Adic Methods to Commutative Algebra

Home , Almost ring, Characteristic (algebra), Melvin Hochster

Recent Applications of 푝-adic Methods to Commutative Algebra

Existence of Big Existence of Small Cohen–Macaulay Cohen–Macaulay Algebras [And18a] Modules

Existence of Big Serre’s Positivity Cohen–Macaulay Conjecture Modules

Monomial Direct Summand Conjecture Theorem [And18a] (Theorem)

Syzygy Intersection Theorem Theorem [PS73, Hoc75, Rob87, Rob89]

Auslander’s Bass’ Question Zerodivisor (Theorem) Conjecture (Theorem)

Linquan Ma and Karl Schwede

820 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 1 Suppose that 푘 is a field and 푅 = 푘[푥1, … , 푥푛]/퐼 is a is the identity. Hence, if 푀 = ker( 푛 ⋅ Tr), then 푅 ≅ commutative polynomial ring over 푘 modulo an ideal 퐼. 퐴 ⊕ 푀 as an 퐴-module. In this case, we say that 푅 has an In other words, it is a finitely generated commutative 푘- 퐴-summand and that 퐴 ⊆ 푅 splits. algebra. Additionally assume that 푅 is also an integral do- Consider now a more general setup. Suppose that main. Emmy Noether’s celebrated Normalization Theo- (*) 퐴 ⊆ 푅 rem proves that inside 푅, there always exists a polynomial subring is a finite extension of Noetherian integral domains. We 퐴 = 푘[푡1, … , 푡푑] ⊆ 푅 ask when 푅 has an 퐴-summand, i.e., when is 푅 ≅ 퐴 ⊕ 푀 for some 푀? In [Hoc73] Hochster conjectured that where the 푡푖 are algebraically independent (have no rela- 1 tions between them) satisfying the following property: the if 퐴 is a Noetherian regular ring and 푅 ⊇ 퐴 is any ex- extension of rings 퐴 ⊆ 푅 makes 푅 into a finitely generated tension ring that is finite as an 퐴-module (henceforth, a finite extension), then 푅 has an 퐴-summand. module over the polynomial ring 퐴 = 푘[푡1, … , 푡푑]. For example, in 푅 = 푘[푥, 푦]/(푦2 − 푥3) we have the subring This was the direct sum- 퐴 = 푘[푥] (or 퐵 = 푘[푦]). mand conjecture, now Andr´e’s Consider the induced finite extension of fraction fields theorem [And18a], and it was one of the central and guiding 푘(푡1, … , 푡푑) = 퐾(퐴) ⊆ 퐾(푅). questions of commutative alge- By viewing 퐾(푅) as a vector space over 퐾(퐴), each ele- bra over the past half-century. 푢 퐾(푅) ment of acts via multiplication Theorem 1 (Direct Summand ×푢 ∶ 퐾(푅) ⟶ 퐾(푅) Theorem). Suppose 퐴 is a Noe- therian regular ring and 푅 ⊇ 퐴 and so we can take its trace, which is then an element of is a finite extension, then 퐴 ↪ 퐾(퐴). This induces a map 푅 splits as a map of 퐴-modules. Tr ∶ 퐾(푅) ⟶ 퐾(퐴). In other words, 푅 has an 퐴- summand. It is not difficult to verify Tr(푅) ⊆ 퐴, and so one obtains Melvin Hochster. a map We observed above that the Tr ∶ 푅 ⟶ 퐴 = 푘[푡1, … , 푡푑]. theorem holds if 퐴 is a polynomial ring over a field of char- Since the trace is a sum of the diagonal matrix entries, the acteristic zero. In fact, the same argument works if 퐴 is any composition regular domain (or even normal2 domain) containing the Tr 퐴 ⊆ 푅 ⟶ 퐴 rational numbers ℚ. Hochster proved in [Hoc73] that The- orem 1 also holds if 퐴 is a regular ring containing the finite is multiplication by the extension degree 푛 ∶= [퐾(푅) ∶ field 픽푝 = ℤ/푝ℤ (e.g., 퐴 = 픽푝[푥1, … , 푥푑]). The methods 퐾(퐴)]. If 푘 has characteristic zero (or more generally if that go into this and the areas of research they spawned are the characteristic does not divide the extension degree 푛 = the topic of ”The Direct Summand Conjecture and Singu- [퐾(푅) ∶ 퐾(퐴)]), then the composition larities in Characteristic 푝.” 1 ⋅Tr 퐴 ⊆ 푅 ⟶푛 퐴 Example 2. The finite extension 퐴 = ℚ[푡2, 푡3] ↪ 푅 = ℚ[푡] does not split. If there was a splitting 휙 ∶ 푅 ⟶ 퐴, Linquan Ma is an assistant professor at Purdue University. His email address it must send 1 ↦ 1 and therefore it must also send 푡2 and is [email protected]. 푡3 to themselves. But 푡3 = 휙(푡3) = 휙(푡2 ⋅ 푡) = 푡2휙(푡) Karl Schwede is a professor at the University of Utah. His email address is 휙(푡) = 푡 퐴 퐴 [email protected]. and so , which does not exist in . Note is not The first named author was supported in part by NSF Grant #1836867/ normal. 1600198. In a recent breakthrough [And18b, And18a], Andr´e The second named author was supported in part by NSF CAREER Grant DMS solved the conjecture in the mixed characteristic3 case, using #1252860/1501102 and NSF grant #1801849. Scholze’s theory of perfectoid algebras and spaces [Sch12]. This material is based upon work supported by the National Science Founda- This will be the topic of the section “Perfectoid Algebras tion under grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the and Ingredients in the Mixed Characteristic Proof.” Spring 2019 semester. 1Regular rings are natural generalizations of polynomial rings over fields and also include Communicated by Notices Associate Editor Daniel Krashen. rings such as ℤ[푥1, … , 푥푑]. For permission to reprint this article, please contact: 2Meaning 퐴 is integrally closed in its fraction field 퐾(퐴). In particular, if 푓 ∈ 퐾(퐴) 퐴 푓 ∈ 퐴 [email protected]. satisfies a monic polynomial with coefficients in , then . 3A ring 퐴 has mixed characteristic if it contains the integers ℤ as a subring, and there is DOI: https://doi.org/10.1090/noti1896 some prime 푝 ∈ ℤ ⊆ 퐴 that is not invertible in 퐴.

