PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS)

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1 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 1. Background Story The starting point of p-adic Hodge theory is the comparison conjectures (now theorems) between p-adic étale cohomology, de Rham cohomology, and (log-)crystalline cohomology. Throughout the notes, K is a finite extension of Q p with residue field k. Let W (k) be the Witt vectors with coefficients in k and let K 0 = Frac W (k). Theorem 1.1. (Hodge-Tate comparison) Let X be a proper smooth variety over K. There exists a canonical isomorphism Het(X ń K, Q p ) Qp C p = H n i (X, Ω i X/K ) K C p ( i) compatible with G K -actions. 0 i n Theorem 1.2. (de Rham comparison) Let X be a proper smooth variety over K. There exists a canonical isomorphism H ń et(x K, Q p ) Qp B dr = H n dr (X/K) K B dr compatible with G K -actions and filtrations. Theorem 1.3. (crystalline comparison) Let X be a proper smooth variety over K. Suppose X has a proper smooth model X over O K. Let X 0 denote the special fiber of X. There exists a canonical isomorphism H ń et(x K, Q p ) Qp B crys = H n crys (X 0 /W (k)) W (k) B crys compatible with G K -actions, Frobenius actions, and filtrations. There is also a semistable comparison relating p-adic étale cohomology to logcrystalline cohomology. The purpose of these notes is to construct and study various period rings including B dr and B crys. Good references for an introduction to p-adic Hodge theory include [BC], [FO], and [Ber]. The problems and examples here are by no means original, most of which are inspired from literatures mentioned above. 2. Witt Vectors Definition 2.1. Let A be a topological ring and let A I 1 I 2 be a decreasing chain of ideals. Assume A/I 1 is an F p -algebra and I n I m I n+m. The topology on A is given by (I n ) n 1. (i) A is called a p-ring if the topology is separated and completed. (ii) A is called a strict p-ring if moreover I n = p n A and p is not a zero divisor in A. 1

2 2 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) Let A be a p-ring with perfect residue ring R = A/I 1. For x R and n N, let x n = x 1/pn. Let x n be a lift of x n to A. The Teichmüller lift of x is defined to be [x] := lim n ( x n ) pn In particular, if A is a strict p-ring with perfect residue ring R, then every a A can be uniquely written as a = p n [a n ] with a n R. (see Problem 2) n=0 Theorem 2.2. If R is a perfect ring of characteristic p. Then there exists a unique strict p-ring W (R) with residue ring R. Roughly speaking, W (R) = { n=0 pn [x n ] x n R}. For an explicit description of W (R), see Problem 4. Theorem 2.3. (Universality) Let R 0 be a perfect ring of characteristic p. Let A be any p-ring with residue ring R. Suppose α : R 0 R is a ring homomorphism and α : R 0 A is a multiplicative lift of α, then there exists a unique homomorphism α : W (R 0 ) A such that α([x]) = α(x). Remark 2.4. Witt vectors can be defined for more general rings, not necessarily F p -algebras. For details, we refer to [Ser]. Problem 1. Which of the following rings are p-rings? Strict p-rings? (a) O K (where K/Q p is a finite extension.) (b) O K (c) O Cp (d) O K [[X 1/p J ]] = lim ( n m=0o K [X 1/pm j ; j J])/p n (J is any index set). (e) R + (where (R, R + ) is a perfectoid algebra of characteristic 0). Problem 2. Let A be a strict p-ring with perfect residue ring R. Show that every a A can be uniquely written as a = p n [a n ] with a n R. n=0

