Witt vectors

Section 3 Witt vectors

References.

As in Section 2. The original paper of Witt is [127]. See also [116], chapter II, section 6.

We now relate the discussion of

δ

-rings to the older construction of

p

-typical Witt vectors. Our main goal is to relate this construction to perfect

δ

-rings Proposition 3.3.6; this only involves evaluating the Witt functor on perfect rings of characteristic

p,

but to develop the theory it is easier to remember that it defines a functor on arbitrary commutative rings.

🔗

Subsection 3.1 $p$ -typical Witt vectors via adjunction

We introduce the

p

-typical Witt vectors, building upon our work with truncated Witt vectors in Subsection 2.3 and the adjunction between rings and

δ

-rings. However, to make this unorthodox development compatible with the more standard treatment (and the big Witt vectors to follow), we must introduce a key change of coordinates.

🔗

Definition 3.1.1.

As indicated in Definition 2.4.5, the forgetful functor

{Ring}_{δ} \to Ring

admits a right adjoint

W .

To identify the image of a ring

A

under this functor, we use the set-theoretic identifications

🔗

\begin{aligned} W (A) & = {Hom}_{Ring} (Z [y], W (A)) \\ = {Hom}_{{Ring}_{δ}} (Z {y}, W (A)) \\ = {Hom}_{Ring} (Z [y_{0}, y_{1}, \dots], A) \\ = A \times A \times \dots . \end{aligned}

This means that each element of

W (A)

has a unique expansion

(y_{0}, y_{1}, \dots)

with each

y_{n} \in A;

we call the

y_{n}

the

y

-coordinates (or Joyal coordinates) of this element of

W (A) .

(This presentation does not directly describe the ring structure on

W (A);

see Remark 4.2.6.)

🔗

In Lemma 3.1.3 below. we will give a second set of generators

x_{0}, x_{1}, \dots

of the polynomial ring

Z [y_{0}, y_{1}, \dots] .

This means that each element of

W (A)

has a unique expansion

(x_{0}, x_{1}, \dots)

with each

x_{n} \in A;

we call the

x_{n}

the

x

-coordinates (or Witt coordinates) of this element of

W (A) .

In these coordinates,

W (A)

will become none other than the ring of

p

-typical Witt vectors over

A

via the translation described in Definition 3.2.1.

🔗

Remark 3.1.2.

Before continuing, we record a statement which will come up repeatedly: for elements

x, y

of a commutative ring,

🔗

\begin{matrix} (3.1) & (x + p y)^{p} \equiv x^{p} (\mod p^{2} y) . \end{matrix}

In particular,

🔗

\begin{matrix} (3.2) & x \equiv y (\mod p^{m}) \Rightarrow x^{p} \equiv y^{p} (\mod p^{m + 1}) . \end{matrix}

🔗

Lemma 3.1.3.

With notation as in Definition 3.1.1, in the ring

Z {y}

there exist elements

🔗

x_{n} \in y_{n} + (y_{1}, \dots, y_{n - 1}) Z [y_{0}, \dots, y_{n - 1}] (n = 0, 1, \dots)

such that

x_{0} = y_{0}, x_{1} = y_{1},

and

🔗

\begin{matrix} (3.3) & ϕ^{n} (x_{0}) = x_{0}^{p^{n}} + p x_{1}^{p^{n - 1}} + \dots + p^{n} x_{n} (n = 0, 1, \dots) . \end{matrix}

🔗

Proof.

Given

x_{0}, \dots, x_{n - 1} \in Z {y}

satisfying (3.3) with

n

replaced by

n - 1,

use (3.1) to write

\begin{aligned} ϕ^{n} (x_{0}) & = ϕ (ϕ^{n - 1} (x_{0})) \\ = ϕ (x_{0})^{p^{n - 1}} + \dots + p^{n - 1} ϕ (x_{n - 1}) \\ = (x_{0}^{p} + p *)^{p^{n - 1}} + \dots + p^{n - 2} (x_{n - 2}^{p} + p *)^{p} + p^{n - 1} ϕ (x_{n - 1}) \\ = x_{0}^{p^{n}} + \dots + p^{n - 2} x_{n - 2}^{p^{2}} + p^{n - 1} (x_{n - 1}^{p} + p δ (x_{n - 1})) + p^{n} * \end{aligned}

where each

*

denotes a quantity in

(y_{1}, \dots, y_{n - 1}) Z [y_{0}, \dots, y_{n - 1}] .

We thus can (and must) take

x_{n} = δ (x_{n - 1}) + * .

Since

x_{n - 1} - y_{n - 1} \in Z [y_{0}, \dots, y_{n - 2}],

we have

δ (x_{n - 1}) = δ (y_{n - 1}) + δ (x_{n - 1} - y_{n - 1}) + * = y_{n} + *

and so

x_{n} - y_{n} \in (y_{1}, \dots, y_{n - 1}) Z [y_{0}, \dots, y_{n - 1}] .

Corollary 3.1.4.

For

w_{n} = \sum_{m = 0}^{n} p^{m} x_{m}^{p^{n - m}} \in Z {y},

we have

🔗

ϕ^{n} (w_{m}) = w_{n + m} (m, n \geq 0) .

🔗

Proof.

By Lemma 3.1.3, both sides are equal to

ϕ^{n + m} (x_{0}) .

Corollary 3.1.5.

In the ring

Z {y},

we have

🔗

ϕ (x_{n}) \equiv x_{n}^{p} (\mod p) .

Consequently, for

A

a ring of characteristic

p,

the map

ϕ

W (A)

coincides with the map induced by functoriality by the Frobenius on

A .

(This is also true in the

y

-coordinates.)

🔗

Proof.

The first assertion is a consequence of the fact that

ϕ

is a Frobenius lift (because

Z {y}

is a

δ

-ring). The second assertion is a direct consequence of the first, but let us spell this out for clarity as the mechanism of the argument will recur in what follows. For a general ring

A,

each of the elements

x_{0}, x_{1}, \dots \in Z {x}

defines a function

W (A) \to A

which is natural in

A .

Similarly, every element of

Z {x}

can be viewed as a “polynomial function” on

W (A)

valued in

A

which is again natural in

A;

that is, we have a map of sets

h : Z {x} \to {Hom}_{Set} (W (A), A) .

This map has the property that

h (ϕ (t)) (u) = ϕ (h (t) (u)) (t \in Z {x}, u \in W (A)) .

In the case where

A

is of characteristic

p,

we have

ϕ (x_{n}) = x_{n}^{p} + p δ (x_{n})

and so for any

u \in W (A),

\begin{aligned} ϕ (h (x_{n}) (u)) & = h (ϕ (x_{n})) (u) \\ = h (x_{n}^{p} + p δ (x_{n})) (u) \\ = h (x_{n}^{p}) (u) + p h (δ (x_{n})) (u) \\ = h (x_{n}) (u)^{p} . \end{aligned}

This shows that the

ϕ

acts via the functorial Frobenius.

We record some consequences of the adjunction between

Ring

and

{Ring}_{δ} .

🔗

Definition 3.1.6.

The identity map in

{Hom}_{{Ring}_{δ}} (W (A), W (A))

corresponds via adjunction to a morphism

W (A) \to A

of rings. In coordinates, this is the map

(x_{0}, x_{1}, \dots) \mapsto x_{0} .

🔗

The identity map in

{Hom}_{Ring} (W (A), W (A))

corresponds via adjunction to a morphism

Δ : W (A) \to W (W (A))

{Ring}_{δ}

which is moreover functorial in

A .

This map is sometimes called the diagonal.

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Definition 3.1.7.

Recall that the action of

δ

Z [y_{0}, y_{1}, \dots]

satisfies

δ (y_{n}) = y_{n + 1};

consequently, if we express an element of

W (A)

in the

y

-coordinates as

(y_{0}, y_{1}, \dots),

it is

δ

-constant if and only if

y_{1} = y_{2} = \dots = 0 .

By Lemma 3.1.3, in the usual coordinates, an element

(x_{0}, x_{1}, \dots)

W (A)

δ

-constant if and only if

x_{1} = x_{2} = \dots = 0 .

🔗

That is, the

δ

-constants are the image of the multiplicative (but not additive; see Exercise 3.6.3) section

[∙] : A \to W (A)

of the projection

W (A) \to A

given by

[x] = (x, 0, 0, \dots) .

We call

[x]

the constant lift (or the multiplicative lift) of

x \in A .

🔗

Remark 3.1.8.

The constant lift is also known as the Teichmüller lift. While this terminology is (fairly) historically accurate, in light of Oswald Teichmüller’s role in the Nazi Party and in denunciation of Jewish mathematicians in Germany during the 1930s

⁵

mathshistory.st-andrews.ac.uk/Biographies/Teichmuller/

, I prefer to use a non-eponymous terminology in this instance.

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Subsection 3.2 Ghost coordinates

The description of

W (A)

we are using does not make it especially clear how the addition and multiplication operations work. To clarify this, we relate back to the more standard presentation of Witt vectors.

🔗

Definition 3.2.1.

Define the elements

w_{n} \in Z {y}

as in Corollary 3.1.4. These define a set-theoretic map

🔗

w : W (A) \to A \times A \times \dots, (x_{n})_{n = 0}^{\infty} \mapsto {(\sum_{m = 0}^{n} p^{m} x_{m}^{p^{n - m}})}_{n = 0}^{\infty} .

which we call the ghost map.

🔗

Note that in general, this map is neither injective (unless

A

p

-torsion-free) nor surjective (unless

A

p

-divisible). Nonetheless, for

x \in W (A),

it will be convenient to refer to the terms of

w (x) = (w_{0}, w_{1}, \dots)

as the ghost coordinates of

x .

By Corollary 3.1.4, the ghost coordinates of

ϕ^{n} (x)

are

(w_{n}, w_{n + 1}, \dots) .

🔗

Now recall the map

x \mapsto w_{0} = x_{0}

is the homomorphism

W (A) \to A

obtained by adjunction. It follows that the map

W (A) \overset{ϕ^{n}}{\to} W (A) \to A

is given by

x \mapsto w_{n} .

That is, the ghost map is a natural transformation of functors of rings!

🔗

Remark 3.2.2.

Using the ghost map, we can now see that

W (A)

agrees with the usual definition of the ring of

p

-typical Witt vectors of

A,

in which the arithmetic operations on Witt vectors are given by certain universal polynomials in the entries of the Witt vectors. We may read off properties of these polynomials using functoriality; this is similar to a more typical proof of the existence of the functor

W

(see for example [73], section 8.10), except that now we don’t need to worry about its existence! This means that we can freely pass from general rings to

p

-torsion-free rings to

Z [p^{- 1}]

-algebras.

🔗

Definition 3.2.3.

For any ring

A,

the Verschiebung map

V : W (A) \to W (A)

is defined by

🔗

V (x_{0}, x_{1}, \dots) = (0, x_{0}, x_{1}, \dots) .

Note that this corresponds to the map on ghost coordinates given by

🔗

(w_{0}, w_{1}, \dots) \mapsto (0, p w_{0}, p w_{1}, \dots) .

Using the ghost map as per Remark 3.2.2, we may deduce that

V

is additive (but not multiplicative) and that

ϕ \circ V

acts via multiplication by

p .

🔗

Definition 3.2.4.

Using the method of Remark 3.2.2, we may show that for each positive integer

n,

there is a natural transformation from

W

to another functor

W_{n}

Ring

which on sets corresponds to the projection

🔗

(x_{0}, x_{1}, \dots) \mapsto (x_{0}, \dots, x_{n - 1})

(and similarly for ghost components). The ring

W_{n} (A)

is called the ring of truncated

p

-typical Witt vectors of length

n

over

A;

for

n = 1

we get

A

itself, while for

n = 2

we recover the construction of Definition 2.3.1. Note that the natural transformation

W \to lim_{n} W_{n}

is an isomorphism.

🔗

The action of

ϕ

W (A)

does not induce an endomorphism of

W_{n} (A)

in general (unless

p = 0

A,

in which case Corollary 3.1.5 applies). However, it does induce a homomorphism

W_{n + 1} (A) \to W_{n} (A)

(the Witt vector Frobenius), from which we can recover

ϕ

as the induced map

🔗

lim_{n} W_{n} (A) = lim_{n} W_{n + 1} (A) \to lim_{n} W_{n} (A) .

🔗

Remark 3.2.5.

In [127], what we call ghost coordinates were instead called Nebenkomponenten, or secondary components. The terminology we use here is quite commonplace but its origins are unclear; the earliest reference we were able to find is Barsotti’s Mathematical Reviews synopsis of [130], but it seems likely that the terminology was in circulation before that.

🔗

Subsection 3.3 Witt vectors and perfect $δ$ -rings

We now focus more closely on Witt vectors valued in a perfect ring of characteristic

p,

and obtain their more familiar ring-theoretic properties.

🔗

Definition 3.3.1.

δ

-ring

A

is perfect if

ϕ

is an isomorphism. By the same token, a ring of characteristic

p

is perfect if

ϕ

is an isomorphism; in this case, injectivity of

ϕ

is equivalent to

A

being reduced.

🔗

Lemma 3.3.2.

Let

A

be a perfect ring of characteristic

p .

Then the ring

W (A)

p

-torsion-free and

p

-adically complete and

W (A) / (p) ≅ A .

🔗

Proof.

By Corollary 3.1.5,

ϕ

is an automorphism of

W (A) .

By Lemma 2.2.8, the ring

W (A)

p

-torsion-free. Since

ϕ \circ V

is multiplication by

p

(Definition 3.2.3) and

ϕ

is bijective, the ideal

p W (A)

coincides with the image of

V,

which in turn equals the kernel of the map

W (A) \to A;

hence

W (A) / (p) ≅ A .

By similar logic, for each positive integer

n,

the ideal

p^{n} W (A)

coincides with the image of

V^{n};

from this, we see that

W (A)

p

-adically complete.

Example 3.3.3.

We have

W (F_{p}) ≅ Z_{p} .

More generally, for any finite extension

F_{q}

F_{p},

W (F_{q}) = Z_{p} [ζ_{q - 1}] .

🔗

Definition 3.3.4.

For

A

a perfect ring of characteristic

p,

Lemma 3.3.2 implies that each element

x

W (A)

can be written uniquely as a

p

-adically convergent sum

\sum_{n = 0}^{\infty} p^{n} [x_{n}]

with

x_{n} \in A,

where

[x_{n}]

denotes the constant lift (Definition 3.1.7). We call this the series representation of

x .

🔗

Lemma 3.3.5.

Let

R

be a perfect ring of characteristic

p .

Let

S

be a

p

-adically complete ring. Then any morphism

R \to S / (p)

lifts uniquely to a morphism

W (R) \to S .

🔗

Proof.

We first use (3.2) to lift

R \to S / (p)

to a multiplicative map

R \to S .

Using the series representations from Definition 3.3.4, we then obtain a set-theoretic map

W (R) \to S

which we must show is an homomorphism; it is enough to check that it induces a homomorphism

W (R) \to S / p^{n}

for each

n .

This is not too onerous to prove by direct computation; see for example [80], Lemma 1.1.7.

A second, more conceptual approach is to apply the fact that if

A

is a perfect ring of characteristic

p,

then the cotangent complex

L_{A / F_{p}}

vanishes; we will revisit this comment once we have introduced the cotangent complex in Subsection 17.1. See Exercise 17.5.1 (and [18], Lecture II, Lemma 3.5).

Proposition 3.3.6.

The following categories are equivalent (via the functors described below).

🔗

The category of $p$ -adically complete, perfect $δ$ -rings.
The category of $p$ -torsion-free, $p$ -adically complete rings whose reductions modulo $p$ are perfect.
The category of perfect rings of characteristic $p .$

The functor from (1) to (2) is the forgetful functor; the functor from (2) to (3) is

A \mapsto A / p A;

the functor from (3) to (1) is

W .

🔗

Proof.

The composition from (3) to (1) to (2) to (3) is an equivalence by Lemma 3.3.2. In particular, the functor from (2) to (3) is essentially surjective. By Lemma 3.3.5, the composition from (3) to (1) to (2) is also essentially surjective; hence (2) and (3) are equivalent. We can now use Lemma 2.2.8 and Corollary 3.1.5 to add (1) to the loop.

🔗

Subsection 3.4 Beyond the perfect case in characteristic $p$

It is not the case that Proposition 3.3.6 can be extended to relate

p

-torsion-free,

p

-adically complete rings whose reductions modulo

p

are not perfect with the image of the functor

W

on nonperfect rings of characteristic

p .

We record some assorted remarks here.

🔗

Definition 3.4.1.

The inclusion of the full subcategory of perfect rings of characteristic

p

into arbitrary rings of characteristic

p

has both left and right adjoints. The left adjoint maps

A

{colim}_{ϕ} A,

which we call the coperfection of

A .

The right adjoint maps

A

lim_{ϕ} A,

which we call the perfection of

A .

🔗

The following examples show that the relationship between the perfection and coperfection can be a bit subtle.

🔗

Example 3.4.2.

For

A = F_{p} [x],

the coperfection equals

F_{p} [x^{p^{- \infty}}]

while the perfection equals

F_{p} .

🔗

Example 3.4.3.

For

A = F_{p} [x^{p^{- \infty}}] / (x),

the coperfection equals

F_{p}

while the perfection equals the

x

-adic completion of

F_{p} [x^{p^{- \infty}}] .

🔗

Remark 3.4.4.

Let

A

be a

p

-torsion-free,

p

-adically complete

δ

-ring. Let

R

be the coperfection of

A / (p)

(Definition 3.3.1). Using Proposition 3.3.6, we obtain a morphism

A \to W (R)

{Ring}_{δ};

this map is injective if

A / (p)

is reduced. If we fix

A

as an underlying ring while varying its

δ

-ring structure, the target

W (R)

remains fixed while the morphism

A \to W (R)

varies.

🔗

Example 3.4.5.

Put

A = Z [x] .

As in Example 2.2.5, for each

y \in A

there is a unique

δ

-ring structure on

A

for which

δ (x) = y .

Each of these gives rise to an injective morphism

A \to W ({colim}_{ϕ} F_{p} [x])

δ

-rings.

🔗

Lemma 3.4.6.

Let

A \to B

be a morphism of

p

-torsion-free,

p

-adically complete rings. Suppose that

A

is equipped with a

δ

-ring structure and that

A / (p) \to B / (p)

is étale. Then

B

admits a unique

δ

-ring structure compatible with

A .

🔗

Proof.

See [18], Lecture II, Lemma 2.9. See also [104] for a supplemental argument that can be used to eliminate the

p

-torsion-free hypothesis.

Remark 3.4.7.

Corollary 3.1.5 implies that when

A

is a reduced ring of characteristic

p,

the map

ϕ

W (A)

is injective. By contrast, if

A

is a nonreduced ring of characteristic

p,

then

ϕ

is not injective: for any nonzero

x \in A

with

x^{p} = 0,

we have

[x] \neq 0

but

ϕ ([x]) = [x^{p}] = 0 .

🔗

A

is a ring not of characteristic

p,

then the map

ϕ

W (A)

is not injective either, but this is somewhat more subtle. See Exercise 3.6.7. (One case which is not subtle: if

p

is invertible in

W (A),

then the ghost map is an isomorphism and so we may see the kernel of

ϕ

on the ghost side, remembering that

ϕ

acts here as the left shift.)

🔗

Remark 3.4.8.

For

A

of characteristic

p,

the map

ϕ

W (A)

is surjective if and only if it is surjective on

A,

i.e., if and only if

A

is semiperfect. However, by contrast with Remark 3.4.7, there are many rings

A

not of characteristic

p

for which

ϕ

is surjective on

W (A) .

There are even more rings for which

ϕ : W_{n + 1} (A) \to W_{n} (A)

is surjective for each

n;

these rings are said to be Witt-perfect in [40], which see for additional characterizations.

🔗

Subsection 3.5 Additional remarks

Proposition 3.5.1.

For any etale morphism

f : A \to B

and any positive integer

n,

the map

W_{n} (f) : W_{n} (A) \to W_{n} (B)

is etale.

🔗

Proof.

This was originally shown by van der Kallen ([122], (2.4)); see also [28], Theorem B. (Both of these references also cover the truncated big Witt vector functors; see Definition 4.1.3.) For the case of a localization, see also Exercise 3.6.5.

Remark 3.5.2.

By Proposition 3.5.1, we may apply the functors

W_{n}

also to schemes. See [29] for some discussion of this construction.

🔗

Exercises 3.6 Exercises

1.

Describe the ring

W (A)

explicitly for

A = F_{p} [x] / (x^{p}),

and show that it is a

δ

-ring with nontrivial

p

-torsion. (This provides a nontrivial example of Lemma 2.2.8.)

🔗

Hint.

Use the fact that

ϕ \circ V

acts as multiplication by

p .

🔗

2.

Let

A

be a

p

-torsion-free,

p

-adically complete ring. Let

R

be the perfection of

A / (p)

(Definition 3.3.1). Show that the natural maps

🔗

lim_{ϕ} W (A) \to lim_{ϕ} W (A / (p)) \to lim_{ϕ} W (R) \to W (R)

are all isomorphisms.

🔗

3.

Show that for any ring

A,

the map

[∙] : A \to W (A)

is multiplicative.

🔗

Hint.

Use the fact that the

δ

-constant elements both form a multiplicative subset and coincide with the image of

[∙] .

🔗

4.

Let

R

be a perfect ring of characteristic

p .

Prove that

R

is noetherian if and only if

R

is a finite (possibly empty) direct product of fields. Consequently,

W (R)

is noetherian if and only if the same conditions hold.

🔗

5.

Let

A

be a ring and let

S

be a multiplicative subset. Let

[S]

be the image of

S

under the constant section. Prove that for each positive integer

n,

there is a natural isomorphism

[S]^{- 1} W_{n} (A) \to W_{n} (S^{- 1} A) .

By contrast, the natural map

[S]^{- 1} W (A) \to W (S^{- 1} A)

is not an isomorphism.

🔗

Hint.

The natural map exists because elements of

[S]

become units in

W_{n} (S^{- 1} A) .

To show that it is surjective, first use the ghost map (and naturality) to figure out how multiplication by a constant lift acts on the Witt components of a general vector.

🔗

6.

Show that in Example 3.4.5, the image of

Z [x]

W ({colim}_{ϕ} F_{p} [x])

need not be generated (as a

Z

-algebra) by multiplicative lifts.

🔗

7.

Let

A

be a ring. Show that if

ϕ : W (A) \to W (A)

is injective, then

p = 0

A .

🔗

Hint.

Show that any multiple of

p

occurs as

x_{0}

in some

x = (x_{0}, x_{1}, \dots) \in W (A)

with

ϕ (x) = 0 .

For more details, see [40], Corollary 2.6.

🔗

8.

Let

R

be a perfect ring of characteristic

p .

Show that

V \circ ϕ

acts on

W (R)

by multiplication by

p

(just as

ϕ \circ V

does for arbitrary

R

🔗

Hint.

Use the fact that

ϕ

agrees with the functorial Frobenius to see that it commutes with multiplication by

p .

🔗

9.

Prove that for any nonzero ring

A,

the characteristic of the ring

W (A)

is either 0 or not divisible by

p .

🔗

Hint.

If any maximal ideal of

A

has characteristic

p,

then

W (A)

maps to a ring of characteristic 0. Otherwise,

W (A)

splits as a product of copies of

A .

🔗

10.

Prove that Figure 3.6.1 is a pullback square.

🔗

Hint.

The corresponding statement with

Z_{p}

replaced by

\hat{Z}

holds because the latter is faithfully flat over

Z .

Now rewrite

\hat{Z}

as the product of

Z_{p}

with a

Z [p^{- 1}]

-algebra and recall that for the latter, the ghost map is an isomorphism.

🔗

11.

Show that for

A \in Ring,

the map

🔗

N : W (A) \to W (A), N (x) = x - V_{p} (δ (x))

is multiplicative and satisfies

🔗

(F \circ N) (x) = x^{p} (x \in W (A)) .

This is called the norm map in [3]; see also Exercise 4.3.8.

🔗

Hint.

Show that the effect on ghost coordinates is given by

(w_{0}, w_{1}, w_{2}, \dots) \mapsto (w_{0}, w_{0}^{p}, w_{1}^{p}, w_{2}^{p}, \dots) .

🔗

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Section 3 Witt vectors

References.

Subsection 3.1 p-typical Witt vectors via adjunction

Definition 3.1.1.

Remark 3.1.2.

Lemma 3.1.3.

Proof.

Corollary 3.1.4.

Proof.

Corollary 3.1.5.

Proof.

Definition 3.1.6.

Definition 3.1.7.

Remark 3.1.8.

Subsection 3.2 Ghost coordinates

Definition 3.2.1.

Remark 3.2.2.

Definition 3.2.3.

Definition 3.2.4.

Remark 3.2.5.

Subsection 3.3 Witt vectors and perfect δ-rings

Definition 3.3.1.

Lemma 3.3.2.

Proof.

Example 3.3.3.

Definition 3.3.4.

Lemma 3.3.5.

Proof.

Proposition 3.3.6.

Proof.

Subsection 3.4 Beyond the perfect case in characteristic p

Definition 3.4.1.

Example 3.4.2.

Example 3.4.3.

Remark 3.4.4.

Example 3.4.5.

Lemma 3.4.6.

Proof.

Remark 3.4.7.

Remark 3.4.8.

Subsection 3.5 Additional remarks

Proposition 3.5.1.

Proof.

Remark 3.5.2.

Exercises 3.6 Exercises

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Subsection 3.1 $p$ -typical Witt vectors via adjunction

Subsection 3.3 Witt vectors and perfect $δ$ -rings

Subsection 3.4 Beyond the perfect case in characteristic $p$