism
of RO( C2)-graded Green functors is given by Lemma 3.5.4:
where deg(bi) = ip. Therefore HF *(MUJR) is flat over HF*" Since the appropriate conditions
hold, we can use the equivariant Bokstedt spectral sequence
where deg(bi) = (0, ip) and deg(zi) = (1, ip ).
Recall that dr: E; ,a ~ Ers -r,a +r - 1. Our spectrum MUJR is commutative, so by [AGH+22,
Proposition 4.2. 8] we can view this as a spectral sequence of HF* -algebras. Consider the differential
47
d2. We know that all the differentials are determined by what the differential does on the generators
of the E2 page, thus since the only generators are in the columns where s = 0, 1 then all of the
differentials on the E2-page are zero and the spectral sequence collapses. □
48
CHAPTER4
HOPF STRUCTURE OF THE BOKSTEDT SPECTRAL SEQUENCE
Throughout this section, let R be a commutative ring, and A a commutative ring spectrum.
Spectral sequences can have algebraic structures, and these structures can be very helpful when
doing computations with said spectral sequences. More specifically, the algebraic structure of a
spectral sequence can help one know more about the differentials of the spectral sequence. As
mentioned in Section 3 .2, the Bokstedt spectral sequence is one of the main tools we have to compute
THH. In this section we will recall results of Angeltveit and Rognes in [AR05] which show that
the Bokstedt spectral sequence has a Hopf algebra structure. These results we will recall extend
the results of [EKMM97] and [MSV97] which demonstrate that for a commutative ring spectrum
A, THH(A) is an A-Hopf algebra. In the future sections, namely Chapter 5 and Chapter 6, we
will prove an equivariant analogue to these results for twisted THH and the equivariant Bokstedt
spectral sequence, so this section is dedicated to recalling these classical results. We will start this
section by recalling the algebraic definition of R-bialgebras and R-Hopf algebras.
Definition 4.0.1. Let R be a commutative ring. An R-bialgebra Mis a unital, associative R-algebra
as well as a counital, coassociative R-coalgebra such that the following diagrams commute:
49
THH(A) AA THH(A)
¢Aid! j ¢
THH(A) AA THH(A) -------THH(A)
to checking the commutativity of:
s1 v s1 v s1 idV ---+-S 1 v s1 • • • • •
¢Yid! !¢
Recall that i
j i.
There are also maps t: Cn+ 1 ---+ Cn+ 1 such that t( yi) is yi+ 1 for j < n and 1 for j = n. It is notable
that dn = do o t: Cn+ 1 ---+ Cn+ 1 for all n.
Angeltveit and Rognes' classical argument, recalled in Chapter 4, requires additional models
of the circle. For our equivariant proof, we will also need additional models and will construct
these using the simplicial edgewise subdivision functor defined by Bokstedt, Hsiang, and Madsen
in [BHM93]. The simplicial r-fold edgewise subdivision functor, sdr(-), is defined so that for a
simplicial object x.,
with face and degeneracy maps di and Si defined by
di = di O di+n+l O • • • 0 di+(r-l)(n+l)
Si = Si+(r-l)(n+2) 0 • • • 0 Si+(n+2) 0 Si
for di and Si the face and degeneracy maps of the simplicial object x •.
Remark 5.0.1. Recall the simplicial relation that di o dj = dj-l o di if i < j and that in sl
dn = do o t: Cn+l ---+ Cn+l· It is also true that in sdr(Sl), dn = do o t. To see this, consider
that do = do o dn+l o ... o d(r-l)n+r-l and using the simplicial relation mentioned above, we
can move do to the front and get that do = dn o d2n+l o ... o d(r-l)n+r-2 o do. Consider
do o t = dn o d2n+l o ... o d(r-l)n+r-2 o do o t = dn o d2n+l o ... o d(r-l)n+r-2 o drn+r-l which is
dn.
Let us start by understanding the 2-fold edgewise simplicial subdivision of the circle, sd2(Sl).
Example 5.0.2. Let us refer to sd2(Sl) as 2s¼. By definition,
sd2(Sl)n = sin+l'
di= di o di+n+l• and
57
Therefore 2S ! is
c6
1 t I t 1
do so J1 s1 d2
.!-I -!-I .!-
c 4
I t I
Jo so J1
-!-I -!-
c2
One can see that the only nondegenerate elements are 1, y E C2 and y, y 3 E C4, where the boundary
of the 1-cell y is defined by
and the boundary of the I-cell y3 is defined by
Therefore 2S ! looks like:
where the C2-action on 2S! is induced from applying the functor sd2(-) to s!. This action sends
. . n
y1 to yi+2, where yn = I in Cn. Therefore the C2-action on 2S! is counter clockwise rotation by
180°.
Remark 5.0.3. If we consider 2S! non-equivariantly, it is not the same as dS! as defined in [AR05]
and discussed earlier in Chapter 4. Non-equivariantly, 2s! is equivalent to d's! as defined in
[AR05, Remark 3.6].
58
Example 5.0.4. We can similarly build 4S! := sd4(S!). Note that this can be constructed by
considering sd4(S!) or sd2(sd2(S!)). By definition,
Therefore sd4 (S!) is
And 4S ! looks like:
sd4(S!)n = sJn+3,
di= di O di+n+l O di+2n+2 ° di+3n+3, and
Si = Si+3n+6° Si+2n+4° Si+n+2° Si.
C12
tit
do so d1 s1 d2
..!- I -!-I ..!-
cs
It I
do so d1
-!-I -!-
c4
where the induced C4-action is counter clockwise rotation by 90° and the induced C2-action is
counter clockwise rotation by 180°.
We can use this process to define mS! for any positive integer m, which will have the Cm-action
of counter clockwise rotation by ( 3!0)°. Notice that in order for mS! to have a simplicial Cn-action
of counter clockwise rotation by ( 3~0)° then m must be a multiple of n. Consider two examples of
C3-equivariant simplicial models of the circle; 3S! and 6S!:
59
where the induced C3-action on both of these simplicial objects is counter clockwise rotation by
120°.
In Section 3.4 we recalled the definition of twisted THH as defined by Angeltveit, Blumberg,
Gerhardt, Hill, Lawson, and Mandell in [ABG+l8]. We also discussed the different perspectives
these authors gave us on twisted THH including, suppressing some change of universe notation,
that for R a commutative ring Cn-spectrum THHcn (R) ~ R ®en S1. The following proposition
demonstrates which simplicial model of the circle is suitable for this perspective.
Proposition 5.0.5. Let R be a commutative ring C p-spectrum indexed on the trivial universe R 00 ,
for p prime. Then R ®cp pS! ~ B~y,Cp (R), the Cp-twisted cyclic bar construction.
Proof. Let μ and T/ be the multiplication and unit maps of R respectively. To show that these
simplicial objects are equivalent we will first show that every level is the same and then we will
show that they have equivalent face and degeneracy maps.
The k-simplicies of R ®cp pS! are defined by the following coequalizer diagram
where the map r is the Cp-action on pS} and f is the induced Cp-action on R.
Let Cp = (y), and pS} = Cpk+p = {l,x, ... ,xPk+p-l }. The induced Cp-action on the set
of elements Cpk+p is defined by yxi = xj such that j = i + k + 1( mod pk+ p). There is a
Cpk+p-action on pS} induced by t: Cpk+p ➔ Cpk+p defined by t(xi) = xj such that j = i + 1(
mod pk+ p).
p-l pk+p-l
As Cp-sets, Cp ®pS} = Cp xCpk+p· Thus R®Cp ®pS} can be written as /\ ( /\ Rxs,yt).
t=O s=O
For ease of notation let us write Rs,t instead of Rxs,yt• Similarly, R ® pS} can be written as
pk+p-l
I\ Rs, With this notation id® r: Rs,t H Rj such that j = s + t(k + 1)( mod pk+ p), and
s=O
f ® id: Rs,t H yt Rs where yt R indicates R which has been acted on by y 1• By definition of the
coequalizer, R ®cp pS} is the quotient space of R ® pS} where the quotient forces these two actions
to agree. Recall that y1 xs = xj for j = s + t(k + 1) ( mod pk+ p). Therefore R ®cp pS} = RA.k+ 1.
60
We will now show that the face and degeneracy maps from R ®cp pS} are equivalent to the
face and degeneracy maps from B~y,Cp (R). We will start by considering the face and degeneracy
maps of pS} and induce the corresponding face and degeneracy maps of R ®cp pSk-
Recall that pS} = sdp(Sl)k with face and degeneracy maps di and Si defined as follows:
di = di O di+k+l O • • • O di+(p-I)(k+I)
Si= Si+(p-l)(k+2) o · · · o Si+(k+2) o Si
where di and Si are the face and degeneracy maps of sl, and 0 ~ i ~ k.
Let us start by finding what the induced face maps are on R ®cp pS}, say 8i: R ®cp pS} ~
R ®cp pS}_1, for 0 ~ i ~ k and k > 0. The map (id® di): R ® pS} ~ R ® pS}_1 applies the
multiplication map to Ri I\ Ri+I as well as Ri+n(k+I) I\ Ri+I+n(k+I) for all 0 ~ n < p. Therefore
8i: R ®cp pS} ~ R ®cp pS}_1 is the map idJ\i Aμ I\ idAk-i-l for 0 ~ i < k.