3 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 3 Problem 3. (Universal Witt polynomials) Consider strict p-ring S = Z p [[X 1/p i, Y 1/p i ]] i 0 with residue ring S = F p [X 1/p i, Y 1/p i ] i 0. (i) Show that there exist polynomials P i, Q i S such that p i [X i ] + p i [Y i ] = p i [P i ] ( p i [X i ] )( p i [Y i ] ) = p i [Q i ] (Hint: Use Problem 2.) (ii) Calculate P 0, P 1, Q 0, Q 1. (iii) Show that P i s and Q i s are universal in the following sense. For any strict p-ring A with perfect residue ring R and any x 0, x 1,..., y 0, y 1,... R, we have p i [x i ] + p i [y i ] = p i [P i (x 0, x 1,..., y 0, y 1,...)] ( p i [x i ] )( p i [y i ] ) = p i [Q i (x 0, x 1,..., y 0, y 1,...)] Problem 4. (Explicit description of W (R)) Let R be a perfect ring of characteristic p. Suppose R has a presentation R = S J /I where S J = F p [X 1/p J ] for some index set J, and I is a perfect ideal of S J. (i) Show that such a presentation always exists. (ii) Consider S J := Z p [[X 1/p J ]] Show that W (R) = S J /W (I) where W (I) = { p i [x i ] x i I }. (iii) Prove Theorem 2.3 using the explicit description above. (Hint: Lift S J A to S J A.) Problem 5.(O C p and W (O C p )) Let (C p, O C p ) be the tilt of the perfectoid field (C p, O Cp ) (see [Sch1]). We briefly review the construction here. Consider O C p := lim x x p O Cp /p equipped with inverse limit topology. This is a perfect ring of characteristic p. Let C p = Frac O C p. The natural projection gives a multiplicative homeomorphism lim O Cp x x p lim O x x Cp /p. p

4 4 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) The inverse is given by lim O Cp /p lim x x p x x p O Cp sending x = (x 0, x 1,...) to (x #, x #(1), x #(2),...) where x #(m) = lim n x pn m n. One can define a valuation on O C p by x := x # Cp. (i) Check that is indeed a non-archimedean valuation on O C p. In particular, x + y max( x, y ), x, y O C p. (ii) Check that O C p is complete and separated with respect to. (iii) Consider ε = (1, ζ p, ζ p 2,...) lim O x x Cp = OC p p For each n N, calculate ε 1/pn 1. (iv) Let K/Q p be a finite extension and let G K = Gal(K/K). Then O C p is equipped with a natural Frobenius action ϕ and an action of G K. More precisely, for x = (x (0), x (1),...) lim O x x p C p = OC p, we define ϕ(x) = ((x (0) ) p, (x (1) ) p,...) and g(x) = (g(x (0) ), g(x (1) ),...), g G K. The ϕ and G K actions also extend naturally to W (O C p ). Find (O C p ) ϕ=1, (O C p ) G K, W (O C p ) ϕ=1, W (O C p ) G K. 3. De Rham period ring B dr Consider the following G K -equivariant ring homomorphism θ : W (O C p ) O Cp p i [x i ] p i x # i It turns out ker(θ) is a principle ideal generated by ξ = [p ] p, where p = (p, p 1/p, p 1/p2,...) lim x x p O Cp = OC p. Define B + dr to be the ker(θ)-adic completion of W (O C )[ 1 p p ]; i.e., The natural projection induces B + dr = lim n W (O C p )[ 1 p ]/(ker θ)n. θ + dr : B+ dr W (O C p )[1 p ]/(ker θ) = C p. In particular, B + dr is a complete discrete valuation ring with maximal ideal m B + dr = (ker θ) and residue field C p. We temporarily equip B + dr with the discrete valuation ring topology. (See Problem 11 for a better topology.)

