MODULI STACKS OF (ϕ, Γ)-MODULES: ERRATA
MATTHEW EMERTON AND TOBY GEEThe following errata and clarifications are for the published version of [EG22].
All references are to [EG22] unless otherwise specified.
We begin with some minor corrections.
• In the notation section (and throughout the book), the residue field of K
is k.
• In Remark 2.1.13, K0 (ζp∞ ) should be replaced by (K0 )cyc , and the claimthat this condition is equivalent to K being abelian over Qp should bedeleted.
The same applies in Definition 3.2.3 and Remark 3.2.4, and the discussion
between them. In particular we should define K basic := K ∩ (K0 )cyc .
(We would like to thank Dat Pham for pointing this out. This issue
was also explained to us by Léo Poyeton some years previously, but we
unfortunately confused K0 (ζp∞ ) and (K0 )cyc in our writeup.)• We should explicitly assume throughout that the coefficient field E contains K. (This is implicitly assumed at several points, but not always
explicitly asserted.)• The definition of the universal unramified character in §5.3 is incorrect if
k 6= Fp ; instead, we need to set ϕ(v) = a0 v where a0 ∈ (F ⊗k A)× hasnorm a. See Dat Pham’s [Pha22, §2]. (Note that this construction is one
place that uses the assumption mentioned in the previous point.)• In Hypothesis 7.3.1, we should take b ≥ peh/(p−1) rather than b ≥ eh/(p−1), because in the following paragraph, the valuation of T 0 should be givenby pv(p)/(p − 1) rather than v(p)/(p − 1).
In the second paragraph of the proof of Lemma 7.3.5, the image of T 0
should be ζps − 1, rather than ζps+1 − 1. Thus the image of T coincides
with the image of the trace of ζps − 1, which is in Kcyc,s and thus fixed by
GKcyc,s .
(This was again pointed out to us by Dat Pham.)• Strictly speaking, the proof of Theorem 8.6.2 is incomplete (but the gap
is easily filled); see the discussion between the statement and proof of[CEGS22, Thm. 7.6] for a complete proof.
• In the first paragraph of the proof of Theorem 6.3.2, the representation ρ (incrys,λthe deformation ring Rρ
) is equal to ρd . (Thanks to Matteo Tamiozzo
for this.)• Contrary to the claim in Section 5.3, it is not always possible to choose thecharacters ψn in such a way that if n, n0 have (ψn ψn−10 )|IK = |IK , then−1
in fact ψn ψn0 = . The only place that this assumption was used was(implicitly) in the definitions of the characters ωk,i before Definition 5.5.11;
accordingly, we simply replace the ωk,i by unramified twists in such a way
1

2

M. EMERTON AND T. GEE

that if kσ,i − kσ,i+1 = p − 1 for all σ, then ωk,i = ωk,i+1 . (Again, we thank
Dat Pham for pointing this out to us.)
We would like to thank Dat Pham for pointing out that it is not clear that theproof of Theorem 5.5.12 is complete. More precisely, without making additional
arguments, it is not obvious that our constructions cover all of Xd,red (Fp ). Since
the proof of Theorem 5.5.12 is quite involved, rather than attempt to describe
the additional arguments needed, we instead give a slightly different and morestreamlined complete proof.
In particular, we allow ourselves to refer forwards to Theorem 6.5.1 , whichstrengthens some of the conclusions of Theorem 5.5.12 , showing that the stacks
small
in the statement of Theorem 5.5.12 are empty. This requires some justiXd,red,Fpfication, because the proof of Theorem 6.5.1 relies on Theorem 5.5.12 ! However,the proof of Theorem 5.5.12 is by induction on d, and the proof of Theorem 6.5.1
for any particular d only uses Theorem 5.5.12 for that value of d, so we are free
is empty for each d0 < d.
to assume in the proof of Theorem 5.5.12 that Xdsmall
0 ,red,Fp

small
Accordingly, we have also removed Xd,red,Ffrom the statement of the followingptheorem.