Before figuring out what 8k must be, recall that pS} has a Cpk+p-action induced by the map
t. Consider (id® t): R ® pS} ~ R ® pS}, this map rotates the last copy of R to the front. This
map also rotates Rk into the position Rk+I was in, this is important as in the quotient R ®cp pS}
we have that the following two are equivalent:
(l' ® id)(Ro,1) = yRo
(id® r)(Ro,1) = Rk+I
so the map that is induced on R ®cp pS} rotates the last copy of R to the front and acts on that
copy of R by y. Let us suggestively refer to this induced map as ak.
Recall from Remark 5.0.1 that dk = do o t, so the last face map 8k is induced from id® dk =
(id® do) o (id® t): R ® pS} ~ R ® pS}_1. The universal property of the coequalizer shows
that the maps (id® do) and (id® t) induce maps on R ®cp pS}, namely 80 and ak respectively.
Further, by the uniqueness property, the map induced from their composition, (id® do) o (id® t)
must be equivalent to the composition of the induced maps. Meaning, 8k = 80 oak.
61
We similarly induce the degeneracy maps of R ®cp pS}, say a-i: R ®cp pS}---+ R ®cp pSl+ 1,
for 0 ~ i ~ k and k ~ 0. By a similar argument as above we can show that these can be written as
U-i = idl\i+ 1 A 77 A idAk-i.
Recall that B?'Cp (R) = RAk+l and the face and degeneracy maps from this level are the
following:
dk = do oak
s j = idAj+ l A 77 A idAk-j
for O ~ i < k and O ~ j ~ k.
Therefore, R ®cp pS! is isomorphic to B~y,Cp (R). □
A result of this proposition is that for R a commutative ring C p-spectrum indexed on C p-universe
iJ, l(It0 fR) ®cp pS!I~ THHcp(R). We can construct similar structures (I"00 3R) ®cp mpS!,
and we will refer to I( I"00 3R) ®cp mpS! I as m THHcp (R).
An equivariant analogue of Angeltveit and Rognes' result [AR05, Lemma 3.8] shows the
following result.
Proposition 5.0.6. Let U be a complete S 1-universe, and let iJ := ic U. Let R be a commutative
p
ring Cp-spectrum indexed on the Cp-universe U,for p prime. Then there is a Cp-weak equivalence
It will also be important to consider simplicial objects that look the same non-equivariantly to
pS!, but have different Cp-actions. For example, one can consider what looks like 5S!
vo vv12o v4 v3
but with the C5-action of counter clockwise rotation by 144°, 216°, or 288° instead of the usual
counter clockwise rotation by 72°. Let us denote the Cp-simplicial space that resembles pS! but
has the Cp-action of counter clockwise rotation by (:360) 0 for 1 < n Yid:3 S! V c3 3S! V c3 3S! ~ 3S! V c3 3S! folds the outer copy of 3S! with the middle copy of
3S! and leaves the inner copy of 3S! alone. Similarly, id V 3: S! V c3 3S! V c3 3S! ~ 3S! V c3 3S!
folds the inner copy of 3S! with the middle copy of 3S! and leaves the outer copy of 3S! alone.
Proposition 5.1.5. Let p be prime. For a commutative ring Cp-spectrum R, THHcp (R) is a
commutative R-algebra in the category of commutative ring Cp-spectra.
Proof. We begin by checking associativity of the product map: THHcp (R) AR THHcp (R) ~
THHcp (R). For ease of notation, let T := THHcp (R). We need to verify that the following
diagram commutes:
It is sufficient to show that the following diagram of C p-simplicial spaces commutes
where pS! Vcp pS! Vcp pS! is the pushout of the span pS! Vcp pS! ~ Cp ~ pS!. This is
thought of as some equivariant analogue to the wedge of three circles, Example 5.1.4 shows this
for p = 3. Note that id V and id fold the inner two copies of pS! together and the outer two
copies of pS! together respectively where folds the two copies of pS! together. Therefore this
diagram commutes.
68
To check unitality and commutativity of the product map, we need to show that the following
diagrams commute:
1 idV17 1 1 17Vid 1
pS. Yep Cp -----+ pS. Yep pS. ----- Cp Yep pS.
pS!
where pS! Yep Cp ~ pS! is the pushout of the span pS! ~ Cp ---+ Cp. The map id Y 7J: pS! Yep
Cp ---+ pS! Yep pS! is the identity on pS! and includes Cp into the second copy of pS!, similarly
7J Yid: Cp Yep pS!---+ pS! Yep pS! is the identity on pS! and includes Cp into the first copy of
p S ! . The map T swaps the first and second copies of p S ! .
For the unitality diagram, note that ➔ 2pS! ---"'--➔ pS! V Cp pS!
~v~l 1¢v¢
where ,fr' and