5 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 5 Let ε be the same as in Problem 5(iii). Consider the element n 1 ([ε] 1)n t = log[ε] = ( 1). n n=1 One can check that t converges to a uniformizer in B + dr. Moreover, t has the nice property that g(t) = χ(g)t, g G K where χ is the cyclotomic character. Finally, we define B dr = B + dr [ 1 t ] = Frac B+ dr, which carries a natural G K-action. One can prove B G K dr = K. In addition, one can put a G K-stable filtration on B dr by setting Fil n B dr := t n B + dr = mn n Z. B dr, + However, the ϕ-action on W (O C p )[ 1 p ] does not extend to B dr. Problem 6. If we identify W (O C p ) with (O C p ) N and equip the product of valuation topology from O C p, show that θ : W (O C p ) O Cp is open. Problem 7. Let k be the residue field of K. Show that θ is actually a morphism of W ( k)-algebras with the natural W ( k)-structures on both sides. Problem 8. For α W (O C p ), let α denote the reduction of α mod p. Show that α ker(θ) is a generator if and only if α = 1. In particular, ξ is a generator. Problem 9. Show that ϕ-action on W (O C p )[ 1 p ] does not extend to B+ dr. (Hint: Consider ϕ-action on ker θ.) Problem 10. Show that [p ] is invertible in B + dr. Problem 11. This is a famous exercise in [BC]. We put a new topological ring structure on W (O C p )[ 1 p ] which extends to one on B+ dr such that the quotient topology on C p through θ + dr is the natural valuation topology! (i) For any open ideal a O C p and N 0, consider U N,a := ( p j W (a pj ) + p N W (O )) C W (O p C p )[ 1 p ]. j> N Prove that U N,a is a G K -stable W (O C p )-submodule of W (O C p )[ 1 p ]. (ii) Define a topological ring structure on W (O C p )[ 1 p ] by making U N,a s a base of open neighborhoods of 0. Show that the topological ring structure is well-defined and the G K -action is continuous under this topology.

6 6 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) (iii) Show that θ : W (O C p )[ 1 p ] C p is continuous and open, where W (O C p )[ 1 p ] is equipped with the new topology and C p with valuation topology. (iv) Show that (ker θ) n = ξ n W (O C p )[ 1 p ] are closed ideals of W (O C p )[ 1 p ]. (v) Equip B + dr with the inverse limit topology of the quotient topologies on each W (O C p )[ 1 p ]/(ker θ)n. Verify that the quotient topology on C p through θ + dr : B+ dr C p coincides with the valuation topology. (vi) Show that the new topology on B + dr is complete. Problem 12. Prove that g(t) = χ(g)t for all g G K. (Hint: Show that both sides are equal to log([ε] χ(g) ). This expression does not converge in discrete valuation topology, but converges in the new topology constructed in Problem 11.) Problem 13.(G K -cohomology of B dr ) (i) Calculate H i (G K, t j B + dr ) for i = 0, 1 and for all j 1. (ii) Calculate (B dr ) G K and (B + dr )G K. (Hint: Use the exact sequence 0 t i+1 B + dr ti B + dr C p(i) 0 and consider the corresponding long exact sequences. The hard part is to prove H 1 (G K, tb + dr ) = 0.) 4. De Rham representations Let K/Q p be a finite extension and let Rep Qp (G K ) denote the category of G K - representations; i.e., finite dimensional Q p -vector spaces with a continuous action of G K. Let Fil K denote the category of filtered K-vector spaces; i.e., finite dimensional K-vector spaces D equipped with an exhaustive and separated filtration {Fil i (D)} i Z. Being exhaustive means Fil i (D) = D for i 0, and being separated means Fil i (D) = 0 for i 0. Consider functor D dr : Rep Qp (G K ) Fil K V (V Qp B dr ) G K The filtration on D dr (V ) is given by Fil i (D dr (V )) := (V Qp t i B + dr )G K. It is always true that dim K D dr (V ) dim Qp V. We say V is de Rham if this is an equality. The subcategory of de Rham representations is denoted by Rep dr Q p (G K ). Theorem 4.1. (i) The functor D dr : Rep dr Q p (G K ) Fil K is exact and faithful (but not full!) Moreover, it respects direct sums, tensor products, subobjects, quotients, and duals.

7 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 7 (ii) If V is de Rham, the natural map α dr,v : D dr (V ) K B dr V Qp B dr is an isomorphism of filtered vector spaces. The notion of Hodge-Tate representations can be defined in the same fashion. The Hodge-Tate period ring is defined to be B HT = n Z C p (n) where C p (n) standards for the Tate twist. For any V Rep Qp (G K ), we can consider functor D HT : Rep Qp (G K ) Vect K. A representation V is called Hodge-Tate if dim K D HT (V ) = dim Qp V. Theorem 4.2. De Rham representations are Hodge-Tate. Important source of de Rham representations: those G K -representations arising from p-adic étale cohomologies of proper smooth varieties over K are de Rham. Problem 14. Prove Theorem 4.2. (Hint: gr D dr (V ) = D HT (V ).) Problem 15. Let V Rep Qp (G K ) and let n Z. Prove that V is de Rham if and only if V Qp Q p (n) is de Rham. Problem 16. Let K /K be a finite extension and let V Rep Qp (G K ). Show that V is de Rham as a G K -representation if and only if it is de Rham viewed as a G K -representations. Problem 17. Suppose V Rep Qp (G K ) is 1-dimensional. Prove that V is de Rham if and only if it is Hodge-Tate. Problem 18. Let η : G K Z p be a continuous character. Show that Q p (η) is de Rham (equivalently, Hodge-Tate) if and only if there exists n Z such that χ n η is potentially unramified. (A character of G K is called potentially unramified if there exists a finite extension L/K such that the image of I L is trivial.) (Hint: Assume 1 a D dr (Q p (η)) = (Q p (η) B dr ) G K. What does a have to satisfy?) Problem 19. (Tate curve) Let K/Q p be a finite extension and let q K be an element such that q < 1. Let q Z = {q n n Z} and consider quotient group E q = K /q Z ( Tate curve )

8 8 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) The abelian group E q has a natural action of G K. For each n 0, let E q [p n ] be the subgroup of p n -torsion elements. Define the Tate module T p (E q ) := lim n E q [p n ] with transition maps being multiplication by p. Inverting p, we obtain the rational Tate module V p (E q ) := T p (E q ) Zp Q p. (i) For each n, choose a primitive p n -th root of unity ζ p n and choose a p n -th root λ n of q in K. Show that (Z/p n Z) 2 E q [p n ] (a, b) ζp a nλb n is an isomorphism. (ii) Show that V p (E q ) is a 2-dimensional Q p -vector space equipped with a continuous action of G K. (iii) Show that V p (E q ) is an extension of Q p by Q p (1). 0 Q p (1) V p (E q ) Q p 0 (iv) V p (E q ) has an explicit basis {e, f} where e = (1, ζ p, ζ p 2,...), f = (q, q 1/p, q 1/p2,...). For any g G K, show that g(e) = χ(g)e, g(f) = f + a(g)e for some a(g) Z p depending on g. (v) Recall that t = log[ε] B + dr. Let q = (q, q 1/p,...) O C p. We can define log[q ] as follows. log[q ] := log p (q) + ( 1) n 1 ([q ]/q 1) n. n n=1 Check that log[q ] converges in B + dr. (vi) Let u = log[q ]. Show that g(u) = u + a(g)t. (vii) Show that V p (E q ) is de Rham. (Hint: Use t and u to modify the basis e 1, f 1 of V p (E q ) Qp B dr into a G K -invariant one.) Problem 20. Let n, m be two positive integers and n m. Let V be any extension in Rep Qp (G K ). Show that 0 Q p (n) V Q p (m) 0 (i) V is Hodge-Tate. (ii) V is de Rham if n > m. (On the other hand, every non-trivial extension 0 Q p V Q p (1) 0 is not de Rham. But this is difficult to prove.)

9 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 9 (Hint: May assume n > 0 and m = 0. Show that the exact sequence of free B + dr -modules 0 Q p (n) Qp B+ dr V Q p B + dr B+ dr 0 has a G K -equivariant splitting.) Problem 21. In this problem, we prove that the functor D dr : Rep dr Q p (G K ) Fil K is not full. (i) For V, W Rep dr Q p (G K ), show that D dr (V ) and D dr (W ) are isomorphic in Fil K if and only if dim K gr i (D dr (V )) = dim K gr i (D dr (W )) for all i. (i.e., they have the same Hodge-Tate weights and Hodge-Tate numbers.) (ii) Show that there exists a non-split extension of Q p by Q p (1) in Rep dr Q p (G K ). (Hint: Already have one from previous problems.) (iii) Conclude that D dr is not full. 5. Crystalline period ring B crys Let K/Q p be a finite extension with residue field k. Let K 0 = W (k)[ 1 p ]. We will construct period ring B crys equipped with both a filtration and a ϕ-action. Recall that ξ = [p ] p. Consider [ ξ A 0 m ] crys = W (O C p ) W (O m! C m 1 p )[ 1 p ]. This is a G K -stable W (O C p )-subalgebra generated by divided-powers. Define A crys = lim A 0 n crys/p n A 0 crys with p-adic topology. We can fill in the top arrow in the commutative diagram A crys B + dr A 0 crys W (O C p )[ 1 p ] and then identify A crys with a subring of B + dr. More precisely, { ξ n } A crys = a n a n W (O n! C p ), a n 0 p-adically n=0 Define B + crys = A crys [ 1 p ] B+ dr. Since t A crys, we can define B crys = B + crys[ 1 t ] B+ dr [1 t ] B dr equipped with subspace topology from the new topology on B dr introduced in Problem 11. Moreover, B crys admits a natural action of G K. There is a G K - equivariant injection K K0 B crys B dr. Consequently, B G K crys = K 0.

10 10 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) The filtrations on B crys are the ones inherited from B dr. Namely, Fil i B crys = Fil i B dr B crys. Unlike B dr, the ϕ-action extends to A 0 crys, and hence on A crys, B crys, + B crys. (One can verify that ϕ(t) = pt.) However, the filtrations are not ϕ-stable. Theorem 5.1. ϕ : A crys A crys is injective. Theorem 5.2. We have fundamental exact sequences 0 Q p B ϕ=1 crys B dr /B + dr 0 0 Q p Fil 0 B crys ϕ 1 B crys 0 The proof of these two fundamental results are difficult. Problem 22. (i) Check that t A crys and t p 1 pa crys. Consequently, tp p! A crys. (ii) Show that for any a ker(a crys O Cp ), we have am m! A crys, m 1. Problem 23. Consider the G K -equivariant injection K K0 B crys B dr. Give left hand side the subspace filtration. Show that the induced map on the graded algebras is an isomorphism. Problem 24. Check that A 0 crys is ϕ-stable. (Hint: Calculate ϕ(ξ) and ϕ(ξ n ).) Problem 25. (i) Show that B + crys Fil 0 B crys. (ii) In this exercise, we show B + crys Fil 0 B crys. Consider α = [ε1/p ] 1 [ε 1/p2 ] 1. 1 Show that α B crys, α B crys B + dr, but 1 1 α Fil0 B crys B crys.) + (Hint: To show 1 α B crys, write 1 α = ϕ(α) [ε1/p2 ] 1 [ε] 1 some a A crys.) ϕ(α) B+ dr. (This implies and show that t = ([ε] 1)a for Problem 26. Show that ϕ on B crys does not preserve filtrations. (Hint: Consider ϕ(ξ).) Problem 27. As pointed out in [Col], the topology on B crys is unpleasant. In particular, the subspace topology inherited from B crys is different from the one on

11 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 11 B + crys. Let ω = ([ε] 1)/([ε 1/p ] 1) and consider ωpn x n = (p n 1)! (i) Show that (x n ) n 0 does not converge to 0 in B + crys. (ii) Show that (ωx n ) n 0 does converge to 0 in B + crys and hence (x n ) n 0 converges to 0 in B crys. Problem 28. A remedy to the topology issue in Problem 27 is to introduce B max. Define { ω n } A max = a n an p n W (O C p ), a n 0 p-adically n=0 and let B max + = A max [ 1 p ] B+ dr, B max = B max[ + 1 t ] B dr. Similar to B crys, the ring B max is equipped with G K -action, ϕ-action, and filtration. (i) Show that B max does not have the issue in Problem 27. (ii) Show that B G K max = K 0. (iii) Show that A ϕ=1 max = Z p. Hence (B max) + ϕ=1 = Q p. (iv) Show that ϕ(b max ) B crys B max. (v) (Hard!) Prove the analogue of Theorem 5.2: The following sequences are exact 0 Q p Bmax ϕ=1 B dr /B + dr 0 0 Q p Fil 0 B max ϕ 1 B max 0 (Hint: To handle the first sequence, first prove the exactness of the following sequence for all i 0: 0 Q p t i (B max ) ϕ=pi B dr /t i B + dr 0. For the second sequence, you need the (nontrvial) fact that ϕ 1 : B max B max is surjective.) 6. Crystalline representations Let MF ϕ K denote the category of triples (D, ϕ D, Fil ) where D is a K 0 -vector space ϕ D : D D is a bijective ϕ-semilinear endomorphism Fil is a filtration on D K = D K0 K such that (D K, Fil ) is an object in Fil K. Objects in MF ϕ K are called filtered ϕ-modules. Consider functor D crys : Rep Qp (G K ) MF ϕ K V ( ) GK V Qp B crys It is always true that dim K0 D crys (V ) dim Qp V. We say V is crystalline if dim K0 D crys (V ) = dim Qp V. The category of crystalline representations is denoted by Rep crys Q p (G K ).

12 12 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) Theorem 6.1. (i) The functor D crys : Rep crys Q p (G K ) MF ϕ K is exact and fully faithful. Moreover, it preserves direct sums, tensor products, subobjects, quotients, and duals. (ii) If V is crystalline, the natural map α crys,v : D crys (V ) K0 B crys V Qp B crys is an isomorphism of filtered ϕ-modules. (iii) If V is crystalline, we can recover V from D crys (V ) by V = Fil 0 (D crys (V ) K0 B crys ) ϕ=1. Theorem 6.2. Crystalline representations are de Rham. Source of crystalline representations: those G K -representations arising from p- adic étale cohomologies of proper smooth varieties over K with good reduction are crystalline. Problem 29. Describe D crys (Q p (n)) explicitly. Problem 30. Let η : G K Q p be a continuous character. Show that Q p (η) is crystalline if and only if there exists n Z such that χ n η is an unramified character. (Hint: Same strategy as Problem 18. Pick b B crys such that 1 b D crys = (Q p (η) B crys ) G K. Show that t n b K un 0 for some n.) Problem 31. Let D be a finite dimensional K 0 -vector space and let ϕ D : D D be an injective ϕ-semilinear morphism. Prove that ϕ D is automatically bijective. Problem 32. Similar to D dr and D crys, we can consider D max : Rep Qp (G K ) MF ϕ K V (V Qp B max ) G K It is always true that dim K0 D max (V ) dim Qp V. We say V is B max -admissible if dim K0 D max (V ) = dim Qp V. Prove that V is B max -admissible if and only if it is crystalline. (Hint: Use Problem 28(iv).) Problem 33. Let V p (E q ) be the representation studied in Problem 19. (i) Is V p (E q ) crystalline? (ii) Describe D crys (V p (E q )) explicitly. Problem 34.(Hard!)

13 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 13 (i) Can you find an extension 0 Q p (1) V Q p 0 so that V is de Rham but not crystalline? (ii) Show that any extension 0 Q p (n) V Q p 0 for n 2 must be crystalline. (Hint: Such an extension is represented by an element c V H 1 (G K, Q p (n)). Then V is crystalline if and only if c ker(h 1 (G K, Q p (n)) H 1 (G K, Q p (n) Qp B crys )) The problems reduce to calculation of cohomologies.) 7. Period sheaves This section dedicates to our first attempt on relative period rings. We define B dr as a sheaf on the pro-étale site of adic spaces. Let X be a locally noetherian adic space over Spa(Q p, Z p ). The objects in the pro-étale site X proét are inverse systems lim i I U i X where U i Xét and the transition maps U j U i are finite étale surjective. The coverings are the topological ones. For details, the readers are referred to [Sch2]. One important property is that affinoid perfectoid objects in X proét form a basis for the pro-étale topology. Let B denote the collection of such objects. To define a sheaf, we only need to define a presheaf on B. To this end, we first define the period rings on affinoid perfectoids. Let (L, L + ) be a perfectoid field of characteristic 0. For any perfectoid affinoid (L, L + )-algebra (R, R + ), we define A inf (R, R + ) := W (R + ) B inf (R, R + ) := A inf (R, R + )[ 1 p ] B + dr (R, R+ ) := lim B inf (R, R + )/(ker θ) n n where θ : A inf (R, R + ) = W (R + ) R + is defined in the same way as in Section 3. Notice that ξ is a generator of θ (see Problem 35). We define B dr (R, R + ) = B + dr (R, R+ )[ 1 ξ ]. The filtration on B dr is given by Fil i B dr = ξ i B + dr, i Z. Back to the pro-étale site. We define a presheaf F Ainf (resp., F Binf, F B +, F BdR ) dr on B by sending U = Spa(R, R + ) to A inf (R, R + ) (resp., B inf (R, R + ), B + dr (R, R+ ), B dr (R, R + )). Finally, define A inf (resp., B inf, B + dr, B dr) to be the coresponding sheafifications. These period sheaves played a central role in proving a de Rham comparison for rigid analytic varieties [Sch2]. Crystalline analogues are studied in [BMS], [TT]. Following the same spirit, sheaf versions of Robba rings and (ϕ, Γ)-modules are studied in [KL1], [KL2].

14 14 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) Problem 35. For any perfectoid affinoid (L, L + )-algebra (R, R + ), show that the kernel of θ : A inf (R, R + ) R + is a principle ideal generated by some ξ A inf (L, L + ). (Hint: Construct ξ = [p ] + i=1 pi [x i ] for some x i L +.) Problem 36. (i) Show that the presheaf F Ainf on B satisfies sheaf properties. In particular, A inf (U) = A inf (R, R + ), and A inf = W (Ô+ Xproét). (ii) Show that H i (U, A inf ) is almost zero for all i > 0. (iii) Show that H i (U, B + dr ) = 0 for all i > 0. (Hint: One need to know the following facts about the tilted structure sheaf ÔX proét: (1) For any affinoid perfectoid U X proét, we have Ô+ X proét(u) = R +. (2) H i (U, Ô+ X proét) is almost zero for all i > 0. By induction on n, one can get descriptions of W (Ô+ X proét)/p n for all n, as well as their almost vanishing of cohomologies. Then take inverse limit (which commutes with cohomology in this situation by [Sch2, Lemma 3.18]) to conclude the statement about A inf. To deal with B + dr, prove the following sequence is almost exact: This amounts to show H 1 (U, B inf ) = 0. B + dr.) 0 B inf ξ i B inf B inf /(ker θ) i 0 Finally, notice that [p ] is invertible in Problem 37. (i) Construct period sheaves A 0 crys, A crys, B + crys, B crys, B + max, B max on X proét in the same way. Repeat Problem 36(i). (ii) Show that H i (U, A 0 crys) and H i (U, A crys ) are almost zero for all i > 0. (iii) Show that H i (U, B + crys) = 0 for all i > 0. References [BC] O. Brinon, B. Conrad, CMI summer school notes on p-adic Hodge theory [Ber] L. Berger, An introduction to the theory of p-adic representations. Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin (2004), [BMS] B. Bhatt, M. Morrow, P. Scholze, Integral p-adic Hodge theory. arxiv: [Col] P. Colmez, Théorie d Iwasawa des représentations de de Rham d un corps local. Annals of Mathematics, Vol. 148, No. 2 (1998), [FO] J.-M. Fontaine, Y. Ouyang, Theory of p-adic represenations. Lecture notes. [KL1] K. Kedlaya, R. Liu, Relative p-adic Hodge theory: foundations. Volume 371, Astérisque, Société Mathématique de France, 2015.

15 PERIOD RINGS AND PERIOD SHEAVES (WITH HINTS) 15 [KL2] K. Kedlaya, R. Liu, Relative p-adic Hodge theory II: imperfect period rings. arxiv: [Sch1] P. Scholze, Perfectoid spaces, Publ. Math. de l IHÉS, 116(1) (2012), [Sch2] P. Scholze, p-adic Hodge theory for rigid-analytic varieties, Forum of Mathematics, Pi, [Ser] J.-P. Serre, Local fields. Graduate Texts in Mathematics 67, Springer-Verlag, [TT] F. Tan, J. Tong, Crystalline comparison isomorphisms in p-adic Hodge theory: the absolutely unramified case. arxiv: