Lecture notes on Witt vectors

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1 Lecture notes on Witt vectors Lars Hesselholt The purpose of these notes is to give a self-containe introuction to Witt vectors. We cover both the classical p-typical Witt vectors of Teichmüller an Witt [4] an the generalize or big Witt vectors of Cartier [1]. In the approach taken here, all necessary congruences are isolate in the lemma of Dwork. A slightly ifferent but very reaable account may be foun in Bergman [3, Appenix]. We conclue with a brief treatment of special λ-rings an Aams operations. We refer the reaer to Langer-Zink [2, Appenix] for a careful analysis of the behavior of the ring of Witt vectors with respect to étale morphisms. Let N be the set of positive integers, an let S N be a subset with the property that, if n S, an if is a ivisor in n, then S. We then say that S is a truncation set. The big Witt ring W S (A) is efine to be the set A S equippe with a ring structure such that the ghost map w : W S (A) A S that takes the vector (a n n S) to the sequence (w n n S), where w n = n a n/, is a natural transformation of functors from the category of rings to itself. Here, on the right-han sie, A S is consiere a ring with componentwise aition an multiplication. To prove that there exists a unique ring structure on W S (A) that is characterize in this way, we first prove the following result. Lemma 1 (Dwork). Suppose that, for every prime number p, there exists a ring homomorphism φ p : A A with the property that φ p (a) a p moulo pa. Then a sequence (x n n S) is in the image of the ghost map w : W S (A) A S if an only if x n φ p (x n/p ) moulo p vp(n) A, for every prime number p, an for every n S with v p (n) 1. Here v p (n) enotes the p-aic valuation of n. Proof. We first show that, if a b moulo pa, then a pv 1 b pv 1 moulo p v A. If we write a = b + pɛ, then a pv 1 = b pv 1 + ( ) p v 1 b pv 1 i p i ɛ i. i 1ip v 1 The author was partially supporte by the National Science Founation. 1

2 In general, the p-aic valuation of the binomial coefficient ( ) m+n n is equal to the number of carriers in the aition of m an n in base p. So (( )) p v 1 v p = v 1 v p (i), i an hence, (( ) p v 1 v p )p i = v 1 + i v p (i) v. i This proves the claim. Now, since φ p is a ring-homomorphism, φ p (w n/p (a)) = φ p (a n/p ) (n/p) which is congruent to (n/p) an/ moulo p vp(n) A. If ivies n but not n/p, then v p () = v p (n), an hence this sum is congruent to n an/ = w n (a) moulo p vp(n) A as state. Conversely, if (x n n S) is a sequence such that x n φ p (x n/p ) moulo p vp(n) A, we fin a vector a = (a n n S) with w n (a) = x n as follows. We let a 1 = x 1 an assume, inuctively, that a has been chosen, for all that ivies n, such that w (a) = x. The calculation above shows that the ifference x n n, n a n/ is congruent to zero moulo p vp(n) A. Hence, we can fin a n A such that na n is equal to this ifference. Proposition 2. There exists a unique ring structure such that the ghost map w : W S (A) A S is a natural transformation of functors from rings to rings. Proof. Let A be the polynomial ring Z[a n, b n n S]. Then the unique ring homomorphism φ p : A A that maps a n to a p n an b n to b p n satisfies that φ p(f) = f p moulo pa. Let a an b be the sequences (a n n S) an (b n n S). Since φ p is a ring homomorphism, Lemma 1 shows immeiately that the sequences w(a) + w(b), w(a) w(b), an w(a) are in the image of the ghost map. It follows that there are sequences of polynomials s = (s n n S), p = (p n n S), an ι = (ι n n S) such that w(s) = w(a) + w(b), w(p) = w(a) w(b), an w(ι) = w(a). Moreover, since A is torsion free, the ghost map is injective, an hence, these polynomials are unique. Let now A be any ring, an let a = (a n n S) an b = (b n n S) be two vectors in W S (A ). Then there is a unique ring homomorphism f : A A such that W S (f)(a) = a an W S (f)(b) = b. We efine a + b = W S (f)(s), a b = W S (f)(p), an a = W S (f)(ι). It remains to prove that the ring axioms are verifie. Suppose first that A is torsion free. Then the ghost map is injective, an hence, the ring axioms are satisfie in this case. In general, we choose a surjective ring homomorphism g : A A from a torsion free ring A. Then W S (g): W S (A ) W S (A ) 2

3 is again surjective, an since the ring axioms are satisfie on the left-han sie, they are satisfie on the right-han sie. If T S are two truncation sets, then the forgetful map R S T : W S(A) W T (A) is a natural ring homomorphism calle the restriction from S to T. If n N, an if S N is a truncation set, then S/n = { N n S} is again a truncation set. We efine the nth Verschiebung map V n : W S/n (A) W S (A) by V n ((a S/n)) m = { a, if m = n, 0, otherwise. Lemma 3. The Verschiebung map V n is aitive. Proof. There is a commutative iagram W S/n (A) w A S/n V n W S (A) w A S V w n where the map V w n is given by V w n ((x S/n)) m = { nx, if m = n, 0, otherwise. Since the map Vn w is aitive, so is the map V n. Inee, if A is torsion free, the horizontal maps are both injective, an hence, V n is aitive in this case. In the general case, we choose a surjective ring homomorphism g : A A an argue as in the proof of Prop. 2 above. Lemma 4. There exists a unique natural ring homomorphism F n : W S (A) W S/n (A) such the iagram W S (A) w A S F n W S/n (A) F w n w A S/n, where F w n ((x m m S)) = x n, commutes. 3

4 Proof. We construct the Frobenius map F n in a manner similar to the construction of the ring operations on W S (A) in Prop. 2. We let A be the polynomial ring Z[a n n S], an let a be the vector (a n n S). Then Lemma 1 shows that the sequence F w n (w(a)) AS/n is the image of a (unique) element F n (a) = (f n, S/n) W S/n (A) by the ghost map. If A is any ring, an if a = (a n n S) is a vector in W S (A ), then we efine F n (a ) = W S/n (g)(f n (a)), where g : A A is the unique ring homomorphism that maps a to a. Finally, since Fn w is a ring homomorphism, an argument similar to the proof of Lemma 3 shows that also F n is a ring homomorphism. The Teichmüller representative is the map efine by [ ] S : A W S (A) ([a] S ) n = { a, if n = 1, 0, otherwise. It is a multiplicative map. Inee, there is a commutative iagram A [ ] S W S (A) A w A S, [ ] w S where ([a] w S ) n = a n, an [ ] w S is a multiplicative map. Lemma 5. The following relations hols. (i) a = n S V n([a n ] S/n ). (ii) F n V n (a) = na. (iii) av n (a ) = V n (F n (a)a ). (iv) F m V n = V n F m, if (m, n) = 1. Proof. One easily verifies that both sies of each equation have the same image by the ghost map. This shows that the relations hol, if A is torsion free, an hence, in general. Proposition 6. The ring W S (Z) of big Witt vectors in the ring of rational integers is equal to the prouct W S (Z) = n S Z V n ([1] S/n ) with the multiplication given by V m ([1] S/m ) V n ([1] S/n ) = c V ([1] S/ ), where c = (m, n) an = mn/(m, n) are the greatest common ivisor an the least common multiple of m an n. 4

5 Proof. The formula for the multiplication follows from Lemma 5 (ii)-(iv). Suppose first that S is finite. If S is empty, the statement is trivial, so assume that S is non-empty. We let m S be maximal, an let T = S {m}. Then the sequence of abelian groups 0 W {1} (Z) Vm W S (Z) RS T W T (Z) 0 is exact, an we wish to show that it is equal to the sequence V 0 Z [1] m {1} Z V n ([1] S/n ) RS T Z V n ([1] T/n ) 0. n T n S The latter sequence is a sub-sequence of the former sequence, an, inuctively, the left-han terms (resp. the right-han terms) of the two sequences are equal. Hence, mile terms are equal, too. The statement for S finite follows. Finally, a general truncation set S is the union of the finite sub-truncation sets S α S, an hence, W S (Z) = lim α W Sα (Z). This proves the state formula in general. The action of the restriction, Frobenius, an Verschiebung operators on the generators V n ([1] S/n ) is easily erive from the relations Lemma 5 (ii) (iv). To give a formula for the Teichmüller representative, we recall the Möbius inversion formula. Let g : N Z be a function, an let f : N Z be the function given by f(n) = n g(). Then the function g is given by f by means of the formula g(n) = n µ()f(n/), where µ: N { 1, 0, 1} is the Möbius function. Here µ() = ( 1) r, if is a prouct of r 0 istinct prime numbers, an µ() = 0, otherwise. Aenum 7. Let m be an integer. Then [m] S = 1 ( µ()m n/) V n ([1] S/n ), n n S n where µ: N { 1, 0, 1} is the Möbius function. Proof. It suffices to prove that the formula hols in W S (Z). We know from Prop. 6 that there are unique integers r, S, such that [m] S = S r V ([1] S/ ). Evaluating the nth ghost component of this equation, we get m n = n r, an the state formula now follows from the Möbius inversion formula. 5

6 Lemma 8. Suppose that A is an F p -algebra, an let ϕ: A A be the Frobenius enomorphism. Then F p = R S S/p W S(ϕ): W S (A) W S/p (A). Proof. We recall from the proof of Prop. 4 that F p (a) = (f p, (a) S/p), where f p, are the integral polynomials efine by the equations f n/ p, = a pn/ pn n for all n S. Let A = Z[a n n S]. We shall prove that for all n S/p, f p,n a p n moulo pa. This is equivalent to the statement of the lemma. If n = 1, we have f p,1 = a p 1 + pa p, an we are one in this case. So let n > 1 an assume, inuctively, that the state congruence has been prove for all proper ivisors in n. Then, if is a proper ivisor in n, f p, a p moulo pa, so f n/ p, apn/ moulo p vp(n)+1 A; compare the proof of Lemma 1. Rewriting the efining equations f n/ p, = a pn/ + a pn/ n pn, n n an noting that if pn an n, then v p () = v p (n) + 1, we fin nf p,n na p n moulo p vp(n)+1 A. Since A is torsion free, we conclue that f p,n a p n moulo pa as esire. We consier the truncation set P = {1, p, p 2,... } N that consists of all powers of a fixe prime number p. The proper non-empty sub-truncation sets of P all are of the form {1, p,..., p n 1 }, for some positive integer n. The rings W (A) = W P (A) W n (A) = W {1,p,...,p n 1 }(A) are calle the ring of p-typical Witt vectors in A an p-typical Witt vectors of length n in A, respectively. We shall now show that, if A is a Z (p) -algebra, the rings of big Witt vectors W S (A) ecompose canonically as a prouct of rings of p-typical Witt vectors. We begin with the following result. Lemma 9. Let m be an integer an suppose that m is invertible (resp. a nonzero-ivisor) in A. Then m is invertible (resp. a non-zero-ivisor) in W S (A). Proof. It suffices to prove the lemma, for S finite. Inee, in general, W S (A) is the limit of W T (A), where T ranges over the finite sub-truncation sets of S. So assume that S is finite an non-empty. Let n S be maximal, an let T = S {n}. Then S/n = {1} an we have an exact sequence 0 A Vn W S (A) RS T W T (A) 0 6

7 from which the lemma follows by easy inuction. Proposition 10. Let p be a prime number, an let A be a Z (p) -algebra. Let S be a truncation set, an let I(S) = {k S p k}. Then the ring W S (A) has a natural iempotent ecomposition W S (A) = W S (A)e k where e k = l I(S),l 1 Moreover, the composite map is an isomorphism. k I(S) ( 1 k V k([1] S/k ) 1 ) kl V kl([1] S/kl ). W S (A)e k W S (A) F k W S/k (A) RS/k S/k P W S/k P (A) Proof. We calculate an hence, w n ( 1 k V k([1] S/k )) = w n (e k ) = { 1, if k S kn, 0, otherwise, { 1, if k S kp, 0, otherwise. It follows that the elements e k, k I(S), are orthogonal iempotents in W S (A). This proves the former part of the statement. To prove the latter part, we note that multiplication by k efines a bijection S/k P = (S kp )/k S kp an that the following iagram commutes: W S (A)e k w R S/k S/k P F k W S/k k (A) A S kp k w A S/k P. We first assume that A is torsion free an has an enomorphism φ p : A A such that φ p (a) a p moulo pa. Then the horizontal maps w are both injective. Moreover, Lemma 1 ientifies the image of the top horizontal map w with the set of sequences (x S kp ) such that x φ p (x /p ) moulo p vp() A. Similarly, the image of the lower horizontal map w is the set of sequences (y S/k P ) such that y φ p (y /p ) moulo p vp() A. Since the right-han vertical map k inuces an isomorphism of these subrings, the left-han vertical map R S/k S/k P F k is an isomorphism in this case. Example 11. Let S = {1, 2,..., n} such that W S (A) is the ring W n (A) of big Witt vectors of length n in A. Then S/k P = {1, p,..., p s 1 } where s = s(n, k) is the unique integer with p s 1 k n < p s k. Hence, if A is a Z (p) -algebra, W n (A) W s (A) 7

8 where the prouct ranges over 1 k n with p k, an where s = s(n, k) is given as above. We now consier the ring W n (A) of p-typical Witt vectors of length n in A in more etail. The ghost map w : W n (A) A n takes the vector (a 0,..., a n 1 ) to the sequence (w 0,..., w n 1 ) where w i = a pi 0 + papi p i a i. If φ: A A is a ring homomorphism with φ(a) a p moulo pa, then Lemma 1 ientifies the image of the ghost map with the subring of sequences (x 0,... x n 1 ) such that x i φ(x i 1 ) moulo p i A, for all1 i n 1. We write for the Teichmüller representative an [ ] n : A W n (A) F : W n (A) W n 1 (A) V : W n 1 (A) W n (A) for the pth Frobenius an pth Verschiebung. Lemma 12. If A is an F p -algebra, then V F = p. Proof. For any ring A, the composite V F is given by multiplication by the element V ([1] n 1 ). Suppose that A is an F p -algebra. The exact sequences 0 A V n 1 W n (A) R W n 1 (A) 0 show, inuctively, that W n (A) is annihilate by p n. Hence, V ([1] n 1 ) is annihilate by p n 1. We show by inuction on n that V ([1] n 1 ) = p[1] n, the case n = 1 being trivial. The formula from Aenum 7 gives that [p] n = p[1] n + p ps p ps 1 p s V s ([1] n s ). 0<s<n Since [p] n = 0, an since, inuctively, V s ([1] n s ) = p s 1 V ([1] n 1 ), for 0 < s < n, we can rewrite this formula as 0 = p[1] n + (p pn 1 1 1)V ([1] n 1 ). But p n 1 1 n 1, so we get p[1] n = V ([1] n 1 ) as state. We now suppose that A is a p-torsion free ring an that there exists a ring homomorphism φ: A A such that φ(a) a p moulo pa. It follows from Lemma 1 that there is a unique ring homomorphism s φ : A W (A) such that the composite A s φ W (A) w A N0 maps a to (a, φ(a), φ 2 (a),... ). We then efine t φ : A W (A/pA) to be the composite of s φ an the map inuce by the canonical projection of A onto A/pA. We recall that the F p -algebra A/pA is sai to be perfect, if the Frobenius enomorphism ϕ: A/pA A/pA is an automorphism. 8

9 Proposition 13. Let A be a p-torsion free ring, an let φ: A A be a ring homomorphism such that φ(a) a p moulo pa. Suppose that A/pA is a perfect F p -algebra. Then the map t φ inuces an isomorphism for all n 1. t φ : A/p n A W n (A), Proof. The map t φ factors as in the statement since V n W (A/pA) = V n W (φ n (A/pA)) = V n F n W (A/pA) = p n W (A/pA). The proof is now complete by an inuction argument base on the following commutative iagram: 0 A/pA pn 1 A/p n A pr A/p n 1 A 0 ϕ n 1 t φ 0 A/pA V n 1 W n (A/pA) t φ R W n 1 (A/pA) 0. The top horizontal sequence is exact, since A is p-torsion free, an the left-han vertical map is an isomorphism, since A/pA is perfect. The statement follows by inuction on n 1. We return to the ring of big Witt vectors. We write (1 + ta[t]) for the multiplicative group of power series over A with constant term 1. Proposition 14. There is a natural commutative iagram W(A) γ (1 + ta[t]) w A N γ w ta[t] t t log where γ(a 1, a 2,... ) = n1(1 a n t n ) 1, γ w (x 1, x 2,... ) = n1 x n t n, an the horizontal maps are isomorphisms of abelian groups. Proof. It is clear that γ w is an isomorphism of aitive abelian groups. We show that γ is a bijection. We have (1 a n t n ) 1 = (1 + b 1 t + b 2 t ) 1 n1 where the coefficient b n is given by the sum b n = ( 1) r a i1... a ir 9

10 that runs over all 1 i 1 < < i r n such that i 1 + 2i ri r = n. This formula shows that the coefficients a n, n 1, are etermine uniquely by the coefficients b n, n 1. Inee, we have the recursive formula a n = b n ( 1) r a i1... a ir, where the sum on the right-han sie ranges over 1 i 1 < < i r < n such that i 1 + 2i ri r = n. To prove that the map γ is a homomorphism from the aitive group W(A) to the multiplicative group (1 + ta[t]), it suffices as usual to consier the case where A is torsion free. In this case the vertical maps in the iagram of the statement are both injective, an hence, it suffices to show that the iagram of the statement commutes. We calculate: t t log( (1 a t ) 1 ) = t t log(1 a t ) = ta t 1 a t = a t a s t s = a q tq = ( a n/ ) t n. 1 s0 1 q1 n1 This completes the proof. Aenum 15. The map γ inuces an isomorphism of abelian groups γ S : W S (A) Γ S (A) where Γ S (A) is the quotient of the multiplicative group Γ(A) = (1 + ta[t]) by the subgroup I S (A) of all power series of the form n NS (1 a nt n ) 1. Proof. The kernel of the restriction map R N S : W(A) W S(A) is equal to the subset of all vectors a = (a n n N) such that a n = 0, if n S. The image of this subset by the map γ is the subset I S (A) Γ. Example 16. If S = {1, 2,..., m}, then I S (A) = (1 + t m+1 A[t]). Hence, in this case, Aenum 15 gives an isomorphism of abelian groups γ S : W m (A) Γ S (A) = (1 + ta[t]) /(1 + t m+1 A[t]). The structure of this group, for A a Z (p) -algebra, was examine in Example 11. Lemma 17. Let p be a prime number, an let A be any ring. Then the ring homomorphism F p : W(A) W(A) satisfies that F p (a) a p moulo pw(a). Proof. We first let A = Z[a 1, a 2,... ] an a = (a 1, a 2,... ). If suffices to show that there exists b W(A) such that F p (a) a p = pb. By Lemma 9, the element is necessarily unique; we use Lemma 1 to prove that it exists. We have w n (F p (a) a p ) = pn a pn/ ( n n a n/ which is clearly congruent to zero moulo pa. So let x = (x n n N) with x n = 1 p (F p(a) a p ). 10 ) p

11 We wish to show that x = w(b), for some b W(A). The unique ring homomorphism φ l : A A that maps a n to a l n satisfies that φ l (f) = f l moulo la, an hence, Lemma 1 shows that x is in the image of the ghost map if an only if x n φ l (x n/l ) moulo l v l(n) A, for all primes l an all n ln. This is equivalent to showing that w n (F p (a) a p ) φ l (w n/p (F p (a) a p )) moulo l v l(n) A, if l p an n ln, an moulo l v l(n)+1 A, if l = p an n ln. If l p, the statement follows from Lemma 1, an if l = p an n ln, we calculate w n (F p (a) a p ) φ p (w n/p (F p (a) a p )) = ( ) p ( + pn, n a pn/ n a n/ (n/p) a n/ ) p. If pn an n, then v p () = v p (n) + 1, so the first summan is congruent to zero moulo p vp(n)+1 A. Similarly, if n an (n/p), then v p () = v p (n), an hence, moulo p vp(n) A. But then ( n n a n/ (n/p) a n/ ) p ( (n/p) a n/ a n/ ) p moulo p vp(n)+1 A; compare the proof of Lemma 1. This completes the proof. Let ɛ: W(A) A be the ring homomorphism that takes a = (a n n N) to a 1. Proposition 18. There exists a unique natural ring homomorphism : W(A) W(W(A)) such that w n ( (a)) = F n (a), for all n N. Moreover, the functor W( ) an the ring homomorphisms an ɛ form a comona on the category of rings. Proof. By naturality, we may assume that A is torsion free. Then Lemma 9 shows that also W(A) is torsion free, an hence, the ghost map w : W(W(A)) W(A) N is injective. Lemma 17 an Lemma 1 show that the sequence (F n (a) a N) is in the image of the ghost map. Hence, the natural ring homomorphism exists. The secon part of the statement means that an W( A ) A = W(A) A : W(A) W(W(W(A))) W(ɛ A ) A = ɛ W(A) A : W(A) W(A). Both equalities are reaily verifie by evaluating the ghost coorinates. Definition 19. A special λ-ring is a ring A an a ring homomorphism λ: A W(A) that makes A a coalgebra over the comona (W( ),, ɛ). 11

12 Let (A, λ: A W(A)) be a special λ-ring. Then the associate nth Aams operation is the ring homomorphism efine to be the composition ψ n : A λ W(A) wn A of the structure map an the nth ghost map. References [1] P. Cartier, Groupes formels associés aux anneaux e Witt généralisés, C. R. Acac. Sci. Paris, Sér. A B 265 (1967), A129 A132. [2] A. Langer an T. Zink, De Rham-Witt cohomology for a proper an smooth morphism, J. Inst. Math. Jussieu 3 (2004), [3] D. Mumfor, Lectures on curves on an algebraic surface, Annals of Mathematics Stuies, vol. 59, Princeton University Press, Princeton, N.J., [4] E. Witt, Zyklische Körper un Algebren er Charakteristik p vom Gra p n, J. reine angw. Math. 176 (1937), Massachusetts Institute of Technology, Cambrige, Massachusetts aress: larsh@math.mit.eu Nagoya University, Nagoya, Japan aress: larsh@math.nagoya-u.ac.jp 12