A review of the foundations of perfectoid spaces

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1 A review of the foundations of erfectoid saces (Notes for some talks in the Fargues Fontaine curve study grou at Harvard, Oct./Nov. 2017) Matthew Morrow Abstract We give a reasonably detailed overview of the various tilting corresondences for erfectoid rings, the almost urity theorem, almost vanishing theorems, etc. We largely follow Scholze s Perfectoid saces and Etale cohomology of diamonds, together with a tiny amount of Bhatt M. Scholze s Integral -adic Hodge theory in order to formulate some results in terms of integral erfectoid rings rather than erfectoid Tate rings. I ve also benefited a lot from Bhatt s Lecture notes for a class on erfectoid saces. Fix a rime number forever. 1 Integral erfectoid rings Let A be a comlete toological ring. We say that A is integral erfectoid if and only if there exists a non-zero-divisor (for simlicity otherwise condition (iii) should be modified, but for our goal of studying erfectoid Tate rings the non-zero-divisor case is sufficient) π A such that (a) the toology on A is the π-adic toology; (b) π A; (c) Φ : A/πA A/π A, a a It is convenient, though not standard, to call any such element π a erfectoid seudo-uniformiser (u). Lemma 1.1. Let A be integral erfectoid, and π A a erfectoid seudo-uniformiser. Then: (i) Every element of A/πA is a th -ower (n.b., this is not a characteristic ring). (ii) If an element a A[ 1 π ] satisfies a A, then a A. (iii) After multilying π by a unit it admits a comatible sequence of -ower roots π 1/, π 1/2, A. Proof. (i): Using the surjectivity of Φ, a simle induction lets us write any a A as an infinite sum a = i 0 a i πi for some a i A; but this is ( i 0 a iπ i ) mod πa. (ii): Let l 0 be the smallest integer such that π l a A. Assuming that l > 0, we get a contradiction by noting that π l a π l A π A, whence π l a πa by condition (c), and so π l 1 a A. (iii): Recall that if B is any -adically comlete ring, then the natural ma lim B x x B/B is an isomorhism of monoids; alying this to the rings A and A/πA, we lim x x 1

2 deduce that the natural ma lim A lim x x x x A/πA is an isomorhism. Since we have just shown that x x is surjective on A/π, it follows that there exists a comatible sequence a, a 1/, a 1/s, A such that a π mod πa; therefore a = uπ for some u 1+A A. Lemma 1.2. Let A be integral erfectoid, and ϖ A any element satisfying (a) and (b). Then ϖ is a erfectoid seudo-uniformiser. Proof. It follows from Lemma 1.1(i) that every element of A/A is a th -ower; hence every element of its quotient A/ϖ A is a th -ower. It remains to rove that Φ : A/ϖA A/ϖ A is injective. Let π A be a erfectoid seudo-unifomiser. The fact that π and ϖ define the same toology easily imlies that ϖ is a non-zero-divisor and that A[ 1 ϖ ] = A[ 1 π ]. If a A satisfies a ϖ A, then (a/ϖ) A[ 1 π ] satisfies (a/ϖ) A, and it then follows from Lemma 1.1(ii) that in fact a ϖa as desired. Lemma 1.3. Suose A is a comlete toological ring such that A = 0. Then A is integral erfectoid if and only if it is erfect and the toology is π-adic for some non-zero-divisor π A. Proof. Condition (b) is clearly vacuous, while (c) imlies by a trivial inductive argument and taking the limit. 1.1 The tilt of an integral erfectoid ring Definition 1.4. The tilt of an integral erfectoid ring A is A := lim ϕ A/A, equied with the inverse limit toology (A/A is of course given the quotient toology). Lemma 1.5. Let A be an integral erfectoid ring. Then: (i) the usual well-defined ma of monoids # : A A, (a 0, a 1,... ) lim i ã i i ã i A are arbitrary lifts of the elements a i A/A) is continuous; (where (ii) the usual isomorhisms of monoids lim A a a A = lim A/A lim A/πA are ϕ ϕ homeomorhisms, where π A is any element defining the toology which divides. (iii) A is also an integral erfectoid ring. Proof. Given (1,..., 1, a n+1, a n+2,... ) lim A/πA, any chosen lifts ã ϕ i satisfy ã i n i 1 mod πa for i > n, whence ã i i 1 mod π n A; taking the limit shows that the shar is 1 mod π n A. This roves that # : lim A/πA A is continuous, from which (i) and (ii) easily follow. ϕ (iii) Let π A be a erfectoid seudo-uniformiser admitting -ower roots; let π = (π, π 1/,..., ) A be the corresonding element of A. Note that if a A satisfies a πa then (aπ ( 1)/ ) π A, whence aπ ( 1)/ πa and so a π 1/ A; similarly, by induction, if a n πa then a π 1/n A. Therefore we have exact sequences 0 π 1 1/n A/πA A/πA π1/n A/πA ϕn A/πA 0 for each n 1, which are comatible over different n (wrt. ϕ acting on the left three terms, and id on the right-most term). Taking lim clearly kills the left-most term (since ϕ(ϕ 1 1/n ) πa) n and so we obtain a short exact sequence 0 A π A mod π A/πA 0. 2

3 Therefore π is a non-zero-divisor of A. Since A lim A/πA is a homeomorhism, a basis of oen neighbourhoods of 0 A is ϕ given by the kernels of (# mod π) ϕ n, for n 1; but these kernels are π n (since π is a non-zero-divisor, A is erfect, and we have roved it when n = 0), thereby showing that the toology on A is the π -adic toology. By the revious lemma, A is integral erfectoid. 1.2 Fontaine s ma θ Proosition 1.6 (Fontaine). Let A be an integral erfectoid ring. (i) There is a unique ring homomorhism which sends [a] to a # for each a A. θ : W (A ) A (ii) θ is surjective and its kernel is generated by a non-zero-divisor. (iii) An element ξ Ker θ is a generator if and only if its Witt vector exansion ξ = (ξ 0, ξ 1,... ) has the roerty that ξ 1 is a unit of A. (iv) W (A ) is Ker θ-adically comlete. Proof. Since any element α W (A ) may be written uniquely as α = i 0 [a i] i = (a 0, a 1, a 2,... ) where a i A, the content of (i) is the assertion that α i 0 a# i i is a ring homomorhism. In rincile this can be checked directly using the olynomial exressions for addition and multilication of Witt vectors; alternatively see Lemma 3.2 of Integral -adic Hodge theory (we do not reroduce the roof here). (ii): Since W (A) is -adically comlete and A is -adically searated, to rove surjectivity it is enough to show that θ is surjective mod. But the Frobenius is surjective mod A and A lim x x A/A, so any element of A is a shar mod ; this is clearly enough. Now we construct a ossible generator of the kernel; let π A be a erfectoid seudouniformiser admitting -ower roots. Since π A and θ is surjective, we may write = π θ( x) for some x W (A ), whence ξ := + [π ] x Ker θ. Since W (A ) is [π ]-adically comlete and A is θ([π ]) = π-torsion-free, one easily sees that θ : W (A )/ξ A is an isomorhism if and only if it becomes an isomorhism when we mod out by [π ]; but then it identifies with A /π A # mod π A/πA (note that ([π ], ξ) = ([π ], )), which we saw was an isomorhism in the roof of Lemma 1.5(iii). We must also show that ξ is a non-zero-divisor, so suose that α W (A ) satisfies ξf = 0. Since ξ divides m + [π ] m x m we deduce ( m + [π ] m x m )α = 0, and so m α [π ] m W (A ); writing α = (a 0, a 1,... ) in Witt coordinates and exanding, this shows that a m i π m+i 1m A for all i 0. But A is erfect so this means a i π i+1m, and then letting m shows a i = 0 for all i 0, i.e., α = 0. (Note: this argument actually showed that W (C) is ξ-torsion-free, where C is any erfect A algebra which is π -adicallly searated; this will be useful later.) (iii): First note that the Witt vector exansion of our element ξ looks like (ξ 0, ξ 1,... ) = + [π ] x = (0, 1, 0, 0,... ) + (π x 0, π 2 x 1,... ) = (π x 0, 1 + π 2 x 1,... ), 3

4 in articular ξ 1 is a unit. Now let ξ = (ξ 0, ξ 1,... ) Ker θ be another element (whence ξ # 0 A and so ξ 0 π A ), and write ξ = αξ = (a 0, a 1,... )(ξ 0, ξ 1,... ) = (a 0 ξ 0, a 1 ξ 0 + a 0 ξ 1,... ). Then ξ is also a generator if and only if α is a unit, which is equivalent to a 0 being a unit, which is equivalent to ξ 1 = a 1ξ 0 + a 0 ξ 1 being a unit (since ξ 0 π A, ξ 1 is a unit, and A is π -adically comlete). (iv): W (A ) is ([π ], )-adically comlete, since each W m (A ) = W (A )/ m is [π ]-adically comlete; therefore it is a fortiori ξ-adicially comlete. 1.3 Tilting corresondence for integral erfectoid rings We begin with some remarks on toologies. Given a toological ring R, we always equi W r (R) with the roduct toology having identified it with R r (if the toology on R is π-adic, then the toology on W r (R) is the [π]-adic toology), and then W (R) = lim W r r (R) with the inverse limit toology (if R is erfect then this is the ([π], )-adic toology). Note that if A is integral erfectoid then, under these toologies, the ma θ : W (A ) A is continuous and oen, since (θ([π ]), ) = πa, i.e., W (A )/ Ker θ A is an isomorhism of toological rings. Given an integral erfectoid ring A, when we seak of an integral erfectoid A-algebra B we mean that B is an A-algebra which is integral erfectoid and whose toology comes from A, i.e., is πb-adic for any π A defining the toology on A (equivalently, by Lemma 1.2, any u of A is also a u of B). Theorem 1.7 (Tilting corresondence integral erfectoid rings). Fix an integral erfectoid ring A. Then tilting induces an equivalence of categories integral erfectoid A-algerbras integral erfectoid A -algebras, B B, with inverse given by sending an integral erfectoid A -algebra C to C # := W (C) W (A ),θ A (toologised as a quotient of W (C) following the above conventions). This equivalence is comatible with almost isomorhisms (i.e., kernel and cokernel killed by all toologically nilotent elements of A, res. A ). Proof. Let π A be a erfectoid seudo-uniformiser admitting -ower-roots, and π the associated u of A ; also let ξ = + [π ] x W (A ) be the generator of Ker(θ : W (A ) A) constructed in the revious roosition. Suose first that B is an integral erfectoid A-algebra. Then the image of ξ in W (B ) certainly lands in the kernel of θ B (i.e., the θ-ma for the integral erfectoid B), and condition (iii) of the revious roosition immediately shows that it is a generator of this ideal (since A B sends units to units); i.e., the diagram W (B ) θ B B W (A ) θa A is a ushout and so B # = B. For the moment this is only an identification of A-algebras, but the next aragrah will shows that the toologies coincide (both have the π-adic toology), i.e., # inverts tilting. 4

5 Now let C be an integral erfectoid A -algebra. As exlained in the aragrah immediately above the theorem, Lemma 1.2 imlies that π is also a u of C, and thus the toology we have ut on C # is (θ([π ]), )-adic toology, i.e., the π-adic toology. We must show that C is an integral erfectoid ring. First we check that π is a non-zero-divisor of C # π, i.e., that W (C)/ξW (C) has no [π ]- torsion. As mentioned arenthetically in the roof of art (ii) of the revious roosition, the argument there shows that ξ is a non-zero-divisor of W (C); since [π ] is also a non-zero-divisor, it enough show that W (C)/[π ]W (C) has no non-zero ξ-torsion. But ξ mod [π ] and is a non-zero-divisor of W (C), so this is equivalent to π being a non-zero-divisor of W (C)/ = C, which is indeed the case. Next, noting that ([π ], ξ) = ([π ], ) and ([π ], ξ) = ([π ], ) in W (C), the ma Φ : C # /πc # C # /π C # identifies with Φ : C/π C C/π C, which is an isomorhism since C is erfectoid. Thus the toological ring C # satisfies all the conditions to be integral erfectoid, excet we do not know yet that it is π-adically comlete (it follows formally that it is derived π-adically comlete, so it would be enough to show it is π-adically searated, but I do not see an exlicit argument...). So let Ĉ# be its π-adic comletion, which is an integral erfectoid A-algebra. Then there are identifications Ĉ# /πĉ# = C # /πc # = C/π C (the second of which comes from the revious aragrah), and taking lim x x shows that Ĉ# = C as integral erfectoid A -algebras. Since we have already roved that # inverts tilting, we aly # to deduce that Ĉ # = C #, i.e., C # was already comlete, and hence it is an integral erfectoid A-algebra. This argument has therefore also shown that C # = C, i.e., tilting inverts #. 5