On the direct summand conjecture and its derived variant
aa r X i v : . [ m a t h . AG ] N ov ON THE DIRECT SUMMAND CONJECTURE AND ITS DERIVED VARIANT
BHARGAV BHATTA
BSTRACT . Andr´e recently gave a beautiful proof of Hochster’s direct summand conjecture in commutative algebra usingperfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma)and a construction of certain faithfully flat extensions of perfectoid algebras where “discriminants” acquire all p -power roots.In this paper, we explain a quicker proof of Hochster’s conjecture that circumvents the perfectoid Abhyankar lemma;instead, we prove and use a quantitative form of Scholze’s Hebbarkeitssatz (the Riemann extension theorem) for perfectoidspaces. The same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de Jong.
1. I
NTRODUCTION
The first goal of this paper is to give a simpler proof of the following recent result of Andr´e, settling the directsummand conjecture:
Theorem 1.1 (Andr´e) . Let i : A ֒ → B be a finite extension of noetherian rings. Assume that A is regular. Thenthe inclusion i is split as an A -module map. When A has characteristic , Theorem 1.1 is easy to prove using the trace map. When dim( A ) ≤ , one canprove Theorem 1.1 using the Auslander-Buchsbaum formula. Hochster conjectured the general case in 1969, andproved it when A has characteristic p in [Ho1]. The first general result in mixed characteristic was Heitmann’s [He],settling the case of dimension . More on the history of this conjecture and its centrality amongst the ‘homologicalconjectures’ in commutative algebra can be found in [Ho3]. The result above is proven by Andr´e [An2] using [An1].In this paper, we give a proof of Theorem 1.1 that avoids [An1] (and is independent of [An2] in terms of exposition).Our approach also adapts to yield the following derived variant, which was conjectured by Johan de Jong in the courseof the author’s thesis work [Bh1, Bh3] as a path towards understanding the direct summand conjecture: Theorem 1.2.
Let A be a regular noetherian ring. Let f : X → Spec( A ) be a proper surjective map. Then themap A → R Γ( X , O X ) splits in the derived category D ( A ) . When A has characteristic , this result is due to Kov´acs [Ko], and is deduced from the fact that Spec( A ) hasrational singularities; see also [Bh1, Theorem 2.12]. The characteristic p case follows from [Bh1, Theorem 1.4 &Example 2.3]. To the best of our knowledge, in mixed characteristic, Theorem 1.2 is new even when dim( A ) = 2 . Remark 1.3.
Again, Theorem 1.1 was proven by Andr´e. Although it is explained in more detail and with more contextin the body of the paper, the main contributions of this paper (as we see it) are:(1) To clearly explain why Andr´e’s relatively simple flatness lemma from [An2, § linear statement about a fixed and explicit system of modules over a perfectoid ring, see Theo-rem 4.2 and its proof); in contrast, the approach in [An2] relies on the perfectoid Abhyankar lemma from[An1] (which is a deep non-linear assertion describing an entire class of algebras over a perfectoid ring).(2) To use these techniques to establish derived version of the direct summand conjecture (i.e., Theorem 1.2).Note that Theorem 1.2 has a range of geometric implications that are inaccessible from Theorem 1.1. Forexample, it implies that passage to alterations of a regular affine scheme can often be lossless for coherentcohomological purposes. In fact, even the special case of Theorem 1.2 where f is birational is a slightlynontrivial assertion about blowups (related to the work in [CR]), and completely orthogonal to Theorem 1.1. ssumption 1.4. In the rest of the introduction, primarily for notational ease, we assume that A := \ W [ x , .., x d ] is the p -adic completion of a polynomial ring over an unramified dvr W of mixed characterisitic (0 , p ) ; there is astandard reduction of Theorem 1.1 to mild variants of such an A , so not much generality is lost.1.1. The strategy of Andr´e’s proof.
Andr´e’s proof of Theorem 1.1 uses perfectoid spaces [Sc1]. To see why thisis natural, we first informally outline the main idea, adapted from [Bh2], in the special case where A [ p ] → B [ p ] is ´etale. The crucial input is Faltings’ almost purity theorem [Fa4] , which asserts: if A ∞ , is the p -adic completionof A [ x p ∞ i , p p ∞ ] , then the p -adic completion of the integral closure B ∞ of B in B ⊗ A A ∞ , [ p ] is almost finite´etale over A ∞ , with respect to the ideal ( p p ∞ ) ⊂ A ∞ , . Concretely, the algebraic obstructions to A ∞ , → B ∞ being finite ´etale — such as the cokernel of the trace map B ∞ → A ∞ , or the Ext -class measuring the failure of A ∞ , → B ∞ to split — are killed by p pk for all k ≥ , and are thus quite ‘small’. The faithful flatness of A → A ∞ , and the noetherianness of A then let one conclude that A → B must actually split. Summarizing, the key ideas are:(1) The construction of the faithfully flat extension A → A ∞ , .(2) The almost splitting after base change to A ∞ , coming from the almost purity theorem.It is now easy to see why perfectoid spaces provide a natural conceptual home for this proof: the ring A ∞ , in (1) is(integral) perfectoid , and the almost purity theorem invoked in (2) is a general fact valid for finite extensions of anyperfectoid algebra that are ´etale after inverting p (due to Kedlaya-Liu [KL] and Scholze [Sc1]).Andr´e’s proof of Theorem 1.1 follows a similar outline to the one sketched above. The first major difference is that A ∞ , is replaced by a larger perfectoid extension A ∞ of A ∞ , coming from the following remarkable construction: Theorem 1.5 (Andr´e) . Fix g ∈ A . Then there exists a map A ∞ , → A ∞ of integral perfectoid algebras that isalmost faithfully flat modulo p such that the element g ∈ A admits a compatible system of p -power roots g pk in A ∞ . Andr´e’s proof of Theorem 1.5 relies crucially on perfectoid geometry, and is explained in §
2. For the application toTheorem 1.1, one chooses g ∈ A to be a discriminant, i.e., an element g such that A [ g ] → B [ g ] is finite ´etale. Theflatness assertions in Theorem 1.5 then reduce us almost splitting the base change A ∞ → B ⊗ A A ∞ .To construct an almost splitting over A ∞ , Andr´e proves a much stronger result, which forms the subject of [An1]:he generalizes the almost purity theorem to describe extensions of A ∞ that are ´etale after inverting g (almost puritycorresponds to g = p ). The output is roughly that the integral closure B ∞ of B ⊗ A A ∞ in B ⊗ A A ∞ [ g ] is almostfinite ´etale over A ∞ , where ‘almost mathematics’ is measured with respect to (( pg ) p ∞ ) ⊂ A ∞ ; the precise statementis more subtle, and we do not formulate it here as we do not need it.1.2. The strategy of our proof.
Our proof uses Theorem 1.5. Thus, the task is to (almost) split A → B after basechange to the ring A ∞ arising from Theorem 1.5. For this, we again use perfectoid geometry. More precisely, for each n ≥ , the general theory gives us the perfectoid ring A ∞ h p n g i of bounded functions on the rational subset U n := { x ∈ X | | p n | ≤ | g ( x ) |} of the perfectoid space X associated to A ∞ . These rings naturally form a projective system as n varies (since U n ⊂ U n +1 ), and can be almost described very explicitly: to get A ∞ h p n g i , one formally adjoins p n g and its p -power roots to A . Their main utility to us is that g divides p n in A ∞ h p n g i , so A → B becomes finite ´etale after base change to Faltings’ theory of almost mathematics, in one incarnation, describes commutative algebra over a ring V equipped with a nonzerodivisor f ∈ V that admits arbitrary p -power roots. More precisely, one works in the quotient of the abelian category of V -modules by the Serre categoryof all ( f p ∞ ) -torsion modules. The study of such ‘almost modules’ was inspired by Tate’s [Ta], and led Faltings to prove fundamental results in p -adic Hodge theory [Fa1, Fa2, Fa3, Fa4]. Following Faltings’ work, a systematic investigation of almost mathematics was carried by Gabber andRamero [GR, GR2]. The relevance of almost mathematics to the direct summand conjecture seems to be first suggested by Roberts [Ro], [GR, § A precise definition is given in § R being perfectoid is that the Frobenius R/p → R/p is surjective, andhas a large but controlled kernel. Under suitable completeness and torsionfreeness hypotheses, this leads to the existence of lots of elements thatadmit arbitrary p -power roots, such as the elements p and x i in R = A ∞ , . The idea of using this tower of rings to study A ∞ also comes from [An1]. ∞ h p n g i [ p ] for any n ≥ ; the almost purity theorem then kicks in to show that for each n ≥ , the base change of A → B to the perfectoid algebra A ∞ h p n g i is almost split. To descend the splitting to A ∞ , we prove the followingquantitative form of Scholze’s Riemann extension theorem [Sc2, Proposition II.3.2]. Theorem 1.6.
Fix an integer m ≥ . The natural map of pro-systems { A ∞ /p m } n ≥ → { A ∞ h p n g i /p m } n ≥ (1) is an almost-pro-isomorphism with respect to ( pg ) p ∞ , i.e., for each k ≥ , the pro-system of kernels and cokernels ispro-isomorphic to a pro-system of ( pg ) pk -torsion modules. Remark 1.7.
On taking limits over n and m in Theorem 1.6, one obtains an almost isomorphism A ∞ a ≃ lim A ∞ h p n g i ,i.e., the following statement from [Sc2, Proposition II.3.2]: any bounded function on the Zariski open set { x ∈ X | g ( x ) = 0 } = ∪ n U n ⊂ X almost extends to X . In other words, this gives a perfectoid analog of the Riemannextension theorem in complex geometry. A similar result in rigid geometry was proven by Bartenwerfer [Ba]. Remark 1.8.
Theorem 1.6 roughly says that the limiting isomorphism A ∞ a ≃ lim A ∞ h p n g i from Remark 1.7 holds truefor ‘diagrammatic’ reasons. Consequently, it remains true after applying A ∞ -linear functors, such as Ext iA ∞ ( N, − ) for any A ∞ -module N , to both sides of (1) and then taking limits. The case i = 0 recovers Scholze’s theorem, thecase i = 1 is essential to Theorem 1.1, and Theorem 1.2 relies on the statement for all i ≥ .Using Theorem 1.6, the proof of Theorem 1.1 proceeds along the lines sketched above, and can thus be summarizedas follows: pass from A to A ∞ using Theorem 1.5 to ensure this passage is lossless, pass from A ∞ to A ∞ h p n g i topush all the ramificiation into characteristic p , construct an almost splitting over A ∞ h p n g i using almost purity, andfinally take a limit over n to get an almost splitting over A ∞ thanks to Theorem 1.6. In particular, it is exactly the laststep (relying on the relatively simple module-theoretic statement in Theorem 1.6) where our approach to Theorem 1.1diverges from that of [An2] (which relies on the sophisticated perfectoid Abhyankar lemma [An1]).To prove Theorem 1.2, we proceed analogously. First, assume that f ramifies only in characteristic p , i.e., f [ p ] isfinite ´etale. Again, it suffices to construct the splitting after going up to a faithfully flat integral perfectoid extensionof A (such as the ring A ∞ , above). After such a base change, the almost purity theorem and a general vanishingtheorem of Scholze settle the question. In general, one first reduces to f being generically finite, and finite ´etale afterinverting some g ∈ A . This case is then deduced from preceding special case using Theorem 1.5 and Theorem 1.6,exactly as was explained above for Theorem 1.1.1.3. Layout.
We begin in § §
4; this depends on the notion of almost mathematics of pro-systems, which is briefly developedin §
3. With these ingredients in place, Theorems 1.1 and 1.2 are proven in § § Notation.
We freely use the language of perfectoid spaces and almost mathematics. Occasionally, we use almostmathematics with respect to different ideals in the same ring; thus we always specify the relevant ideal, sometimes atthe beginning of each section. The letter K denotes a perfectoid field , and K ◦ ⊂ K is the ring of integers. We fix anelement t ∈ K ◦ which admits arbitrary p -power roots t pk , and such that | t | = | p | if K has characteristic . A K ◦ -algebra A is called integral perfectoid if it is flat, t -adically complete, satisfies A = A ∗ , and satisfies the following:Frobenius induces an isomorphism A/t p ≃ A/t . The category of such algebras is equivalent to usual category ofperfectoid K -algebras by [Sc1, Theorem 5.2]; the functors are A A [ t ] and R R ◦ respectively. At first pass, not much is lost if one simply sets K := \ Q p ( p p ∞ ) in characteristic (with t = p ), and K := \ F p (( t p ∞ )) in characteristic p . Concretely, the assumption A = A ∗ means that if f ∈ A [ t ] is such that t pk · f ∈ A for all k ≥ , then f ∈ A . This condition is not reallyserious and can often be ignored: if A is a t -adically complete and flat K ◦ -algebra with Frobenius inducing an almost isomorphism A/t p ≃ A/t ,then A ′ := A ∗ := Hom K ◦ ( t p ∞ , A ) is an integral perfectoid K ◦ -algebra in the sense introduced above by [Sc1, Lemma 5.6], and A → A ∗ isan almost isomorphism. cknowledgements. This paper is obviously inspired by Andr´e’s preprints [An1, An2]; I thank heartily him forsharing them, and for his kind words about this paper. I am also very grateful to Peter Scholze for patiently explainingbasic facts about perfectoid spaces, for discussions surrounding Theorem 1.5, and for convincing me that ‘almost-pro-isomorphism’ is a better name than ‘pro-almost-isomorphism’ in §
3. I am equally indebted to my former PhDadvisor Johan de Jong for talking through the contents of this manuscript, and his very prescient suggestion (slightlyover seven years ago!) that I pursue the mathematics surrounding Theorem 1.2. Finally, I thank Brian Conrad, RayHeitmann, Mel Hochster, Kiran Kedlaya, Linquan Ma, Peter Scholze, Karl Schwede and an anonymous referee formany useful comments on the first version of this paper. I was partially supported by NSF Grant DMS
DJOINING ROOTS OF THE DISCRIMINANT
Notation 2.1.
Let A be an integral perfectoid K ◦ algebra. Fix g ∈ A . Set X := Spa( A [ t ] , A ) and Y :=Spa( A h T p ∞ i [ t ] , A h T p ∞ i ) ; these are perfectoid spaces. All occurrences of almost mathematics in this section arewith respect to t p ∞ .The main goal of this section is to construct an almost faithfully flat extension A → A ∞ of perfectoid algebrassuch that g acquires arbitrary p -power roots in A ∞ . For this, we essentially set T = g in A h T p ∞ i . More precisely, toget a perfectoid algebra, we approximate bounded functions on the Zariski closed space Z := V ( T − g ) := { y ∈ Y | T ( y ) = g ( y ) } ⊂ Y using bounded functions on rational open neighbourhoods Y h T − gt ℓ i := { y ∈ Y | | T ( y ) − g ( y ) | ≤ | t ℓ |} ⊂ Y of Z for varying integers ℓ . Definition 2.2.
Set A ∞ to be the integral perfectoid ring of functions on the Zariski closed subset of Y defined bythe ideal ( T − g ) , in the sense of [Sc2, § II.2]. Explicitly, we have A ∞ = [ colim ℓ ∈ N B ℓ where B ℓ := O + Y ( Y h T − gt ℓ i ) , and the completion appearing on the left is t -adic.Note that T = g in A ∞ as ( T − g ) is divisible by t ℓ in B ℓ , and thus in A ∞ , for all ℓ . Thus, g has a distinguishedsystem of p -power roots g pk := T pk in A ∞ . The main theorem is (see [An2, § Theorem 2.3 (Andr´e) . For each ℓ > , the map A → B ℓ is almost faithfully flat modulo t . Consequently, the map A → A ∞ is almost faithfully flat modulo t .Proof. It is enough to show the first statement modulo t ǫ for some ǫ > . The approximation lemma for perfectoidspaces (as in [KL, Corollary 3.6.7] or [Sc1, Corollary 6.7]) gives an f ∈ (cid:0) A h T p ∞ i (cid:1) ♭ such that(1) f ♯ ≡ T − g mod t p .(2) We have an equality Y h T − gt ℓ i = Y h f ♯ t ℓ i of subsets of Y .The explicit description of O + Y ( Y h f ♯ t ℓ i ) from [Sc1, Lemma 6.4] identifies B ℓ (almost) with the t -adic completion of colim k (cid:16) A h T p ∞ i [ u pk ] / (cid:0) ( u · t ℓ ) pk − ( f ♯ ) pk (cid:1)(cid:17) = A h T p ∞ i [ u p ∞ ] / (cid:0) ∀ k : ( u · t ℓ ) pk − ( f ♯ ) pk (cid:1) . (2)Thus, it is enough to show that the A -algebra C ℓ,k := A h T p ∞ i [ u p ∞ ] / (cid:0) ( u · t ℓ ) pk − ( f ♯ ) pk (cid:1) The ring A ∞ defined here might not be integral perfectoid, but is almost isomorphic to one (by passing to ( A ∞ ) ∗ ), so we ignore the distinction. s faithfully flat over A after reduction modulo t ǫ for some ǫ . We take ǫ = p k +1 . For this choice, we have t ℓpk ≡ t ǫ , so the relation above simplifies to ( f ♯ ) pk = 0 modulo t ǫ . As f ♯ ≡ T − g mod t p , the k -fold Frobeniusidentifies the A/t ǫ -algebra C ℓ,k /t ǫ with the A/t p -algebra A [ T p ∞ , u p ∞ ] / ( t p , T − g ) . The latter is faithfully flat (evenfree) over A/t p , so the claim follows. (cid:3) Remark 2.4.
Theorem 2.3 is proven in [An2] under a more restrictive setup (but with a stronger conclusion). I amgrateful to Scholze for pointing out that the same proof goes through in the above generality.
Remark 2.5.
One might worry that the presentations from [Sc1, Lemma 6.4] used above are only valid in the non-derived sense, and thus do not play well with reduction modulo t or t -adic completion. More precisely, one may ask if(2) is also true if one imposes the corresponding relations in the derived sense (i.e., one works with the correspondingKoszul complexes). While answering this question is not necessary for our purposes, the answer is indeed ‘yes’, andwe record it here for psychological comfort, especially since such presentations are also important later. Lemma 2.6.
Let A be an integral perfectoid K ◦ -algebra. Choose f , ..., f n , g ∈ A ♭ , and set B to be the direct limitof the Koszul complexes Kos( A [ T p ∞ i ]; ( g ♯ · T i ) pm − ( f ♯i ) pm ) . Then the Koszul complex Kos( B ; t ) is almost discrete.Thus, the derived t -adic completion of B is almost isomorphic to the perfectoid algebra A h f g , ..., f n g i .Proof. Note that A [ T p ∞ i ] has no t -torsion. Thus, the complex Kos( B ; t ) is identified with M := colim m (cid:16) Kos(
A/t [ T p ∞ i ]; ( g ♯ · T i ) pm − ( f ♯i ) pm ) (cid:17) since, at level m , freely imposing the relations t = 0 and ( g ♯ · T i ) pm − ( f ♯i ) pm = 0 in the derived sense on the ring A [ T p ∞ i ] can be done in any order. But now M looks the same for both A and A ♭ , so we may assume that A hascharacteristic p (and so f i = f ♯i , g = g ♯ ). In this case, M identifies with Kos( R ; t ) , where R := colim m (cid:16) Kos( A [ T p ∞ i ]; ( g · T i ) pm − ( f i ) pm ) (cid:17) . But R is discrete: it is the perfection of the derived ring Kos( A [ T p ∞ i ]; g · T i − f ) , which is always discrete by [BS,Lemma 3.16 or Proposition 5.6]. As M ≃ Kos( R ; t ) , we are reduced to showing that the t -torsion of R is almost zero.But this follows from perfectness: if α ∈ R and t · α = 0 , then t · α p n = 0 for all n ≥ , which, by perfectness, gives t pn · α = 0 for all n ≥ , so α is almost zero. (cid:3) In particular, all operations in the proof of Theorem 2.3 can be interpreted in the derived sense.
Remark 2.7.
One may upgrade the above techniques to show the following (see [Bh4, Corollary 9.4.7]): for anyintegral perfectoid K ◦ -algebra A , there exists a functorial map A → B ( A ) of integral perfectoid K ◦ -algebras thatis almost faithfully flat modulo t such that B ( A ) is absolutely integrally closed, i.e., each monic polynomial has asolution. In particular, any b ∈ B ( A ) admits a compatible system { b pn } n ≥ of p -power roots.3. A LMOST - PRO - ZERO MODULES
We introduce the relevant notion of almost mathematics in the pro-category necessary for Theorem 1.6.
Notation 3.1.
Let A be a ring equipped with a nonzerodivisor t together with a specified collection { t pk } of compat-ible p -power roots. All occurrences of almost mathematics in this section are with respect to t p ∞ .There is an intrinsic notion of almost mathematics of pro- A -modules: one might simply work with pro-objects inthe almost category. For example, a projective system { M n } n ≥ of A -modules is ‘almost-zero’ as a pro-object if forany n ≥ , there exists some m = m ( n ) ≥ n such that the map M m → M n has image annihilated by t pk for all k .This intrinsic notion is too strong for our purposes, and we use the following weakening, where m depends on k : efinition 3.2. A pro- A -module { M n } n ≥ is said to be almost-pro-zero if for any k ≥ and any n ≥ , thereexists some m = m ( n, k ) ≥ n such that im( M m → M n ) is killed by t pk ; equivalently, for each k ≥ , the map { M n [ t pk ] } n ≥ → { M n } n ≥ is a pro-isomorphism in the usual sense. A map of pro-objects in D b ( A ) is said to be an almost-pro-isomorphism if the cohomology groups of cones form an almost-pro-zero system.We begin with an example illustrating the novel features of this notion: Example 3.3.
Consider the system { M n } where M n = A/ ( t pn ) , and M n +1 → M n is the injective map definedby t pn − pn +1 . Then { M n } is almost-pro-zero (in fact, each M m is killed by t pk for m ≥ k ), even though thecorresponding pro-object of the almost category is not zero.The next few lemmas record the stability properties of this notion: Lemma 3.4. If { M n } n ≥ is an almost-pro-zero pro- A -module, then the complex R lim( { M n } n ≥ ) is almost zero,i.e., it has almost zero cohomology groups.Proof. Fix k ≥ . Then the inclusion { M n [ t pk ] } n ≥ → { M n } n ≥ is a pro-isomorphism, so both sides have the same R lim . In particular, the cohomology groups of R lim( { M n } n ≥ ) are killed by t pk . (cid:3) Lemma 3.5. If { N n } n ≥ → { M n } n ≥ is an almost-pro-isomorphism in D b ( A ) , then R lim( { N n } n ≥ ) → R lim( { M n } n ≥ ) is an almost isomorphism.Proof. This follows by applying Lemma 3.4 to the cone. (cid:3)
Lemma 3.6. If { M n } n ≥ is an almost-pro-zero pro- A -module, and F : Mod A → Mod A is an A -linear functor, then { F ( M n ) } n ≥ is also almost-pro-zero.Proof. Fix k ≥ , n ≥ . Then M m → M n factors over M n [ t pk ] ⊂ M n for some m ≥ n . But then F ( M m ) → F ( M n ) factors over F ( M n [ t pk ]) → F ( M n ) , and hence over F ( M n )[ t pk ] ֒ → F ( M n ) , by the A -linearity of F . (cid:3)
4. A
QUANTITATIVE FORM OF THE R IEMANN EXTENSION THEOREM
Notation 4.1.
Let A be an integral perfectoid K ◦ -algebra with associated perfectoid space X := Spa( A [ t ] , A ) . Fixan element g ∈ A that admits a compatible system of p -power roots g pk . Assume that g is a nonzerodivisor modulo t m in the almost sense (with respect to t p ∞ ).In this section, we prove Theorem 1.6. Thus, we study the rings A h t n g i := O + X ( X h t n g i ) and their variation with n .More precisely, we show the following quantitative form of Scholze’s Hebbarkeitssatz [Sc2, Proposition II.3.2]: Theorem 4.2.
For each m ≥ , consider the natural projective system of maps { f n : A/t m → A h t n g i /t m } n ≥ in almost mathematics with respect to t p ∞ . Then we have:(1) Each ker( f n ) is almost zero.(2) The pro-system { coker( f n ) } n ≥ is uniformly almost-pro-zero with respect to g p ∞ , i.e., for any k ≥ , thereexists some c ≥ such that for any n ≥ , the image of the c -fold transition map coker( f n + c ) → coker( f n ) is killed by g pk . (In fact, c = p k m works.)In particular, the projective system { f n } n ≥ is an almost-pro-isomorphism with respect to ( tg ) p ∞ . This assumption is not actually necessary, and can be dropped a posteriori ; see Remark 4.4. It is also harmless in applications. roof. Fix some integer m ≥ . We use the explicit presentations for A h t n g i /t m coming from the perfectoid theory.By [Sc1, Lemma 6.4], there is almost isomorphism (with respect to t p ∞ ) M n := A [ u p ∞ n ] / (cid:0) t m , ∀ k : ( u n · g ) pk − t npk (cid:1) a ≃ A h t n g i /t m . defined by viewing u pk n as the function ( t n g ) pk . It is thus enough show the assertions in the theorem for the pro-system { f n : A/t m → M n } n ≥ of obvious maps. As g is a nonzerodivisor modulo t m , the same holds true for g pk . It is then easy see that each f n isinjective, so the kernels are on the nose. For the cokernels, fix some k ≥ . We shall show that any c ≥ p k · m works,i.e., for such c , the element g pk · u en + c ∈ M n + c maps into A/t m ⊂ M n under the c -fold transition map M n + c → M n for all exponents e ∈ N [ p ] . By construction, the transition map carries g pk · u en + c to g pk · t ce · u en ∈ M n , so we must show this last expression lies in A/t m ⊂ M n for c ≥ p k m and all e ∈ N [ p ] . There are two cases: • If e ≥ p k , then t m | t ce as c ≥ p k m , so the above expression is zero as we work modulo t m . • If e < p k , then the above expression can be written as g pk · t ce · u en = g pk − e · g e · t ce · u en = g pk − e · t ce · ( g · u n ) e = g pk − e · t ce · t ne = g pk − e · t ( n + c ) e ∈ A/t m ⊂ M n , as wanted. (cid:3) Remark 4.3.
Theorem 4.2 shows that the map { A/t m } n ≥ → { A h t n g i /t m } n ≥ is a uniform almost-pro-isomorphismwith respect to ( tg ) p ∞ , i.e., the constant c appearing in the theorem is independent of n . It formally follows that forany A/t m -complex K , the kernel and cokernel pro-systems of the induced { H i ( K ) } n ≥ → { H i ( K ⊗ LA/t m A h t n g i /t m ) } n ≥ are both uniformly almost-pro-zero in the preceding sense and with the same implicit constants. In particular, whenapplied to Koszul complexes arising from regular sequences of elements in A/t m , we learn that the homology of thecorresponding pro-system of Koszul complexes on { A h t n g i /t m } is uniformly almost-pro-zero in nonzero degrees. Remark 4.4.
The assumption that g is a nonzerodivisor modulo t m in Notation 4.1 can be dropped without affectingthe conclusion of the final statement of Theorem 4.2. Indeed, consider first the universal case R := K ◦ h T p ∞ i with g = T . This falls under the case that is already treated, so we have an almost-pro-isomorphism { R/t m } n ≥ → { R h t n T i /t m } n ≥ with respect to ( tT ) p ∞ . For general A and g , there is a unique map R → A carrying T pk to g pk for all k . By basechange, we have an almost-pro-isomorphism { A/t m } n ≥ → { R h t n T i ⊗ LR A/t m } n ≥ with respect to ( tg ) p ∞ . The explicit description of [Sc1, Lemma 6.4] shows that R h t n T i ⊗ R A/t m a ≃ A h t n g i /t m . In particular, applying H to the almost-pro-isomorphism above gives the desired statement. . T HE DIRECT SUMMAND CONJECTURE
In this section, we prove Theorem 1.1. We begin by collecting some preliminaries that shall be useful in the proof.The following proposition is borrowed from [BMS, Lemma 4.30 and Remark 4.31], and is presumably well-known:
Proposition 5.1.
Let A → B be a map of commutative rings with A is noetherian. Assume that there exists some π ∈ A such that both A and B are π -torsionfree and π -adically complete, and A/π → B/π is (faithfully) flat. Then A → B is (faithfully) flat.Proof. For flatness: we must check that M ⊗ LA B lies in D ≥ for any finitely generated A -module M . As A isnoetherian, we can choose a resolution P • → M with each P i being finite free. The complex M ⊗ LA B is thencomputed by P • ⊗ A B . As B is π -adically complete, we have P • ⊗ A B ≃ lim n P • ⊗ A B/π n at the level of complexes.The transition maps in the system on the right are termwise surjective, so we can write this more intrinsically as M ⊗ LA B ≃ R lim n ( M ⊗ LA B/π n ) ≃ R lim n (( M ⊗ LA A/π n ) ⊗ LA/π n B/π n ) . As M is finitely generated, the pro- A -complex { M ⊗ LA A/π n } is pro-isomorphic to { M/π n } : the obstruction is thepro-system { M [ π n ] } , which is pro-zero as the π ∞ -torsion of M is bounded by finite generation. Thus, we obtain M ⊗ LA B ≃ R lim n ( M/π n ⊗ LA/π n B/π n ) . As A/π → B/π is flat, the same holds true for
A/π n → B/π n as π is a nonzerodivisor on both A and B . Inparticular, the terms showing up inside the limit lie in D ≥ , so the same holds true for the limit, as wanted.For faithful flatness, we must check that Spec( B ) → Spec( A ) is surjective if A → B is flat and A/π → B/π isfaithfully flat. As the image is stable under generalizations by flatness, it suffices to check that all closed points liein the image; equivalently, we must show that A/ m ⊗ A B = 0 for any maximal ideal m in A . But π ∈ m as A is π -adically complete, so A/ m ⊗ A B ≃ A/ m ⊗ A/π
B/π , which is nonzero by faithful flatness of
A/π → B/π . (cid:3) Next, we explain why regular local rings admit faithfully flat covers by perfectoids.
Proposition 5.2.
Let A be a p -torsionfree noetherian regular local ring whose residue characteristic is p . Then thereexists a map A → A such that(1) The ring A admits the structure of an integral perfectoid K -algebra for K = \ Q p ( p p ∞ ) .(2) The map A → A is “almost faithfully flat” in the following sense: for any A -module M , we have(a) Tor A i ( M, A ) is almost zero for i > .(b) If M ⊗ A A is almost zero, then M = 0 . In the unramified case, we can also arrange for A → A to be faithfully flat. The proof below shows that it ispossible to achieve the same in general provided we make either one of the following modifications: (a) relax (1) aboveto only requiring either that A is an integral perfectoid ring in a generalized sense (i.e., one that does not necessarilycontain a perfectoid field, as elaborated in the proof below), or (b) only require A to be a p -adically complete and p -torsionfree K ◦ -algebra that is almost isomorphic to an integral perfectoid K ◦ -algebra. Related constructions occurin [Sh, Proposition 4.9] or [An1, Example 3.4.6 (3)]. Proof.
We are free to replace A by noetherian regular local rings that are faithfully flat over it. Thus, we may assumethat A is complete for the topology defined by powers of the maximal ideal, and has an algebraically closed residuefield k . Let W = W ( k ) be the Witt vectors of k , and write m ⊂ A for the maximal ideal. Write d = dim( A ) .Assume p / ∈ m (which is the so-called unramified case). Then p is part of a basis of m / m , and thus A isisomorphic to W J x , ..., x d K . In this case, we may simply take A to be the p -adic completion of A [ p p ∞ , x p ∞ i ] . Inthis case, the map A → A is faithfully flat by Proposition 5.1 (and thus also almost faithfully flat by the argumentgiven at the end of this proof for the ramified case). Note that this case suffices Theorem 1.1 by [Ho2, Theorem 6.1].Assume p ∈ m (which is the so-called ramified case). By choosing d generators for m , we obtain a surjection ψ : P := W J x , ..., x d K → A . Using the regularity of A and the assumption p ∈ m , it is easy to see that ker( ψ ) is enerated by an element of the form p − f where f = f ( x i ) ∈ ( p, x , ..., x d ) is a power series. Moreover, as A is p -torsionfree and p -adically complete, we may also conclude that p ∤ f and f has no constant term. Now write P m = P [ x pm i ] , and consider the ring A ′ obtained as the p -adic completion of (colim m P m ) ⊗ P A ≃ colim m P m / ( p − f ) . As P → P m is faithfully flat, it is easy to see that A → A ′ is also faithfully flat. Moreover, the element g = σ − ( f )( x p i ) ∈ P satisfies g p = f + ph for some h ∈ P ; here σ is the (unique) lift of the Frobeniusautomorphism of k to W , and σ − ( f ) is the power series obtained by applying σ − to the coefficients of f . As f and g have no constant terms, nor does h . In A ′ , this gives g p = p + ph = p (1 + h ) = pu for some unit u ∈ A ′ .In particular, the ring A ′ equipped with the p -adic topology is integral perfectoid in a generalized sense , i.e., thetopological ring A ′ [ p ] (topologized by making p n A ′ a neighbourhood basis of ) is a perfectoid Tate ring in the senseof [SW, Definition 6.1.1], and A ′ is a ring of integral elements in A ′ [ p ] . The proof of [SW, see Lemma 6.2.2] gives anelement π ∈ A ′ admitting a compatible system of p -power roots such that π p = pv for some unit v ∈ A ′ . Henceforth,almost mathematics over A ′ is measured with respect to ( π p ∞ ) ; this also coincides with the ideal p ( p ) , and is thusindependent of the choice of π or its roots.Now the theory of perfectoid spaces extends to the generalized setting, see [SW, § § § A ′ [ p ] → A ′ [ p , v p ∞ ] obtained by formally extracting p -power roots of v from A ′ [ p ] . By the almost purity theorem [Sc1, Theorem 7.9 (iii)], the π -adic completion A of the integral closure of A ′ inthis extension of A ′ [ p ] is the π -adic completion of an ind-(almost finite ´etale) extension of A ′ . In particular, A is anintegral perfectoid ring in the generalized sense, there is no π -torsion in A , and the map A ′ → A is almost faithfullyflat modulo π . By construction, the element p = π p v − ∈ A ′ admits a compatible system of p -power roots, so A ′ canalso be viewed as an integral perfectoid algebra over K ◦ for K = \ Q p ( p p ∞ ) in the sense used elsewhere in this article(at least after application of ( − ) ∗ , which is harmless for for our purposes). Note that ideals ( p p ∞ ) and ( π p ∞ ) in A are identical (and both coincide with p ( p ) ), so there is a natural notion of almost mathematics over A .It remains to check that the composite map A → A satisfies (2). This map factorizes as A a −→ A ′ b −→ A !! c −→ A where ( − ) !! is defined as in [GR, Definition 2.2.23]. By construction, the map a is faithfully flat and the map c is aninjective almost isomorphism. In particular, A !! is p -adically complete and p -torsionfree as A is so. As the formation of ( − ) !! commutes with reduction modulo p m (see [GR, Remark 2.2.28 (ii)]), it follows from the almost faithful flatnessmodulo p m of A ′ → A and [GR, Remark 3.1.3 (ii)] that b is faithfully flat modulo p m for any m ≥ . The composite b ◦ a is then faithfully flat by Proposition 5.1. As c is an almost isomorphism, this verifies (a) in (2).For (b) in (2), say M is an A -module with M ⊗ A A almost zero. As c is an almost isomorphism, this is equivalentto asking M ⊗ A A !! is almost zero. We want to show M = 0 . As A → A !! is faithfully flat, we may filter M toreduce to the case where M = A /I for some ideal I ⊂ A . The hypothesis M ⊗ A A !! being almost zero thentranslates to π pn ∈ IA !! for all n ≥ . But this implies p = π p v − ∈ I p n A !! for all n ≥ . By faithful flatness of A → A !! , we must have p ∈ I p n for all n ≥ . But Krull’s intersection theorem implies ∩ n I p n = 0 if I is non-trivial.As p = 0 on A , we must therefore have I = A , and thus M = 0 as wanted. (cid:3) Finally, we recall a slightly non-standard consequence of the Artin-Rees lemma.
Lemma 5.3.
Let R be a noetherian ring equipped with an ideal I . For any pair M, N of finitely generated R -modules,the pro- R -modules { Hom R ( M, N ) /I n } n ≥ and { Hom R ( M, N/I n N ) } n ≥ are pro-isomorphic via the natural map.In particular, lim Hom R ( M, N/I n N ) = 0 .Proof. We shall use the Artin-Rees lemma in the following form: the functor P
7→ {
P/I n } is an exact functor fromfinitely generated R -modules P to pro- R -modules. To apply this, pick a presentation F → F → M → with F i being finite free. Applying Hom R ( − , N ) gives an exact sequence → Hom R ( M, N ) → Hom R ( F , N ) → Hom R ( F , N ) . he previously mentioned form of the Artin-Rees lemma then yields an exact sequence of pro- R -modules of the form → { Hom R ( M, N ) /I n } n ≥ → { Hom R ( F , N ) /I n } n ≥ → { Hom R ( F , N ) /I n } n ≥ . Repeating this analysis using the functor
Hom R ( − , N/I n N ) instead gives an exact sequence of pro- R -modules → { Hom R ( M, N/I n N ) } n ≥ → { Hom R ( F , N/I n N ) } n ≥ → { Hom R ( F , N/I n N ) } n ≥ . Comparing the sequences yields the lemma as
Hom R ( F i , N/I n N ) ≃ Hom R ( F i , N ) /I n since F i is finite free. (cid:3) We can now prove the promised theorem.
Theorem 5.4.
Let A be a regular ring. The map A → B of A -modules is split.Proof. We may assume that A is a noetherian regular local ring of mixed characteristic (0 , p ) . Choose g ∈ A coprime to p such that A → B is finite ´etale after inverting pg . Let A → A ∞ , be the extension provided byProposition 5.2, and let A ∞ , → A ∞ be the extension resulting from applying Theorem 2.3 to the integral perfectoidring A ∞ , equipped with the element g .Consider the canonical exact triangle A → B → Q of A -modules. The boundary map α ∈ Hom A ( Q , A [1]) is the obstruction to this sequence being split. We wouldlike this show this obstruction vanishes. We change subscript to denote derived base change to either A ∞ , or A ∞ ;for example, Q ∞ , := Q ⊗ LA A ∞ , , α ∞ := α ⊗ LA A ∞ , etc.First, it suffices to show that α /p m ∈ Hom A ( Q , A /p m [1]) vanishes for all m ≫ . Indeed, we have Hom A ( Q , A [1]) ≃ lim m Hom A ( Q , A /p m [1]) , as A is p -adically complete and { Hom A ( Q , A /p m ) } m ≥ has vanishing lim by Lemma 5.3.Choose m ≥ such that α /p m = 0 , so Ann A /p m ( α /p m ) = A /p m ; if no such m exists, then α /p m = 0 forall m , so we are done. Otherwise, by Krull’s theorem, there exists k ≥ such that p g / ∈ (cid:0) Ann A /p m ( α /p m ) (cid:1) p k ;here we use that m ≥ and that = g ∈ A /p . As both A /p m → A ∞ , /p m and A ∞ , /p m → A ∞ /p m are almostfaithfully flat with respect to p p ∞ , we get pg / ∈ (cid:0) Ann A ∞ /p m ( α ∞ /p m ) (cid:1) p k , so ( pg ) pk / ∈ Ann A ∞ /p m ( α ∞ /p m ) ;here we lose a power of p in passing to almost mathematics. It is thus enough (via contradiction) to show that α ∞ /p m ∈ Hom A ∞ ( Q ∞ , A ∞ /p m [1]) is almost zero with respect to ( pg ) p ∞ .Consider the tower { A ∞ h p n g i} from §
4. As g divides p n in A ∞ h p n g i , the base change A ∞ h p n g i → B ⊗ LA A ∞ h p n g i of A → B is finite ´etale after inverting p . Almost purity [Sc1, Theorem 7.9 (iii)] then implies that this base changecan be dominated by an almost finite ´etale cover of A ∞ h p n g i , and is thus almost split with respect to p p ∞ (see [Bh2,Lemma 2.7]). The same then holds modulo p m , so the image of α ∞ /p m under can : Hom A ∞ ( Q ∞ , A ∞ /p m [1]) → lim n Hom A ∞ ( Q ∞ , A ∞ h p n g i /p m [1]) is almost zero with respect to p p ∞ . It is now enough to show that the above map is an almost isomorphism with respectto ( pg ) p ∞ . By Theorem 4.2, the only obstruction is lim of { Hom A ∞ ( Q ∞ , A ∞ h p n g i /p m ) } n ≥ . This pro-system isalmost-pro-isomorphic to a constant pro-system by Lemma 3.6 and Theorem 4.2, so we are done by Lemma 3.5. (cid:3) Remark 5.5.
The proof given above goes through for any noetherian ring A that admits a faithfully flat extensionwhich is integral perfectoid. Thus, one may ask: does this condition characterize regularity? In other words, is there a p -adic analog of Kunz’s theorem characterizing regularity in characteristic p as the flatness of Frobenius? A positiveanswer to this question will appear in forthcoming work of the author with Iyengar and Ma. . T HE DERIVED DIRECT SUMMAND CONJECTURE
The goal of this section is to prove Theorem 1.2
Theorem 6.1.
Let A be a regular noetherian ring, and let f : X → Spec( A ) be a proper surjective map. Thenthe map A → R Γ( X , O X ) splits in D ( A ) .Proof. We may assume A is a regular local ring. By taking the closure of a suitable generically defined multisection,we may assume that X is integral and f is generically finite. Then we can choose g ∈ A coprime to p such that f is finite ´etale after inverting pg . Construct A ∞ , and A ∞ as Theorem 5.4. Repeating the argument in the proof ofTheorem 5.4, we must show that for fixed m, n ≥ , the map A ∞ h p n g i /p m → R Γ( X , O X ) ⊗ LA A ∞ h p n g i /p m is almost split with respect to p p ∞ . As the base change X × Spec( A ) Spec( A ∞ h p n g i ) → Spec( A ∞ h p n g i ) is properand finite ´etale after inverting p (as g divides p n on the base), Proposition 6.2 and a diagram chase finish the proof. (cid:3) The following special case of Theorem 6.1 is the crucial one:
Proposition 6.2.
Let A be an integral perfectoid K ◦ -algebra, and set S = Spec( A ) . Let f : Y → S be a propermorphism such that f [ p ] is finite ´etale. Then A → R Γ( Y, O Y ) is almost split.Proof. Let B = H ( Y, O Y ) , so B is an integral extension of A which is finite ´etale after inverting p , and Y is naturallya B -scheme. Almost purity [Sc1, Theorem 7.9 (iii)] gives a map B → C which is an isomorphism after inverting p such that the induced map A → C is an almost finite ´etale cover. In particular, C is integral perfectoid, and A → C is almost split. Thus, on replacing A with C and Y with Y ⊗ B C , we may assume that f [ p ] is an isomorphism. Butthen the p -adic completion b f of f can be dominated by an admissible blowup of S . Set S η = Spa( A [ p ]) to be theassociated affinoid perfectoid space, so the natural map ( S η , O + S η ) → S factors through every admissible blowup of b S . In particular, it factors as ( S η , O + S η ) → Y → S. Taking cohomology of the structure sheaf gives A b −→ R Γ( Y, O Y ) a −→ R Γ( S η , O + S η ) . Now a ◦ b is an almost isomorphism by Scholze’s vanishing theorem [Sc1, Proposition 6.14], so b is almost split. (cid:3) R EFERENCES[An1] Y. Andr´e,
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