Theorem (Theorem 5.5.12 ).
(1) The Ind-algebraic stack Xd,red is an algebraic stack, of finite presentation
over F.
(2) Xd,red is equidimensional of dimension [K : Qp ]d(d − 1)/2. We can writekk(Xd,red )Fp as a union of irreducible components Xd,red,F , where each Xd,red,Fpp
is generically maximally nonsplit of niveau one and weight k, and the corresponding eigenvalue morphism is dominant (i.e. has dense image in (Gm )dk ).
(3) If we fix an irreducible representation α : GK → GLa (Fp ) (for some a ≥ 1),then the locus of ρ in Xd,red (Fp ) for which dim HomGK (ρ, α) ≥ r (for any
r ≥ 1) is (either empty, or) of dimension at most
[K : Qp ]d(d − 1)/2 − dr (a2 + 1)r − a /2e.
(4) If we fix an irreducible representation α : GK → GLa (Fp ) (for some a ≥1), then the locus of ρ in Xd,red (Fp ) for which dim Ext2GK (α, ρ) ≥ r is ofdimension at most[K : Qp ]d(d − 1)/2 − r.
Proof. Recall that a closed immersion of reduced algebraic stacks that are locallyof finite type over Fp which is surjective on finite-type points is necessarily an isomorphism. As recalled above, Xd,red is an inductive limit of such stacks (indeed, byaaLemma A.9 , we have Xd,red = lim Xd,h,s,red, where the Xd,h,s
are as in Section 3.4 ),
−→
k
small
and so if we produce closed algebraic substacks Xd,red,F and Xd,red,F of Xd , theppunion of whose Fp -points exhausts those of Xd,red , then Xd,red,Fp will in fact be anksmall
algebraic stack which is the union of its closed substacks Xd,red,Fand Xd,red,F .
pp

Thus (1) is an immediate consequence of (2) (where the “union” statement in (2)is now to be understood on the level of Fp -points). In fact, it suffices to constructk
the closed substacks Xd,red , and to show that the remaining Fp -points of Xd,red
small
are contained in a closed substack Xd,red,F, which is a finite union of finite typep

MODULI STACKS OF (ϕ, Γ)-MODULES: ERRATA

3

algebraic substacks of dimension less than [K : Qp ]d(d − 1)/2; indeed, this obviously suffices to prove (1), and it also suffices to prove (2) by an application ofTheorem 6.5.1 . (See the discussion before the theorem for an explanation of why
this is not a circular argument.)
Claim (4) follows from (3) (with α replaced by α ⊗ ) by Tate local duality andthe easily verified inequalitydr (a2 + 1)r − a /2e ≥ r.
Thus it is enough to prove (2) and (3), which we do simultaneously by induction
on d.
As recalled in Remark 5.3.4 , there are up to twist by unramified characters
only finitely many irreducible Fp -representations of GK of any fixed dimension.
Accordingly, we let {αi } be a finite set of irreducible continuous representations
αi : GK → GLdi (Fp ), such that any irreducible continuous representation of GK
over Fp of dimension at most d arises as an unramified twist of exactly one of the αi .
We let the 1-dimensional representations in this set be the characters ψn defined in
Section 5.3 .
Each αi corresponds to a finite type point of Xdi ,red , whose associated residual
gerbe is a substack of Xdi ,red of dimension −1: the morphism Spec Fp → Xdi ,redcorresponding to αi factors through a monomorphism [Spec Fp /Gm ] → Xdi ,red .
It follows from Lemma 5.3.2 that for each αi there is an irreducible closed zerodimensional algebraic substack of Xdi ,red of finite presentation over Fp which contains a dense open substack whose Fp -points are the unramified twists of αi .
kIn particular, if d = 1, then we let X1,red be the zero-dimensional stack constructed in the previous paragraph; this satisfies the required properties by definition, so (2) holds when d = 1. For (3), note that if r > 0 then we must have a = 1,
and then the locus where HomGK (ρ, α) is non-zero (equivalently, 1-dimensional) is
exactly the closed substack of dimension −1 corresponding to α, so the required
bound holds.
We now begin the inductive proof of (2) and (3) for d > 1. In fact, it will be
helpful to simultaneously prove additional statements (2’) and (4’), which we begin
by stating. (It is trivial to verify that the discussion above also proves (2’) and (4’)
when d = 1.)
k,fixed(2’) is as follows: for each k, there is a closed irreducible algebraic substack Xd,red,Fpof (Xd,red )Fp of finite presentation over Fp and dimension [K : Qp ]d(d − 1)/2 − 1,which is generically maximally nonsplit of niveau 1 and weight k, and furthermorekhas the properties that the corresponding character ν1 is trivial, and that Xd,red,Fpk,fixedis obtained from Xd,red,F by unramified twisting. (The assumption that there ispk,fixeda dense open substack of Xd,red whose Fp -points correspond to representations
which are of niveau 1 and are maximally nonsplit implies in particular that thecorresponding family is twistable).
×
We now turn to (4’). For each character α : GK → Fp , and each k, let Xαkbe the locally closed substack consisting of those ρ in Xd,red,F (Fp ) for whichpdim Ext2GK (α, ρ) 6= 0. Then by (4), the dimension of Xα is at most [K : Qp ]d(d −1)/2 − 1. Then (4’) is as follows: if Xα has dimension [K : Qp ]d(d − 1)/2 − 1,then after replacing α by an unramified twist, Xα contains a dense open substack4

M. EMERTON AND T. GEEk,fixedkk,fixedUd,red,F of Xd,red,F , and the complement Xα \ Ud,red,F has dimension at mostppp

[K : Qp ]d(d − 1)/2 − 2.
We now prove the inductive step, so we assume that (2), (2’), (3) and (4’) hold in
dimension less than d. Let k d−1 be the Serre weight in dimension (d − 1) obtainedby deleting the first entry in k. By hypothesis there are corresponding (dense)
kd−1 ,fixed
kd−1
kd−1 ,fixed
kd−1
open substacks Ud−1,redand Ud−1,red,Fof Xd−1,red
and Xd−1,red,F
respecpptively, which are maximally nonsplit of weight k. By our inductive construction,
kd−1
kd−1 ,fixed
we can and do assume that Ud−1,red,F
by unramified
is obtained from Ud−1,redptwisting. (These substacks are of course not uniquely determined, and we will allow
ourselves to further shrink them in the below.)
Set α := 1−d ωk,1 . If kσ,1 − kσ,2 = p − 1 for all σ then we say that we arek,fixedd−1
in the très ramifiée case, and we let U denote Ud−1,red,F; otherwise, we let U
p

k

d−1
. In the latter case, it follows from our inductive hypothesis (moredenote Ud−1,red,Fpprecisely, from the assumption that (2) and (3) hold in dimension (d − 1)), together
with Tate local duality, that after possibly replacing U with an open substack we
can and do assume that for each Fp -point u of U, we have Ext2GK (α, r̄u ) = 0,
where r̄u is the (d − 1)-dimensional representation corresponding to u. If we arein the très ramifiée case then it follows similarly that after replacing U with anopen substack we can and do assume that for each Fp -point u of U, Ext2GK (α, r̄u )is 1-dimensional. (Note that the condition that Ext2GK (α, r̄u ) is 1-dimensional is apriori only locally closed, but by hypothesis we can pass to an open substack where
the dimension is at least 1, and the condition that the dimension is exactly 1 isthen open.)k,fixedWe let T be an irreducible scheme which smoothly covers U, and we let Xd,red,Fp

be the irreducible closed substack of (Xd,red )Fp constructed as the scheme-theoretic
image of V in the notation of Proposition 5.4.4 . Part (2) of that proposition, tok,fixedgether with the inductive hypothesis, implies that Xd,red,F has the claimed dimenp

sion. (Note in particular that if K = Qp , then condition (2d) of Proposition 5.4.4
holds. Indeed, if we are in the très ramifiée case then condition (2d)(iii) holds, andotherwise the inductive hypothesis that the image of the eigenvalue morphism isdense in (Gm )d−1
implies that condition (2d)(i) holds after possibly further shrinkk
d−1kk,fixeding U.) We then let Xd,red,F be the substack obtained from Xd,red by twisting bypunramified characters, which has the claimed dimension by Lemma 5.3.2 .
k,fixedk
By construction, we see that the stacks Xd,red,F and Xd,red,F satisfy the proppperties required of them by (2) and (2’) (other, of course, than the claim that theyexhaust the irreducible components of Xd,red,Fp ); except that in the très ramifiéek,fixedkcase, we need to check that Xd,red,F (and consequently Xd,red,F ) is genericallyppmaximally nonsplit of niveau 1 and weight k. More precisely, in this case weneed to check the condition that the final extension is not just nonsplit, but is(generically) très ramifiée. This follows from Proposition 5.4.4 (3). Indeed by our
kd−1
inductive assumption that Xd−1,red,Fis generically maximally nonsplit of niveau 1pand weight k d−1 , and the image of the eigenvalue morphism is dense, we see that

MODULI STACKS OF (ϕ, Γ)-MODULES: ERRATA

5

we are in case (3)(b)(i) unless  is trivial and furthermore we have kσ2 − kσ3 = p − 1
for all σ, in which case we satisfy (3)(b)(ii).
small
small
. Let Xd−1,red,F
To complete the proof of (2), we need to construct Xd,red,Fppdenote the union of the closed substacks
k

k

d−1
d−1
Xd−1,red,F\ Ud−1,red,F.
pp

By Proposition 5.4.4 , Tate local duality, upper semi-continuity of the fibre dimension, and the inductive hypothesis, we see that for each 1 ≤ i ≤ d, each αi (ofdimension ai , say), and each s ≥ 0 there is a finitely presented closed algebraic
substack Xs,αi ,Fp of (Xd,red )Fp , whose Fp -points contain all the representations of
the form 0 → ρd−ai → ρ → αi → 0 for which dimFp Ext2GK (αi , ρd−ai ) = s, andwhose dimension is at most[K : Qp ](d − ai )(d − ai − 1)/2 − ds((ai 2 + 1)s − ai )/2e + [K : Qp ]ai (d − ai ) + s − 1.
small
Furthermore, if a = 1, then the locus where ρd−1 is an Fp -point of Xd−1,red,F
is ofpdimension strictly less than this. (Here we use (4’), which shows in particular thatsmallof points where dimFp Ext2GK (αi , ρd−1 ) = 1 has dimension
the locus in Xd−1,red,Fp
at most [K : Qp ](d − 1)(d − 2)/2 − 2.)
These stacks are only nonzero for finitely many values of s. For fixed ai , we seethat as a function of s, this quantity is maximised by s = 0, as well as by s = 1 whenai = 1. (To see this, we have to maximise the quantity s − ds((ai 2 + 1)s − ai )/2e.
Suppose firstly that ai > 1. Then if s = 0 we have 0, while if s > 0 we haves−ds((ai 2 +1)s−ai )/2e ≤ s−s((ai 2 +1)s−ai )/2 ≤ s−s(ai 2 +1−ai )/2 ≤ s−3s/2 < 0.
Meanwhile if ai = 1, then for s = 0 we have 0, for s = 1 we have 1 − d1/2e = 1,
while for s > 1 we have s − ds(2s − 1)/2e ≤ s − s(2s − 1)/2 ≤ s − 3s/2 < 0.) It
follows that as a function of ai the bound is maximised at ai = 1 and s = 0 or 1,
when it is equal to [K : Qp ]d(d − 1)/2 − 1, and it is otherwise strictly smaller.
By Lemma 5.3.2 , it follows that the locus in (Xd,red )Fp of representations of
the form 0 → ρd−ai → ρ → α0 → 0, with α0 an unramified twist of αi for which
dimFp Ext2GK (α0 , ρd−ai ) = s, is of dimension at most [K : Qp ]d(d − 1)/2, withequality holding only if ai = 1 and s = 0 or 1.
small
Putting this together, we claim that (2) holds in dimension d if we take Xd,red,Fp
to be the union of the twists by unramified characters of the substacks Xs,αi forwhich dim αi > 1 or s > 1, together with the union of the twists by unramified characters of the substacks of the Xs,αi ,Fp for which dim αi = 1, s = 0 or 1,
small
small
and ρd−1 is an Fp -point of Xd−1,red,F. Indeed, by construction, Xd,red,F
is cerpptainly a finite union of closed substacks of dimension less than [K : Qp ]d(d − 1)/2,
and in order to see that it exhausts the remaining points of Xd,red,Fp (Fp ), weonly need to verify that if k is in the très ramifiée case, and ρ corresponds to a
kd−1
point of Ud−1,red,Fwith dimFp Ext2GK (α, ρ) = 1, then every extension of α by ρpkis contained in Xd,red,F . By construction, it is enough to check that such a ρ ispk,fixedkk,fixedd−1
d−1
d−1
automatically a point of Ud−1,red,F. But Ud−1,red,F
is obtained from Ud−1,redppby unramified twisting, and since ρ is maximally nonsplit, the hypothesis thatkd−1 ,fixed
Ext2GK (α, ρ) 6= 0 forces ρ to be a point of Ud−1,red. This completes the proofof (2).

6

M. EMERTON AND T. GEE

We now prove (3) in dimension d. In the case r = 0, there is nothing to prove.
If dim HomGK (ρ, α) ≥ r ≥ 1, then we may place ρ in a short exact sequence
0 → θ → ρ → α⊕r → 0,
where θ is of dimension d − ra < d. We may apply part (2) so as to find that
Xd−ar,red,Fp has dimension at most [K : Qp ](d − ar)(d − ar − 1)/2. Let Us be thelocally closed substack of Xd−ar,red,Fp over which dim H 2 (GK , θ ⊗ α∨ ) = s; by the
inductive hypothesis, thislocus has dimension at most [K : Qp ](d − ar)(d − ar −1)/2 − s (a2 + 1)s − a /2, and over this locus we may construct a universal familyof extensions
0 → θ → ρUs → α⊕r → 0.
The locus of ρ we are interested in is contained in the scheme-theoretic image ofthis family in (Xd,red )Fp , and Proposition 5.4.4 shows that this scheme-theoretic
image has dimension bounded above by[K : Qp ](d − ar)(d − ar − 1)/2 − s((a2 + 1)s − a)/2 + r([K : Qp ]a(d − ar) + s) − r2=[K : Qp ]d(d − 1)/2 − (r(a2 + 1)r − a)/2 − (r − s)2 /2 − (ar(ar − 1))/2≤[K : Qp ]d(d − 1)/2 − (r(a2 + 1)r − a)/2.
Since this conclusion holds for each of the finitely many values of s (and since
the dimension is an integer, allowing us to take the floor of this upper bound), we
have proved (3).
Finally, we prove (4’) in dimension d. Assume that Xα has dimension [K :Qp ]d(d−1)/2−1. By Tate local duality, the condition Ext2GK (α, ρ) 6= 0 is equivalentto HomGK (ρ, α ⊗ ) 6= 0; so if ρ is a point of Xα , then ρ admits α ⊗  as a Jordan–Hölder factor. It follows immediately that Xα is essentially twistable, so that theksubstack of Xd,red,F given by unramified twists of Xα has dimension [K : Qp ]d(d −pk1)/2, so is dense in Xd,red,F . In particular, it contains a dense open substack ofpkUd,red,F .
pkRecalling that Ud,red,F is maximally nonsplit, it follows that after replacing αpk,fixedby an unramified twist, and possibly shrinking Ud,red,F , the stack Xα contains
pk,fixedk

Ud,red,F . Furthermore if Xα0 := Xα \ Ud,red,F had dimension [K : Qp ]d(d − 1)/2 − 1ppthen we could repeat this argument with Xα replaced by Xα0 , and conclude thatk,fixedXα0 contains a dense open substack of an unramified twist of Ud,red,F , which is apcontradiction. This completes the proof of the theorem.

References[CEGS22] Ana Caraiani, Matthew Emerton, Toby Gee, and David Savitt, The geometric Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate Galois representations, 2022.
[EG22]
Matthew Emerton and Toby Gee, Moduli stacks of étale (ϕ, Γ)-modules and the existence of crystalline lifts, Annals of Math. Studies (to appear) (2022).
[Pha22]
Dat Pham, The Emerton–Gee stack of rank one (ϕ, Γ)-modules, 2022.

MODULI STACKS OF (ϕ, Γ)-MODULES: ERRATA

7Email address: emerton@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S. University Ave.,
Chicago, IL 60637
Email address: toby.gee@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK