ring of Witt vectors

This article is about of groups of Witt vectors and rings of Witt vectors; which are often called just Witt groups and Witt rings. However, there is also a different notion of Witt group? and Witt ring.



Rings of Witt vectors are the co-free Lambda-rings. Depending on whether one defines the latter via Frobenius lifts at a single prime number pp one speaks of pp-typical Witt vectors, or of big Witt vectors if all primes are considered at once.

In arithmetic geometry the impact of rings of Witt vectors W(R)W(R) of a given ring RR is that they are like rings formal power series on Spec(R)Spec(R), such as rings of p-adic numbers. For more on this see at arithmetic jet space and at Borger's absolute geometry.

In components, a Witt vector is an infinite sequence of elements of a given commutative ring kk. There is a ring structure on the set W(k)W(k) of Witt vectors of kk and W(k)W(k) is therefore called the Witt ring of kk. The multiplication is defined by means of Witt polynomials w iw_i for every natural number ii. If the characteristic of kk is 00 the Witt ring of kk is sometimes called universal Witt ring to distinguish it from the case where kk is of prime characteristic and a similar but different construction is of interest.

A p-adic Witt vector is an infinite sequence of elements af a commutative ring of prime characteristic pp. There exists a ring structure whose construction parallels that in characteristic 00 except that only Witt polynomials w p lw_{p^l} whose index is a power of pp are taken.

More abstractly, the ring of Witt vectors carries the structure of a Lambda-ring and the construction W:kW(k)W \colon k\mapsto W(k) of the Witt ring W(k)W(k) on a commutative ring kk is right adjoint to the forgetful functor from Lambda-rings to commutative rings. Hence rings of Witt vectors are the co-free Lambda-rings.

Moreover W()W(-) is representable by Symm, the ring of symmetric functions which is a Hopf algebra and consequently WW is a group scheme. This is explained at Lambda-ring.

The construction of Witt vectors gives a functorial way to lift a commutative ring AA of prime characteristic pp to a commutative ring W(A)W(A) of characteristic 0. Since this construction is functorial, it can be applied to the structure sheaf of an algebraic variety. In interesting special cases the resulting ring W(A)W(A) has even more desirable properties: If AA is a perfect field then W(A)W(A) is a discrete valuation?. This is partly due to the fact that the construction of W(A)W(A) involves a ring of power series and a ring of power series over a field is always a discrete valuation ring.

There is a generalization to non-commutative Witt vectors, however these only carry a group- but no ring structure.

The Lubin-Tate ring in Lubin-Tate theory is a polynomial ring on a ring of Witt vectors and this way Witt vectors control much of chromatic homotopy theory.


In an expansion of a pp-adic number a=Σa ip ia=\Sigma a_i p^i the a ia^i are called digits. Usually these digits are defined to be taken elements of the set {0,1,,p1}\{0,1,\dots,p-1\}.

Equivalently the digits can be defined to be taken from the set T p:={x|x p1=1}{0}T_p:=\{x|x^{p-1}=1\}\cup \{0\}. Elements from this set are called Teichmüller digits or Teichmüller representatives.

The set TT is in bijection with the finite field F pF_p. The set W(F p)W(F_p) of (countably) infinite sequences of elements in F pF_p hence is in bijection to the set p\mathbb{Z}_p of pp-adic integers. There is a ring structure on W(F p)W(F_p) called Witt ring structure such that all ‘’truncated expansion polynomials’‘ Φ n=X p n+pX p n1+p 2X p n2++p nX\Phi_n=X^{p^n}+pX^{p^{n-1}}+p^2X^{p^{n-2}}+\dots +p^n X called Witt polynomials are morphisms

Φ n:W(F p) p\Phi_n:W(F_p)\to \mathbb{Z}_p

of groups.


We first give the

and then discuss the

In components

The ring structure


Let kk be a commutative ring.

If the characteristic of kk is 00 then the Witt ring W(k)W(k) of kk is defined defined by the addition

(a 1,a 2,,)+(b 1,b 2,):=(Σ 1(a 1,b 1),Σ 2(a 1,a 2,b 1,b 2),)(a_1, a_2, \ldots, )+(b_1, b_2, \ldots) :=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)

and the multiplication

(a 1,a 2,)(b 1,b 2,):=(Π 1(a 1,b 1),Π 2(a 1,a 2,b 1,b 2),)(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots ) :=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )

If kk is of prime characteristic pp we index the defining formulas by p 1,p 2,p 3,p^1,p^2,p^3,\dots instead of 1,2,3,1,2,3,\dots.

Here the Σ i\Sigma_i are called addition polynomials and the Π i\Pi_i are called multiplication polynomials, these are described below.

The Witt polynomials, the Σ i\Sigma_i, the Π i\Pi_i, phantom components

let x 1,x 2,x_1, x_2, \ldots be a collection of variables. We can define an infinite collection of polynomials in [x 1,x 2,]\mathbb{Z}[x_1, x_2, \ldots ] using the following formulas:

w 1(X)=x 1w_1(X)=x_1

w 2(X)=x 1 2+2x 2w_2(X)=x_1^2+2x_2

w 3(X)=x 1 3+3x 3w_3(X)=x_1^3+3x_3

w 4(X)=x 1 4+2x 2 2+4x 4w_4(X)=x_1^4+2x_2^2+4x_4

and in general w n(X)= d|ndx n n/d\displaystyle w_n(X)=\sum_{d|n} dx_n^{n/d}. The value w n(w)w_n(w) of the nn-th Witt polynomial in some element wW(k)w\in W(k) of the Witt ring of kk is sometimes called the nn-th phantom component of ww or the nn-th ghost component of ww.

Now let ϕ(z 1,z 2)[z 1,z 2]\phi(z_1, z_2)\in\mathbb{Z}[z_1, z_2]. This just an arbitrary two variable polynomial with coefficients in \mathbb{Z}.

We can define new polynomials Φ i(x 1,x i,y 1,y i)\Phi_i(x_1, \ldots x_i, y_1, \ldots y_i) such that the following condition is met ϕ(w n(x 1,,x n),w n(y 1,,y n))=w n(Φ 1(x 1,y 1),,Φ n(x 1,x n,y 1,,y n))\phi(w_n(x_1, \ldots ,x_n), w_n(y_1, \ldots , y_n))=w_n(\Phi_1(x_1, y_1), \ldots , \Phi_n(x_1, \ldots x_n, y_1, \ldots , y_n)).

In short we’ll notate this ϕ(w n(X),w n(Y))=w n(Φ(X,Y))\phi(w_n(X),w_n(Y))=w_n(\Phi(X,Y)). The first thing we need to do is make sure that such polynomials exist. Now it isn’t hard to check that the x ix_i can be written as a \mathbb{Q}-linear combination of the w nw_n just by some linear algebra.

x 1=w 1x_1=w_1, and x 2=12w 2+12w 1 2x_2=\frac{1}{2}w_2+\frac{1}{2}w_1^2, etc. so we can plug these in to get the existence of such polynomials with coefficients in \mathbb{Q}. It is a fairly tedious lemma to prove that the coefficients Φ i\Phi_i are actually in \mathbb{Z}, so we won’t detract from the construction right now to prove it.

The addition- and multiplication polynomials

Define yet another set of polynomials Σ i\Sigma_i, Π i\Pi_i and ι i\iota_i by the following properties:

w n(Σ)=w n(X)+w n(Y)w_n(\Sigma)=w_n(X)+w_n(Y), w n(Π)=w n(X)w n(Y)w_n(\Pi)=w_n(X)w_n(Y) and w n(ι)=w n(X)w_n(\iota)=-w_n(X).

We now can construct W(A)W(A), the ring of generalized Witt vectors over AA. Define W(A)W(A) to be the set of all infinite sequences (a 1,a 2,)(a_1, a_2, \ldots) with entries in AA. Then we define addition and multiplication by (a 1,a 2,,)+(b 1,b 2,)=(Σ 1(a 1,b 1),Σ 2(a 1,a 2,b 1,b 2),)(a_1, a_2, \ldots, )+(b_1, b_2, \ldots)=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots) and (a 1,a 2,)(b 1,b 2,)=(Π 1(a 1,b 1),Π 2(a 1,a 2,b 1,b 2),)(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots )=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots ).

Universal characterization


The assignment

W:kW(k) W \;\colon\; k\mapsto W(k)

is a functor

W:CRingΛRing W \;\colon\; CRing \longrightarrow \Lambda Ring

from the category of commutative rings to that of Lambda-rings.

Composed with the forgetful functor

U:ΛRingCRing U \;\colon\; \Lambda Ring \longrightarrow CRing

this is the unique endofunctor W:CRingCRingW \;\colon\; CRing \longrightarrow CRing such that all Witt polynomials

w n:{W(A)A aw n(a) w_n : \begin{cases} W(A)\to A \\ a\mapsto w_n(a) \end{cases}

are homomorphisms of rings.


There is a nice trick to prove that W(A)W(A) is a ring when AA is a \mathbb{Q}-algebra. Just define ψ:W(A)A \psi: W(A)\to A^\mathbb{N} by (a 1,a 2,)(w 1(a),w 2(a),)(a_1, a_2, \ldots) \mapsto (w_1(a), w_2(a), \ldots). This is a bijection and the addition and multiplication is taken to component-wise addition and multiplication, so since this is the standard ring structure we know W(A)W(A) is a ring. Also, w(0,0,)=(0,0,)w(0,0,\ldots)=(0,0,\ldots), so (0,0,)(0,0,\ldots) is the additive identity, W(1,0,0,)=(1,1,1,)W(1,0,0,\ldots)=(1,1,1,\ldots) which shows (1,0,0,)(1,0,0,\ldots) is the multiplicative identity, and w(ι 1(a),ι 2(a),)=(a 1,a 2,)w(\iota_1(a), \iota_2(a), \ldots)=(-a_1, -a_2, \ldots), so we see (ι 1(a),ι 2(a),)(\iota_1(a), \iota_2(a), \ldots) is the additive inverse.

We can actually get this idea to work for any characteristic 00 ring by considering the embedding AAA\to A\otimes\mathbb{Q}. We have an induced injective map W(A)W(A)W(A)\to W(A\otimes\mathbb{Q}). The addition and multiplication is defined by polynomials over \mathbb{Z}, so these operations are preserved upon tensoring with \mathbb{Q}. We just proved above that W(A)W(A\otimes\mathbb{Q}) is a ring, so since (0,0,)(0,0,)(0,0,\ldots)\mapsto (0,0,\ldots) and (1,0,0,)(1,0,0,)(1,0,0,\ldots)\mapsto (1,0,0,\ldots) and the map preserves inverses we get that the image of the embedding W(A)W(A)W(A)\to W(A\otimes \mathbb{Q}) is a subring and hence W(A)W(A) is a ring.

Lastly, we need to prove this for positive characteristic rings. Choose a characteristic 00 ring that surjects onto AA, say BAB\to A. Then since the induced map again preserves everything and W(B)W(A)W(B)\to W(A) is surjective, the image is a ring and hence W(A)W(A) is a ring.


The construction of the ring of Witt vectors W(k)W(k) on a given commutative ring kk is the right adjoint to the forgetful functor UU from Lambda-rings to commutative rings

(UW):CRingWUΛRing. (U \dashv W) \;\colon\; CRing \stackrel{\overset{U}{\leftarrow}}{\underset{W}{\longrightarrow}} \Lambda Ring \,.

Hence rings of Witt-vectors are the co-free Lambda-rings.

This statement appears in (Hazewinkel 08, p. 87, p. 97).


On the other hand, the free Lambda-ring (on one generator) (hence the left adjoint construction to the forgetful functor) is the ring of symmetric functions.

This statement appears in (Hazewinkel 08, p. 98).


Operations on the p-adic Witt vectors

On untruncated pp-adic Witt vectors there are two operations, the Frobenius morphism and the Verschiebung morphism? satisfying relations (Lemma 1) being constitutive for the definition of the Dieudonné ring: In fact the Dieudonné ring is generated by two objects satisfying these relations.

Also the nn-truncations of a Witt ring are rings since by definition the ring operations (addition and multiplication) of the first nn components only involve the first nn components. We have Wlim nW nW\simeq lim_n W_n and the projection map W(A)W n(A)W(A)\to W_n(A) is a ring homomorphism. We also have operations on the truncated Witt rings.

The shift map

For W(k)W(k) as for every kk-scheme we have the Verschiebung morphism?. It is defined to be the adjoint operation to the Frobenius morphism. For W(k)W(k) the Verschiebung morphism coincides with the shift (a 0,a 1,)(0,a 0,a 1,)(a_0, a_1,\dots)\mapsto (0, a_0, a_1,\dots)

For the truncated Witt rings and the shift operation V:W n(k)W n+1(k)V:W_n(k)\to W_{n+1}(k) the Verschiebung morphism equals the VR=RVVR=RV where RR is the restriction map.

The restriction map

The restriction map R:W n+1(A)W n(A)R: W_{n+1}(A)\to W_n(A) is given by (a 0,,a n)(a 0,,a n1)(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1}).

The Frobenius morphism

The Frobenius endomorphism F:W n(A)W n(A)F: W_n(A)\to W_n(A) is given by (a 0,,a n1)(a 0 p,,a n1 p)(a_0, \ldots , a_{n-1})\mapsto (a_0^p, \ldots, a_{n-1}^p). This is also a ring map, but only because of our necessary assumption that AA is of characteristic pp.

Just by brute force checking on elements we see a few relations between these operations, namely that V(x)y=V(xF(R(y)))V(x)y=V(x F(R(y))) and RVF=FRV=RFV=pRVF=FRV=RFV=p the multiplication by pp map.

Duality of finite Witt groups

For a kk-ring RR let W (R)W^\prime(R) denote the ideal in W(R)W(R) consisting of sequences x=(x n) nx=(x_n)_n of nilpotent elements in W(R)W(R) such that x n=0x_n=0 for large nn.

Let EE denote the Artin-Hasse exponential?. Then we have E(x,1)E(x,1) is a polynomial for xW (R)x\in W^\prime(R) and

E(,1):{W μ k wE(w,1)E(-,1):\begin{cases}W^\prime\to \mu_k \\ w\mapsto E(w,1)\end{cases}

is a morphism of group schemes to the multiplicative group scheme μ k\mu_k.


a) W (R)W^\prime (R) is an ideal in W(R)W(R).

b) E(,1):W μ kE(-,1): W^\prime\to \mu_k is an morphism of group schemes.

c) The morphism

{W×W μ k (x,y)E(xy,1)\begin{cases} W\times W^\prime\to \mu_k \\ (x,y)\mapsto E(xy,1) \end{cases}

is bilinear and gives an isomorphism of group schemes

W D(W)W^\prime\to D(W)

where DD denotes the Cartier dual of WW (maybe it is equivalently the Pontryagin dual of the underlying group of the (plain) ring WW). That this map is a morphism of group schemes follows from the definition of the Cartier dual.

d) For xW(R)x\in W(R) and yW (R)y\in W^\prime (R) we have E(xy,1)R ×E(xy,1)\in R^\times and

E(V nxy,1)=E(T n(xF ny),1)=E(xF ny,1)E(V^n x y,1)=E (T^n(x F^n y),1)=E(x F^n y,1)
Definition and Theorem

Let ker(F n m):=ker(F m:W nkW nk)ker(F_n^m):=ker (F^m:W_{nk}\to W_{nk}) denote the kernel of mm-times iterated Frobenius endomorphism of the nn-truncated Witt ring.Let

σ n:{W nkW k (α 0,,α n1)(α 0,,α n1,0,)\sigma_n:\begin{cases} W_{nk}\to W_k \\ (\alpha_0,\dots,\alpha_{n-1})\mapsto(\alpha_0,\dots,\alpha_{n-1},0,\dots) \end{cases}

be the section of the restriction R n:W kW nkR_n:W_k\to W_{nk}. σ n\sigma_n sends ker(F n m)ker(F_n^m) in W W^\prime. Note that σ n\sigma_n is not a morphism of groups.

We define the bilinear map

<,>:{ker(F n m)×ker(F m n)R × <x,y>E(σ n(x)σ m(y),1)\lt-,-\gt:\begin{cases} ker(F^m_n)\times ker(F^n_m)\to R^\times \\ \lt x,y\gt\mapsto E(\sigma_n(x)\sigma_m(y),1) \end{cases}

then <x,y>\lt x,y\gt is bilinear and gives an isomorphism

ker(F n m)D(ker(F m n)ker(F^m_n)\simeq D(ker(F^n_m)

and satisfies

<x,ty>=<fx,y>\lt x,t y\gt=\lt f x,y\gt
<x,ry>=<ix,y>\lt x,r y\gt=\lt i x,y\gt

where the morphisms are

ker(F n m) t ker(F n+1 m) f r ker(F n m1) i ker(F n m)\array{ ker(F^m_n) &\stackrel{t}{\to}& ker(F^m_{n+1}) \\ \downarrow^f&&\downarrow^r \\ ker(F^{m-1}_n) &\stackrel{i}{\hookrightarrow}& ker(F^m_n) }

where ii is the canonical inclusion, and r,f,tr,f,t are induced by R,F,TR,F,T where FF is Frobenius, TT is Verschiebung and R:WW nR:W\to W_n is restriction. ii and tt are monomorphisms, ff and rr are epimorphisms, and for ker(F m n)ker(F^n_m) we have F=ifF=if and V=rtV=rt.

References: Pink, §25, Demazure, III.4

Properties of the Witt group

The group of universal (i.e. not pp-adic) Witt vectors equals W(k)=1+k[[X]]W(k)= 1+k [ [ X] ] i.e. the multiplicative group of power series in one variable XX with constant term 11.

Properties of the Witt ring


Let kk be a perfect field of prime characteristic pp.

Then W(k)W(k) is a discrete valuation ring with maximal ideal generated by pp. From the above we see that pW(k)=(0,a 0 p,a 1 p,)pW(k)=(0, a_0^p, a_1^p, \ldots ). This clearly gives W(k)/pW(k)kW(k)/pW(k)\simeq k.

Also, W(k)/p nW(k)W n(k)W(k)/p^nW(k)\simeq W_n(k). Thus the completion of W(k)W(k) with respect to the maximal ideal is just limW n(k)W(k)lim W_n(k)\simeq W(k) which shows that W(k)W(k) is a complete discrete valuation ring.


Basic examples

  • W p (𝔽 p) pW_{p^\infty}(\mathbb{F}_p)\simeq \mathbb{Z}_p the pp-adic integers.

  • W p (𝔽 p n)W_{p^\infty}(\mathbb{F}_{p^n}) is the unique unramified extension of p\mathbb{Z}_p of degree nn.

Lubin-Tate ring

The Lubin-Tate ring in Lubin-Tate theory is a power series ring over a Witt ring and this way Witt rings govern much of chromatic homotopy theory.

As an Abelian group W(A)W(A) is isomorphic to the group of curves in the one-dimensional multiplicative formal group. In this way there is a Witt-vector-like Abelian-group-valued functor associated to every one-dimensional formal group. For special cases, such as the Lubin–Tate formal groups, this gives rise to ring-valued functors called ramified Witt vectors. (eom)


Original texts and classical surveys

witt vectors were introduced in

  • Ernst Witt, Zyklische Körper und Algebren der Characteristik pp vom Grad p np^n. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p np^n, J. Reine Angew. Math. , 176 (1936) pp. 126–140, (web)

In the context of formal group laws they were used in

  • Jean Dieudonné, Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique p>0p \gt 0 VII“ Math. Ann. , 134 (1957) pp. 114–133

See also

  • Michiel Hazewinkel, Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials, Transactions of the AMS (1980)

Surveys incluce

Modern surveys

Review in the context of the Kummer-Artin-Schreier-Witt exact sequence is in

  • Ariane Mézard, Matthieu Romagny, Dajano Tossici, section 2 of Sekiguchi-Suwa theory revisited (arXiv:1104.2222)

Further development of the theory

In the context of Borger's absolute geometry:

Revised on July 23, 2014 21:49:09 by Urs Schreiber (