2. The Carlitz-Witt suite 6
2.4. Carlitz-Witt vectors
Parroting Definition 2.3, we define:
Definition 2.16. Let Nbe aq-nest. Let Abe a commutativeFq[T]-algebra. The Carlitz ghost ring of A will mean the Fq[T]-algebra AN with componentwise Fq[T]-algebra structure (i. e., a direct product of Fq[T]-algebras A indexed over N). TheCarlitz N-ghost map wN : AN → AN is the map defined by
wN (xP)P∈N=
∑
D|P
D P
D
(xD)
P∈N
for all (xP)P∈N ∈ AN. ThisN-ghost map isFq-linear but (generally) neither multiplicative norFq[T] -linear.
From the equivalenceC1 ⇐⇒ D1 in Theorem 2.13, we can obtain:5
4Non-rhetorical question. Please let me know! (darijgrinberg[at]gmail.com)
5I’m not going to show the proof, as I don’t think you will have any trouble reconstructing it.
One has to set A=Fq[T] [Ξ], whereΞis a family of indeterminates, and define morphisms ϕP by ϕP(Q) = Q([P] (Ξ)), where [P] (Ξ) means the family obtained by applying [P] to
Theorem 2.17. Let N be a q-nest. There exists a unique functor WN : CRingF
q[T] →CRingF
q[T] with the following two properties:
– We haveWN(A) = AN as a set for every commutativeFq[T]-algebra A.
– The map wN : AN → AN regarded as a mapWN(A) → AN is anFq[T] -algebra homomorphism for every commutativeFq[T]-algebra A.
This functorWN is called theCarlitz N-Witt vector functor. For everyFq[T] -algebra A, we call theFq[T]-algebraWN(A)theCarlitz N-Witt vector ring over A.
The map wN : WN(A) → AN itself becomes a natural transformation from the functor WN to the functor CRingFq[T] → CRingFq[T], A 7→ AN. We will call this natural transformationwN as well.
This theorem, of course, yields that the sum and the product of two Carlitz ghost-Witt vectors over any commutative Fq[T]-algebra is a Carlitz ghost-Witt vector, and that any Fq[T]-multiple of a Carlitz ghost-Witt vector is a Carlitz ghost-Witt vector.
But this result is not optimal. In fact, it still holds in the more general setup of Remark 2.15. This can no longer be proven using Theorem 2.17, since the polynomial ring Fq[T] [Ξ] is a free commutative Fq[T]-algebra but not (in a reasonable way) a free object in the category of Fq[T]-modules A with an Fq -linear Frobenius mapF : A→ Awhich satisfies (2). I will lose some more words on this in Subsection 2.5.
Remark 2.18. Let N be a q-nest. The Fq-vector space structure on theFq[T] -algebraWN(A) is just componentwise. Thus,wN is anFq-vector space homo-morphism when considered as a map AN → AN. As a consequence, the zero of theFq[T]-algebraWN(A) is the family(0)P∈N.
The unity of the Fq[T]-algebraWN(A) is not as simple as it was in Theorem 2.4.
We have only usedC1⇐⇒ D1so far. What about C1 ⇐⇒ D2 ?
Definition 2.19. Let N be a q-nest. Let A be a commutative Fq[T]-algebra.
TheCarlitz tilde N-ghost map weN : AN → AN is the map defined by
weN (xP)P∈N =
∑
D|P
DxqDdeg(P/D)
P∈N
for all (xP)P∈N ∈ AN. This tilde N-ghost map is Fq-linear but (generally) neither multiplicative nor Fq[T]-linear.
each variable in the familyΞ. Alternatively, one could define morphisms ϕP by ϕP(Q) = Q
ΞqdegP
; these are different morphisms but they also work here.
From the equivalenceC1 ⇐⇒ D2 in Theorem 2.13, we get:
Theorem 2.20. Let N be a q-nest. There exists a unique functor WeN : CRingFq[T] →CRingFq[T] with the following two properties:
– We haveWeN(A) = AN as a set for every commutativeFq[T]-algebra A.
– The map weN : AN → AN regarded as a mapWeN(A) → AN is anFq[T] -algebra homomorphism for every commutativeFq[T]-algebra A.
This functor WeN is called the Carlitz tilde N-Witt vector functor. For every Fq[T]-algebra A, we call the Fq[T]-algebra WeN(A) the Carlitz tilde N-Witt vector ring over A. The zero of thisFq[T]-algebraWeN(A)is the family(0)P∈N, and its unity is the family(δP,1)P∈N (whereδu,v is defined to be
(1, if u=v;
0, if u6=v for any two objectsu andv).
The map weN : WeN(A) → AN itself becomes a natural transformation from the functor WeN to the functor CRingF
q[T] → CRingF
q[T], A 7→ AN. We will call this natural transformationweN as well.
But we have not really found two really different functors...
Theorem 2.21. Let Nbe a q-nest. The functorsWN andWeN are isomorphic by an isomorphism which forms a commutative triangle withwN and weN. This is again proven using Theorem 2.13 and universal polynomials.
The following theorem allows us to prove functorial identities by working with ghost components:
Theorem 2.22. Let N be a q-nest. For any commutative Fq(T)-algebra A, the maps wN : WN(A) → AN and weN : WeN(A) → AN are Fq[T]-algebra isomorphisms.
We have an “almost-universal property” again, following from exponent lift-ing and the implicationC1 =⇒ D1 in Theorem 2.13:
Theorem 2.23. Let N be a q-nest. Let A be a commutative Fq[T]-algebra such that no element ofN is a zero-divisor in A. For everyP∈ N, letσP be an Fq[T]-algebra endomorphism of A. Assume thatσP◦σQ =σPQ for anyP∈ N and Q ∈ N satisfying PQ ∈ N. Also assume that σ1 = id. Finally, assume that σπ(a) ≡ [π] (a)modπA (or, equivalently, σπ(a) ≡ aqdegπmodπA) for every monic irreducible π ∈ N and every a ∈ A. Then, there exists a unique Fq[T]-algebra homomorphism ϕ: A→WN(A) satisfying
(wN ◦ϕ) (a) = (σP(a))P∈N for everya∈ A. (3)
A similar result holds forWeN and weN. What about Frobenius operations?
Theorem 2.24. Let N be aq-nest.
(a)Let M∈ Fq[T]+ be such that everyP ∈ N satisfies MP∈ N. Then, there exists a unique natural transformation fM : WN → WN of set-valued (not Fq[T]-algebra-valued) functors such that any commutative Fq[T]-algebra A and anyx ∈WN(A) satisfy
wN(fM(x)) = (MP-th coordinate ofwN(x))P∈N, wherefM is short forfM(A).
(b) This natural transformation fM is actually a natural transformation WN →WN ofFq[T]-algebra-valuedfunctors as well. That is, fM : WN(A)→ WN(A) is an Fq[T]-algebra homomorphism for every commutative Fq[T] -algebra A. (Here, again, fM stands short for fM(A).) We call fM the M-th FrobeniusonWN.
(c)We have f1 =id. Any P ∈ Fq[T]+ and Q ∈ Fq[T]+ such that fP and fQ are well-defined satisfyfP◦fQ =fPQ.
(d) Let π ∈ Fq[T] be a monic irreducible such that every P ∈ N satisfies πP ∈ N. We havefπ(x) ≡ [π] (x)modπWN(A) (inWN(A)) for every com-mutative Fq[T]-algebra Aand every x∈ WN(A).
Corollary 2.25. Consider the setting of Theorem 2.23. Then (from Theorem 2.23) we know that there exists a unique Fq[T]-algebra homomorphism ϕ : A → WN(A) satisfying (3). Consider this ϕ. Let M ∈ N be such that every P∈ Nsatisfies MP∈ N. Then,
ϕ◦σM =fM◦ϕ for every M ∈ N.
Corollary 2.26. Consider the setting of Theorem 2.23. Assume thatNis closed under multiplication (i.e., we have MP ∈ N for every M ∈ N and P ∈ N).
Furthermore, let B be a commutative Fq[T]-algebra such that no element of N is a zero-divisor in B. Let projB : WN(B) → B be the map sending every u ∈ WN(B) to the 1-st coordinate of wN(u) ∈ BN. This projB is an Fq[T] -algebra homomorphism (sincewN is an Fq[T]-algebra homomorphism).
Let g : A → B be an Fq[T]-algebra homomorphism. Then, there exists a unique Fq[T]-algebra homomorphism G : A → WN(B) with the properties that w1◦G =g and that
G◦σM =fM◦g for every M∈ N.
This G can be constructed as follows: Theorem 2.23 shows that there exists a unique Fq[T]-algebra homomorphism ϕ : A → WN(A) satisfying (3). Con-sider this ϕ. SinceWN is a functor, theFq[T]-algebra homomorphismg: A →
Bgives rise to an Fq[T]-algebra homomorphismWN(g) : WN(A) →WN(B). Now, the Gis constructed as the composition WN(g)◦ϕ.
A Verschiebung exists too:
Theorem 2.27. Let N be aq-nest.
(a) Let M ∈ Fq[T]+. Then, there exists a unique natural transformation VM : WN → WN of set-valued (not Fq[T]-algebra-valued) functors such that any commutativeFq[T]-algebra Aand any x∈ WN(A)satisfy
wN(VM(x)) =
M·
P
M-th coordinate ofwN(x)
, if M| P;
0, if M- P
P∈N
,
whereVM is short forVM(A).
(b) This natural transformation VM is actually a natural transformation WN → WN of abelian-group-valued functors as well. More precisely, VM : WN(A)→WN(A) is a homomorphism of additive groups for every commu-tative Fq[T]-algebra A. (Here, again, VM stands short for VM(A).) We call VM the M-th Verschiebungon WN.
(c) We have V1 = id. Any two P ∈ Fq[T]+ and Q ∈ Fq[T]+ satisfy VP◦ VQ =VPQ.
(d) Actually, VM (xP)P∈N =
(xP/M, if M| P;
0, if M-P
!
P∈N
for any P ∈ Fq[T]+, any commutative Fq[T]-algebra Aand any (xP)P∈N ∈ WN(A). And here is a Carlitz analogue of the Artin-Hasse exponential:
Theorem 2.28. Let N be a q-nest. Assume that PQ ∈ N for all P ∈ N and Q ∈ N.
(a) There exists a unique natural transformation AH : WN → WN ◦WN (of functors CRingF
q[T] → CRingF
q[T]) such that every commutative Fq[T] -algebra A, every P ∈ N and every x∈ WN(A)satisfy
(P-th coordinate ofwN(AH(x))) =fP(x)
(wherewN this time stands for the natural transformationwN evaluated at the Fq[T]-algebraWN(A); thus, wN(AH(x))is an element of (WN(A))N).
(b) Let P ∈ N, and let A be a commutative Fq[T]-algebra. Let wP : WN(A) → A be the map sending each x ∈ WN(A) to the P-th coordinate ofwN(x). Then,WN(wP)◦AH=fP.