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 821 Previously, the best case that Existence of Big Existence of Small was known was the case when Cohen–Macaulay Cohen–Macaulay 퐴 is a regular ring of di- Algebras [And18a] Modules mension ≤ 3, which is due to Heitmann [Hei02]. In Existence of Big Serre’s Positivity the mixed characteristic setting, Cohen–Macaulay Conjecture Bhargav Bhatt and Ofer Gabber Modules also made substantial contribu- tions to these circles of ideas Monomial Direct Summand [Bha14a,Bha18, Gab18]. Conjecture Theorem [And18a] The methods of Andr´e’s proof (Theorem) have also been used to prove Yves Andr ´e. generalizations of the direct summand theorem, notably the Syzygy Intersection Theorem existence of big Cohen–Macaulay algebras and the derived Theorem [PS73,Hoc75, Rob87,Rob89] direct summand theorem, see [And18a, And18c, Bha18,

Gab18, HM18, Shi17]. We expect that the existence of Auslander’s big Cohen–Macaulay algebras will stimulate further study Bass’ Question Zerodivisor 푓 (Theorem) Conjecture of in mixed characteristic: In fact, they can be thought (Theorem) of as a tool that replaces certain aspects of Hironaka’s resolution of singularities from characteristic zero alge- ture was proved by Peskine–Szpiro [PS73]. We note that braic geometry, as explained in [MS18b]. We will dis- many of the early homological conjectures are solved in cuss big Cohen–Macaulay algebras and singularities in mixed characteristic, thanks to Roberts’ proof of the Inter- mixed characteristic in the section “Big Cohen–Macaulay section Theorem using localized Chern characters [Rob87, Algebras and Singularities in Mixed Characteristic.” As an Rob89]. We also mention that there are various stronger application of these ideas, in our final section, “An Appli- forms of some of these conjectures that are proved based cation to Symbolic Powers,” we discuss a result on uniform on Andr´e’s work, see for example [And18c,AIN18,Gab18, growth of symbolic powers of ideals [MS18a]. HM18]. Homological Conjectures. The Homological Conjectures in We want to highlight that, despite the recent commutative algebra are a network of conjectures relating breakthroughs in mixed characteristic, Serre’s Positivity various homological properties Conjecture on intersection multiplicity is still wide open of a commutative ring with its in the ramified mixed characteristic case. To this date, the internal ring structure. They most important progress to- have generated a tremendous wards Serre’s Conjecture is due amount of activity over the to Gabber, see [Hoc97]. We last fifty years. The follow- refer the reader to [Hoc17] for ing is a diagram of homo- a recent extensive survey on logical conjectures, which is Serre’s Conjecture and other part of Hochster’s 2004 dia- (old and new) homological gram [Hoc04] (one sees that conjectures and theorems. the Direct Summand Conjec- We end the introduction by ture/Theorem lies in the heart). briefly discussing one of the ho- Most of these conjectures are mological theorems in the dia- Ofer Gabber. now completely resolved thanks gram above. to the work of Andr´eand oth- Theorem 3 (The Syzygy Theo- Paul C. Roberts. ers. rem). Let (푅, 픪) be a Cohen– Some of these implications are highly nontrivial: For Macaulay (or even regular) local domain and let 푀 be a non- example, the fact that the Direct Summand Theorem im- free finitely generated 푅-module. If 푀 is a 푘-th syzygy module plies the Syzygy Theorem and the Intersection Theorem of finite projective dimension,5 then the rank of 푀 is at least 푘. was due to Hochster [Hoc83],4 and that the Intersection Theorem implies Bass’ Question and Auslander’s Conjec- For instance, the first syzygy module is a submodule of

5 훿푘 This means 푀 ≅ Image(훿푘) in a finite free resolution 0 ⟶ 퐹푛 ⟶ ⋯ ⟶ 퐹푘 ⟶ 4 See also [Dut87] for other interesting connections between the homological conjectures. ⋯ ⟶ 퐹0 ⟶ 푁 ⟶ 0 of a finitely generated 푅-module 푁.

822 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 a free module 퐹0, so its rank is at least one (i.e., it cannot for every nonzero 푐 ∈ 퐴, there exists an 푒 > 0 such that the 푒 be torsion). Theorem 3 is a huge generalization to higher 1↦푐1/푝 푒 퐴-module map 퐴 ⟶ 퐴1/푝 splits. syzygy modules. Theorem 3 was first proved by Evans–Griffith when 푅 Proof. For any nonzero 푐 ∈ 퐴, by looking at terms of 푐 of contains a field [EG81] based on earlier work of Hochster minimal degree, there exists an 푒 so that [Hoc75]. The fact that it follows from Theorem 1 in mixed 푝푒 푝푒 characteristic is a result of Hochster [Hoc83]. So this is 푐 ∉ (푥1 , … , 푥푑 ). now a theorem by Andr´e’s work. 1/푝푒 1/푝푒 In other words, 푐 ∉ (푥1, … , 푥푑) ⋅ 퐴 . Thus the Theorem 3 itself has many unexpected consequences. 푒 image of 푐1/푝 is nonzero in the 푘-vector space For instance, it has connections to Horrock and 푒 Hartshorne’s question on the cohomology of vector bun- 퐴1/푝 푛 . dles of small rank on ℙ [Har79]. 1/푝푒 (푥1, … , 푥푑) ⋅ 퐴 The Direct Summand Conjecture and Singulari- By Nakayama’s lemma, 푐 can be chosen as part of a basis 푒 ties in Characteristic 푝 for the free6 퐴-module 퐴1/푝 . Hence there is a map 휓 ∶ 푒 푒 Suppose now that 퐴 is a regular Noetherian domain and 퐴1/푝 ⟶ 퐴 such that 휓(푐1/푝 ) = 1, which proves the 푅 ⊇ 퐴 is a finite extension that is also a domain. The lemma. □ fraction field extension Rings that satisfy the conclusion of Lemma 4 are called 퐾(퐴) ⊆ 퐾(푅) strongly 퐹-regular (see Definition 7). We will discuss them is separable if and only if the map Tr ∶ 푅 ⟶ 퐴 is nonzero. more in what follows. So to solve the direct summand conjecture, we cannot ex- Theorem 5. If 퐴 is a Noetherian regular ring of characteristic pect to use the field trace as we did when 퐴 contains ℚ. 푝 > 0, then any finite ring extension 퐴 ⊆ 푅 splits. For an arbitrary ring 퐴 that contains 픽푝, we have the Frobenius map (which is a ring homomorphism): Proof. By standard commutative algebra techniques, we / // J K 퐹 ∶ 퐴 퐴 푎 푎푝. may assume that 퐴 ≅ 푘 푥1, … , 푥푑 where 푘 is perfect and that 푅 is an integral domain. 푒 We can iterate the Frobenius map: we label the 푒-fold self- 푒 > 0 퐴 휙 ∶ 퐴1/푝 푒 푝푒 For any integer and any -linear map composition 퐹 ∶ 퐴 ⟶ 퐴 and observe it sends 푎 ↦ 푎 . ⟶ 퐴 we consider the commutative diagram: One of the first issues one runs into when working with 1/푝푒 1/푝푒 / the Frobenius is that it can be difficult to distinguish the Hom퐴1/푝푒 (푅 , 퐴 ) Hom퐴(푅, 퐴) source and target of the map as they are the same ring. We eval@1 eval@1 explain one way to handle this issue. In the case that 퐴 is 1/푝푒 푒 퐴 푝 푒 휙 / an integral domain, we let denote the ring of th 퐴1/푝 퐴. roots of elements of 퐴 embedded inside the algebraic closure of the fraction field, in other words: The top horizontal map is obtained by restricting to 푅 the 푒 푒 푒 푒 1/푝 1/푝 퐴1/푝 ∶= {푥 ∈ 퐾(퐴) | 푥푝 ∈ 퐴}. domain of an element 휓 ∈ Hom퐴1/푝푒 (푅 , 퐴 ) and 휙 1/푝푒 then post-composing with . Since the vertical maps are The ring 퐴 is abstractly isomorphic to 퐴 via the map evaluation at 1, to show that 퐴 ⟶ 푅 splits it is enough 1/푝푒 푝푒 퐴 ⟶ 퐴 which sends 푏 ↦ 푏 . This isomorphism to show that the right vertical map surjects (then there identifies the inclusion exists 휓 ∈ Hom퐴(푅, 퐴) so that 휓(1) = 1). Because 1/푝푒 eval@1 퐴 ⊆ 퐴 Hom퐴(푅, 퐴) ⟶ 퐴 surjects generically, i.e., it is surjec- with Frobenius, and it provides us with a convenient way tive if we tensor with the fraction field of 퐴, we can choose eval@1 of distinguishing the source and target of the Frobenius a nonzero 푐 in the image of Hom퐴(푅, 퐴) ⟶ 퐴. It fol- 푒 map. lows that 푐1/푝 is in the image of the left vertical map (for The proof of the direct summand conjecture in charac- any 푒 > 0). Next, using Lemma 4, we choose 푒 > 0 and an 푒 푒 teristic 푝 > 0 we present here follows from [Hoc73] in 퐴-linear map 휙 ∶ 퐴1/푝 ⟶ 퐴 sending 푐1/푝 ⟶ 1. The spirit. We begin by proving the following lemma (we use commutative diagram implies that the composition from formal power series instead of the polynomial ring, but the upper left to the lower right surjects, and hence so does the idea is the same); this lemma also motivates further eval@1 Hom (푅, 퐴) ⟶ 퐴 □ investigations on characteristic 푝 > 0 singularities. 퐴 .

푒 푒 Lemma 4. Suppose 푘 is a perfect field of positive characteristic 6 푗1/푝 푗푑/푝 One common choice of basis is that made up of all elements 푥1 ⋯ 푥푑 where each 푒 and 퐴 = 푘J푥1, … , 푥푑K is the formal power series ring. Then 푗푖 varies between 0 and 푝 − 1.

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 823 Philosophy. We take a step back to think through this Example 8. In Example 6, we saw that the affine cone over proof. In characteristic 푝 > 0, the trace map does not al- an elliptic curve is not a splinter. The ring 픽푝[푥, 푦, 푧]/ ways lead to splittings, even for separable extensions due (푥푦 − 푧2) is strongly 퐹-regular, and hence it is a splin- to the presence of what is called wild ramification. Roughly ter. The fact that it is strongly 퐹-regular most easily follows speaking, we get around this by choosing an element 푐 ∈ from the fact that it is a direct summand of a regular ring. 퐴 over which 퐴 ⊆ 푅 is ramified. By taking 1/푝푒th roots of 푐 (or in other words using Frobenius), we minimize the Strongly 퐹-regular singulari- ramification until it almost goes away and so that exten- ties are intimately tied to singu- sion 퐴 ⊆ 푅 splits. larities that appear in complex We also give a new example of a ring with a non-split algebraic geometry. Roughly extension. speaking, a complex algebraic variety 푋 has rational singulari- Example 6. Consider 푘 = 픽푝, the algebraic closure of the ties if its line bundles have the finite field 픽푝 = ℤ/푝ℤ. We form the ring 퐴 = 푘[푥, 푦, 푧]/ same sheaf cohomology as the (푧푦2 − 푥(푥 − 푧)(푥 + 휆푧)), defining the affine cone over pullbacks of those line bundles a (projective) elliptic curve 퐸 (for example, one could take to a resolution of singularities.8 휆 = 1). If the characteristic 푝 > 0 is such that the curve A refinement of rational sin- is supersingular, then the (absolute) Frobenius map 퐹 ∶ gularities is log terminal sin- 퐴 ⟶ 퐴 (which sends 푟 ↦ 푟푝 for all 푟 ∈ 푅) does not gularities (rational singularities split. If 푝 > 0 is such that the curve is ordinary, then there Karen E. Smith. whose finite étale9 in codimen- exists a degree 푝 étalemap 퐸′ ⟶ 퐸 between elliptic curves sion 1 covers also have rational [Sil09, Chapter V, Theorem 3.1]. This gives a finite exten- singularities). Rational singularities are exactly the same sion 퐴 ↪ 퐴′ of the rings corresponding to the cones. This as log terminal singularities on hypersurfaces (and more extension is not étaleat the cone point(s) and in fact is not generally, on Gorenstein varieties). split. In either case 퐴 has a finite extension 퐴 ↪ 푅 that is What is really surprising, given their completely disjoint not split. definitions, is that log terminal singularities are essentially the same as strongly 퐹-regular singularities, modulo reduc- Singularities in characteristic 푝 > 0. The techniques we tion to characteristic 푝 > 0. Specifically, suppose that a used to prove the direct summand conjecture in character- chart 푈 on 푋 is given as the spectrum of 푅ℂ=ℂ[푥1, … , 푥푑] istic 푝 > 0 have led to a vigorous study of singularities /퐼ℂ. Suppose for simplicity that all the coefficients of the in characteristic 푝 > 0 (typically under the names tight generators of the ideal 퐼ℂ live in ℤ. We can reduce 푅ℂ to closure theory and Frobenius splitting theory). characteristic 푝 > 0 by taking the coefficients of the gener- 퐼 푝 > 0 Definition 7 (Strongly 퐹-regular singularities). A Noether- ators of the ideal ℂ modulo some . For example: 1/푝 ian domain 푆 containing 픽푝 such that 푆 is a finitely /o /o mod/o /o 5/o / 푓 = 푥2 + 101푥푦 − 7 /o /o 푥2 + 푥푦 + 3. generated 푆-module is called strongly 퐹-regular if for any 1/푝푒 nonzero 푐 ∈ 푆, there is an 푒 > 0 and 휙 ∶ 푆 ⟶ 푆 For each 푝 > 0, this gives us a ring 푅푝 =픽푝[푥1, … , 푥푑]/퐼푝. 푒 such that 휙(푐1/푝 ) = 1. In this case, the singularities of Theorem 9 ([HW02,Har98,Smi97,MS97]). 푈 = Spec 푅 Spec 푆 are called strongly 퐹-regular singularities. ℂ has log terminal singularities if and only if for all 푝 ≫ 0, 푅푝 The proof we gave in Theorem 5 shows that strongly 퐹- has strongly 퐹-regular singularities. regular rings are direct summands of all their finite exten- In fact, this connection is a small part of a large dic- sions. In fact, a Noetherian domain that is a direct sum- tionary where notions from higher dimensional complex mand of every finite extension is called a splinter. Thus we algebraic geometry correspond to concepts involving the have: Frobenius map. (Strongly 퐹-regular ring) ⇒ (Splinter) Definition 10 (Derived splinters). A Noetherian domain The converse is open except in the case that the ring is 푆 is called a derived splinter if for every proper surjective (close to) Gorenstein [HH94,Sin99]. In fact, the converse map 휋 ∶ 푌 ⟶ 푋 = Spec 푆, the induced map 풪푋 ⟶ would imply arguably the most studied question in char- 푅휋∗풪푌 splits in the derived category of 풪푋-modules. acteristic 푝 > 0 commutative algebra: that a weakly 퐹- regular ring7 is strongly 퐹-regular. 8A resolution of singularities is a proper map 휋 ∶ 푌 ⟶ 푋 of varieties such that 푌 is non- singular and 휋 is birational, which means it is an isomorphism “almost everywhere” (i.e., 7A ring is called weakly 퐹-regular if all of its ideals are tightly closed [HH90]; we will not on a Zariski open and dense subset). delve into these definitions here, however. 9essentially covering spaces

824 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 It is a theorem of Bhatt that 푅 such that the set of power-bounded elements10 푅∘ ⊆ 푅 for rings of finite type over is bounded and the Frobenius is surjective on 푅∘/푝.A fields of characteristic zero, ra- 퐾∘-algebra 푆 is called integral perfectoid if it is 푝-adically tional singularities are exactly complete, 푝-torsion free, and the Frobenius induces an derived splinters [Bha12] (and isomorphism 푆/푝1/푝 ⟶ 푆/푝. also see [Kov00]). But, in [Bha12], Bhatt also shows that If 푅 is a perfectoid 퐾-algebra, then the ring of power- ∘ derived splinters in character- bounded elements 푅 is integral perfectoid, and if 푆 is in- istic 푝 > 0 are exactly the tegral perfectoid, then 푆[1/푝] is a perfectoid 퐾-algebra, same as splinters. Both these see [Sch12, Theorem 5.2]. results strongly suggest that reg- We give examples of integral perfectoid algebras. It is ular rings should be derived important to note that these algebras are never Noetherian. As before, •̂ denotes the 푝-adic completion of •. Kei-ichi Watanabe. splinters in general. This derived statement is an extension 퐾∘ = ℤ ̂[푝1/푝∞ ] of the direct summand conjec- Example 13. (a) 푝 . ture (since any finite extension of ring 퐴 ⊆ 푅 induces a (b) Spec 푅 ⟶ Spec 퐴 1/푝∞ 1/푝∞ proper surjective map ). In [Bha18], 퐾∘⟨푥 , … , 푥 ⟩ applying some of Andr´e’s ideas, Bhatt gives a simplified 2 푑 ⋀ ∞ ∞ proof of Theorem 1 and also proves this derived version. 1/푝∞ 1/푝 1/푝 ∶= ℤ푝[푝 ][푥2 , … , 푥푑 ].

Theorem 11 (The Derived Direct Summand Theorem). (c) 푅̂+, where (푅, 픪) is a Noetherian complete local Any Noetherian regular ring is a derived splinter. domain of mixed characteristic (0, 푝) and 푅+ is the integral closure of 푅 in an algebraic closure of its frac- Perfectoid Algebras and Ingredients in the Mixed tion field. Characteristic Proof Now we move to the discus- Although we will not use it in this survey, a key idea in sion of the proof of the Di- the theory of perfectoid algebras (and spaces) is that nu- rect Summand Theorem (The- merous questions can be studied via tilting. This can turn orem 1) in mixed characteris- a mixed characteristic question (or ring) into one in pos- tic, which uses perfectoid tech- itive characteristic, and vice versa. This principle is used niques. Consider a Noetherian extensively (behind the scenes) in what follows. For more local ring 퐴 with maximal ideal discussion, see for instance [Bha14b,Sch12]. 픪. As before, we say that 퐴 has A key part of the general theory of perfectoid rings is that mixed characteristic (0, 푝) if 퐴 we can talk about “almost mathematics” (appearing orig- has characteristic 0 and 퐴/픪 inally in the work of Faltings, [Fal88], also see the work has prime characteristic 푝 > 0. of Gabber and Ramero [GR03]). Roughly speaking, we For example, the ring of 푝-adic treat modules that are annihilated by the ideal (푝, 푝1/푝, 2 ∞ integers ℤ푝 (the ring of formal 푝1/푝 , … ) =∶ (푝1/푝 ) ⊆ 퐾∘ as if they were zero (since 푒 Bhargav Bhatt. power series in 푝) has maximal 푝1/푝 is almost 1 for 푒 ≫ 0). ideal generated by 푝, its residue ∘ field is 픽푝, while its fraction field ℚ푝 has characteristic 0. Definition 14. Let 푆 be an integral perfectoid 퐾 -algebra. ∞ We use ℚ푝 to construct our first example of a perfec- 1/푝 푒 (a) An 푆-module 푀 is almost zero if (푝 )푀 = 0. toid ring. Begin by adjoining all 푝 th roots of 푝 to ℚ푝 (b) An 푆-module 푀 is almost flat if 1/푝∞ to form ℚ푝(푝 ). We 푝-adically complete and call the 1/푝∞ 푆 ̂1/푝∞ (푝 )Tor (푀, 푁) = 0 resulting ring 퐾 = ℚ푝(푝 ). This field 퐾 is a typical 1 example of a perfectoid field. The field 퐾 contains a subring for all 푆-modules 푁. ∘ ̂1/푝∞ ∘ 퐾 = ℤ푝[푝 ], which is its “ring of integers.” 퐾 is a (c) A short exact sequence of 푆-modules 0 ⟶ 푀 ⟶ typical example of an integral perfectoid ring. For the rest 푁 ⟶ 푁/푀 ⟶ 0 represented by a class 휂 ∈ 퐾 퐾∘ 1 of this section, we fix and as above. We now give a Ext푆(푁/푀, 푀) is almost split if definition of a perfectoid algebra. 1/푝∞ 1 (푝 )휂 = 0 in Ext푆(푁/푀, 푀). Definition 12 (Perfectoid algebras [Sch12, BMS18, And18c]). A perfectoid 퐾-algebra is a Banach 퐾-algebra 10Elements 푥 such that the norm of 푥푛 is bounded independent of 푛.

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 825 The most crucial result that is relevant to the proof of 퐾(푅) is finite étale, i.e., separable), we can invert some Theorem 1 is the following theorem, proved by Scholze other element to make 퐴 ⟶ 푅 étale. Thus we sup- [Sch12] and independently by Kedlaya–Liu [KL15]. Spe- pose that 퐴[1/푝푔] ⟶ 푅[1/푝푔] is finite étalefor some cial cases were first obtained by Faltings [Fal88,Fal02] and nonzero element 푔. The main obstruction to “running” Gabber–Ramero [GR04]. the above argument is that Theorem 15 no longer works: 퐴 [1/푝] ⊗ 푅 is no longer finite étaleover 퐴 [1/푝]. We Theorem 15 (The Almost Purity Theorem). Suppose 푆 is ∞ ∞ only know that it becomes finite étalewhen we further integral perfectoid and 푆[1/푝] ⟶ 푇 is a finite étaleextension. invert 푔. To overcome this difficulty, Andréproved two Then the integral closure 푆′ of 푆 in 푇 is an almost finite étale remarkable theorems using Scholze’s theory of perfectoid extension of 푆. In particular, 푆 ⟶ 푆′ is almost split. spaces: Here, almost finite étale roughly means that the obstruc- ∞ Theorem 16 ([And18a]). Suppose 푆 is an integral perfectoid tions to being finite étaleare annihilated by (푝1/푝 ). For 푔 ∈ 푆 푆 ⟶ 푆̃ our purposes however, we will only need the weaker fact algebra and . Then there exists a map of integral ′ 푔 that the map 푆 ⟶ 푆 is almost split. perfectoid algebras, such that admits a compatible system of 1/푝푒 ∞ ̃ 1/푝푒+1 푝 1/푝푒 We sketch the proof of the Direct Summand Theorem 푝-power roots {푔 }푒=1 in 푆 (i.e., (푔 ) = 푔 for (Theorem 1) in mixed characteristic. For simplicity, we set all 푒) and that 푆 ⟶ 푆̃ is almost faithfully flat modulo powers ⋀ 푝 퐴 = ℤ푝[푥2, … , 푥푑]. Our goal is to show that every finite of . extension 퐴 ⊆ 푅 splits. Theorem 17 ([And18b]). Suppose 푆 is integral perfectoid The case 퐴[1/푝] ⟶ 푅[1/푝] is finite étale. We let such that 푔 ∈ 푆 has a compatible system of 푝-power roots

⋀ ∞ ∞ 1/푝∞ 1/푝 1/푝 in 푆, and 푆[1/푝푔] ⟶ 푇 is a finite étaleextension. Then the 퐴 = ℤ [푝 ][푥 , … , 푥 ]; ∞ ∞ 푝 2 푑 integral closure of 푆 in 푇 is (푝푔)1/푝 -almost finite étaleover this is an integral perfectoid algebra. Consider the follow- 푆 modulo powers of 푝. ing diagram: We return to the proof of Theorem 1. As before, we / ⋀ 퐴 푅 NN ∞ 1/푝∞ 1/푝∞ NNN set 퐴 = ℤ [푝1/푝 ][푥 , … , 푥 ]. We apply Theo- NNN ∞ 푝 2 푑 NNN ̃ N' rem 16 to construct 퐴∞ ⟶ 퐴∞,∞ ∶= 푆 such that 푔 has / / ′ a compatible system of 푝-power roots in 퐴∞,∞ and such 퐴∞ 퐴∞ ⊗ 푅 (퐴∞ ⊗ 푅) that 퐴∞,∞ is almost faithfully flat over 퐴∞ mod powers of ′ where (퐴∞ ⊗ 푅) denotes the normalization of 퐴∞ ⊗ 푅 푝. Consider the following diagram: in 퐴 [1/푝] ⊗ 푅. Since 퐴 [1/푝] ⟶ 퐴 [1/푝] ⊗ 푅 is ∞ ∞ ∞ / P a finite étaleextension by base change, Theorem 15 says 퐴 푅 PPP PPP that the composition PPP PPP ′ ' 퐴∞ ⟶ 퐴∞ ⊗ 푅 ⟶ (퐴∞ ⊗ 푅) / / ′ 퐴∞,∞ 퐴∞,∞ ⊗ 푅 (퐴∞,∞ ⊗ 푅) is almost split and hence 퐴∞ ⟶ 퐴∞ ⊗ 푅 is almost split. ′ This implies 퐴 ⟶ 푅 is split, since 퐴 is Noetherian and where (퐴∞,∞ ⊗푅) denotes the normalization of 퐴∞,∞ ⊗ 퐴 ⟶ 퐴∞ is faithfully flat (we 푅 in (퐴∞,∞ ⊗ 푅)[1/푝푔]. Applying Theorem 17 we find omit the details here, this fol- that the composition map lows from a standard commuta- ′ 퐴∞,∞ ⟶ 퐴∞,∞ ⊗ 푅 ⟶ (퐴∞,∞ ⊗ 푅)

tive algebra argument). ∞ The argument above was is (푝푔)1/푝 -almost split modulo powers of 푝. It follows 1/푝∞ first observed by Bhatt that 퐴∞,∞ ⟶ 퐴∞,∞ ⊗ 푅 is (푝푔) -almost split mod [Bha14a]. Notice that we powers of 푝. This is enough to conclude that 퐴 ⟶ 푅 is only used Theorem 15 for split by the Noetherianity and 푝-adic completeness of 퐴, the integral perfectoid algebra the faithful flatness of 퐴∞ over 퐴, and the almost faithful ⋀ ∞ 1/푝∞ 1/푝 flatness of 퐴∞,∞ over 퐴∞ mod powers of 푝 (again some 퐴∞ = ℤ푝[푝 ][푥2 ,…, 1/푝∞ work is required here, but we omit the details). 푥푑 ]. This version is due to Faltings [Fal02]. Big Cohen–Macaulay Algebras and Singularities Craig Huneke. General case. We now assume in Mixed Characteristic that 퐴[1/푝] ⟶ 푅[1/푝] is Let (푅, 픪) be a Noetherian local ring. Recall that a sys- not necessarily ´etale. But since 퐴 ⟶ 푅 is “generically tem of parameters 푥1, … , 푥푑 in (푅, 픪) is a collection of ´etale”(i.e., the extension of the fraction field 퐾(퐴) ⟶ 푑 = dim 푅 elements that generate the maximal ideal up

826 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 to radical, i.e., 픪 = √(푥1, … , 푥푑). An 푅-algebra 퐵 is the extension called big Cohen–Macaulay if every system of parameters 푘 ′ 푥 = 푥1, … , 푥푑 of 푅 is a regu- 푇 ⟶ 푇 = 푇[푋1, … , 푋푘]/ ⎛푡푘+1 − ∑ 푥푖푋푖⎞ . lar sequence on 퐵. This means ⎝ 푖=1 ⎠ ′ 푥푖+1 is a nonzero divisor on We have forced 푡푘+1 to be inside (푥1, … , 푥푘)푇 , and so the ′ 퐵/(푥1, … , 푥푖) and that 픪퐵 ≠ bad relation is trivialized. Such a 푇 is called an algebra 퐵. modification of 푇. Repeat this process to trivialize all bad Now suppose that 퐴 ⟶ relations on 푇 and we obtain a total algebra modification of 푅 is a module-finite extension 푇; call it 푇1. Now we might have new bad relations on of Noetherian local rings such 푇1, but we can repeat the whole process above and take a that 퐴 is regular and 푅 ad- (huge) direct limit. More precisely, we set mits a big Cohen–Macaulay al- 퐵 ∶= lim(푅 = 푇 ⟶ 푇 ⟶ 푇 ⟶ ⋯). gebra. We claim that the map −→ 1 2 퐴 ⟶ 푅 splits. So suppose that The above construction guarantees that for every sys- 퐵 is a big Cohen–Macaulay 푅- tem of parameters 푥1, … , 푥푑 of 푅, 푥푖+1 is a nonzero di- Raymond C. Heitmann. algebra; it is easy to see that 퐵 is visor on 퐵/(푥1, … , 푥푖−1). However, one must show that also a big Cohen–Macaulay 퐴- 픪퐵 ≠ 퐵. In characteristic 푝 > 0, this can be proved us- algebra (since every system of parameters of 퐴 becomes ing the Frobenius map. If 푅 contains ℚ, a reduction to a system of parameters of 푅). Because 퐵 is big Cohen– characteristic 푝 > 0 technique can be applied (basically Macaulay and 퐴 is regular, 퐵 is faithfully flat over 퐴 (see by noticing that if 픪퐵 = 퐵 then this must happen at a [HH92, p.77]). It then follows from the factorization finite level). Later Hochster [Hoc02] essentially observed 퐴 ⟶ 푅 ⟶ 퐵 that 퐴 ⟶ 푅 is split (this is similar to that 픪퐵 ≠ 퐵 as long as we can map 푇 to a certain “almost the final arguments of the last section). Cohen–Macaulay algebra”11 (e.g., in characteristic 푝 > 0, ∞ Therefore the existence of big Cohen–Macaulay algebras we have 푅 ⟶ 푅1/푝 ). Finally in mixed characteristic, 1/푝∞ ′ implies Theorem 1 for local rings. The general case of The- Andr´ereplaced 푅 by (퐴∞,∞ ⊗푅) , which is the object orem 1 follows from the local case: the evaluation at 1 that appears in the argument in the proof of Theorem 1, map Hom퐴(푅, 퐴) ⟶ 퐴 is surjective if and only if it is to prove 픪퐵 ≠ 퐵. surjective locally. This explains that the existence of big It turns out that big Cohen–Macaulay algebras have Cohen–Macaulay algebras sits at the top of the diagram at deep connections with singularities. In fact, as we men- the end of the introduction. tioned at the start of the section, if 퐴 is regular, then a In the case that 푅 contains a field ℚ or ℤ/푝ℤ, the ex- big Cohen–Macaulay 퐴-algebra 퐵 is faithfully flat over 퐴. istence of big Cohen–Macaulay algebras was established From one perspective the role of 퐵 is analogous to a resolu- by Hochster–Huneke [HH92, HH95]. Hochster had also tion of singularities in equal characteristic zero. Suppose shown that big Cohen–Macaulay algebras exist in mixed that 푆 is (essentially) of finite type over a field 푘 of charac- characteristic in dimension three [Hoc02], using ideas of teristic zero. Let 휋 ∶ 푌 ⟶ 푋 = Spec 푆 be a resolution of Heitmann’s proof of the direct summand conjecture in di- singularities. Grauert–Riemenschneider vanishing [GR70] mension three [Hei02]. Andr´e[And18a] completed this (a relative version of Kodaira or Kawamata–Viehweg van- program and showed they exist in mixed characteristic in ishing [Kaw82,Vie82,EV92]) tells us that the higher direct all dimensions. images of the canonical sheaf We roughly sketch the strategy of the construction of 푖 big Cohen–Macaulay algebras following [Hoc94, Hoc02, 푅 휋∗휔푌 = 0 And18a, HM18]. Suppose that 푇 is an 푅-algebra and that vanish for 푖 > 0. By local duality [Har66], this vanishing 푥1, … , 푥푘+1 is part of a system of parameters for 푅. Further is equivalent to the following vanishing of local cohomol- assume that 푡1, … , 푡푘+1 are elements of 푇 satisfying ogy, 푗 퐻푥 (푅휋∗풪푌) = 0 푘 푗 where 푗 < dim 푋, 푥 ∈ 푋 is any closed point, and ℍ푥 de- (⋆) 푥푘+1푡푘+1 = ∑ 푥푖푡푖 but 푡푘+1 ∉ (푥1, … , 푥푘)푇. 푖=1 notes sheaf cohomology with support at 푥. In other words, the local cohomology of the complex 푅휋∗풪푌 vanishes except in the top degree. For finitely generated modules, this In other words, 푥 is a zero divisor in 푇/(푥 , … , 푥 )푇 푘+1 1 푘 property of vanishing local cohomology is equivalent to and so 푇 is not (big) Cohen–Macaulay over 푅. We there- ⋆ fore call ( ) a bad relation. 11This means systems of parameters on 푅 are “almost regular sequences”; we omit the de- Let 푋1, … , 푋푘 be indeterminates over 푇. We consider tailed definition.

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 827 the Cohen–Macaulay property. In other words, the com- Suppose 퐴 is a Noetherian regular ring (e.g., a polyno- plex mial ring over a field or over ℤ). Suppose 푄 ⊆ 퐴 is a 푅휋∗풪푌 prime ideal. For any integer 푛 > 0 we define the 푛th sym- is a Cohen–Macaulay algebra, except that it is not an alge- bolic power of 푄 to be bra: it lives in the derived category! This was first observed (푛) 푛 푄 ∶= (푄 퐴푄) ∩ 퐴. by Roberts [Rob80] (and we omit the non-triviality condi- tion analogous to 픪퐵 ≠ 퐵 for simplicity). Many common In other words, 푄(푛) is the set of elements of 퐴 (or func- local applications of Grauert–Riemenschneider vanishing tions on Spec 퐴) that vanish to order 푛 at the generic point can be proved using the vanishing of local cohomology of of 푉(푄) ⊆ Spec 퐴. 푛 (푛) 푅휋∗풪푌. Tied up closely with this vanishing are log termi- Evidently, 푄 ⊆ 푄 but they are not always equal. A nal singularities, which we define in a special case: very extensively explored question in commutative algebra (푛) 푛 Definition 18. A Gorenstein variety 푋 in characteristic studies the difference between 푄 and 푄 . For example, zero is called log terminal if the canonical map when 푄 is generated by (part of) a regular sequence, a clas- 푄푛 = 푄(푛) 푑 푑 sical result in commutative algebra says that 퐻 (풪 ) ⟶ 퐻 (푅휋 풪 ) (푛) 푛 푥 푋 푥 ∗ 푌 for all 푛. However 푄 can be much bigger than 푄 . injects for every 푥 ∈ 푋 and for some (equivalently every) 푅 = 푘[푥, 푦, 푧]/(푥푦 − 푧푚) 푄 = resolution of singularities 휋 ∶ 푌 ⟶ 푋. Example 20. Let . Then (푥, 푧) is a prime ideal of height one and 푥 ∉ 푄푚 (in fact, Of course, for a big Cohen–Macaulay algebra 퐵 over a 푥 is not even in 푄2). However, we see that 푥 ∈ 푄(푚) 푚 local ring 푆, we also have the vanishing because 푥 ∈ 푄 푅푄 (푦 is a unit in 푅푄). 퐻푖 (퐵) = 0 픪 Although the above example shows that 푄푛 and 푄(푛) for the maximal 픪 ∈ Spec 푆, and all 푖 < dim 푆. Further- can be quite different, a surprising result was obtained by more, it follows from work of Smith [Smi94] that in char- Swanson [Swa00] (see also [HKV09]), who proved essen- acteristic 푝 > 0, a Gorenstein local ring (푆, 픪) is strongly tially that, for any complete local domain (푅, 픪) and any 퐹-regular if and only if prime ideal 푄 ⊆ 푅, there is a constant 푘 (depending on 푑 푑 (푘푛) 푛 퐻픪(푆) ⟶ 퐻픪(퐵) 푅 and 푄) such that 푄 ⊆ 푄 for all positive integers 푛. In other words, the difference between 푄푛 and 푄(푛) is is injective for every big Cohen–Macaulay 푅-algebra 퐵. bounded “linearly.” Inspired by this, we introduce the following definition For complete regular local rings, we have an even in [MS18b]. stronger result. The following theorem was proved when Definition 19. Let (푅, 픪) be a Gorenstein local ring of di- our ring contains a field, by Hochster–Huneke [HH02] mension 푑 and let 퐵 be a big Cohen–Macaulay 푅-algebra. (see also Ein–Lazarsfeld–Smith [ELS01]), and recently it Then we say 푅 is big-Cohen–Macaulay-regular with respect to was extended to mixed characteristic in [MS18a]. Com- 퐵 (or more compactly is BCM퐵-regular) if the natural map pared with Swanson’s result mentioned above, the theo- 푑 푑 퐻픪(푅) ⟶ 퐻픪(퐵) is injective. rem shows that for regular rings the constant 푘 can be chosen to be the dimension of the ring. In particular it is in- It turns out that BCM -regular singularities share many 퐵 dependent of the ideal 푄! analogous properties of log terminal singularities in equal characteristic 0 or strongly 퐹-regular singularities in equal Theorem 21. Let 퐴 be a complete regular local ring of dimen- characteristic 푝 > 0 (again, all based on this vanishing). sion 푑. Then for every prime ideal 푄 ⊆ 퐴 and every 푛, we Furthermore, we apply this to study singularities when the have 푄(푑푛) ⊆ 푄푛. characteristic varies, e.g., families of singularities defined over Spec ℤ. We refer to [MS18b] for more results in this We briefly explain the strategy of the proof ofTheo- direction. rem 21 in mixed characteristic. The idea is to construct a multiplier ideal like object in mixed characteristic and An Application to Symbolic Powers then use the same strategy as in Ein–Lazarsfeld–Smith 12 We discuss another commutative algebraic application of [ELS01]. integral perfectoid big Cohen–Macaulay algebras in mixed For the moment suppose that 퐴 is a regular ring of finite characteristic [MS18a]. In fact, our proof strategy is di- type over a field of characteristic 0 (e.g., 퐴=ℚ[푥1, … , 푥푑]). rectly inspired by the connection between big Cohen– Suppose that 픞 ⊆ 퐴 is an ideal and 푡 ∈ ℝ≥0 a formal Macaulay algebras and resolution of singularities and the exponent for 픞. In this setting we can take a log resolution vanishing theorems discussed above. Let us state the problem. 12A similar object exists in characteristic 푝 > 0 and is called the test ideal.

828 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 13 휋 ∶ 푌 ⟶ Spec 퐴 with 픞풪푌 = 풪푌(−퐺) and define the In the multiplier ideal definition, we extend 픞 to 풪푌 multiplier ideal at a maximal ideal 픪 ⊆ 퐴14: and then multiply the associated divisor by 푡 (and round as appropriate). We do the same thing here: we define 푡 푑 풥(퐴픪, 픞 ) = Ann퐴 {휂 ∈ 퐻픪(퐴) ∣ 휂 ↦ 0 픪 휏(퐴, (푔)푡) 푑 ∈ ℍ픪(푅휋∗풪푌(⌊푡퐺⌋))}. 푑 1/푝∞ 푎/푝푒 푑 =Ann퐴 {휂∈퐻픪(퐴) ∣ 푝 푔 휂=0 in 퐻픪(퐴∞,∞)} This is an ideal of 퐴픪 that measures the singularities of 푉(픞) ⊆ Spec 퐴, scaled by 푡, at 픪 ∈ Spec 퐴. Crucially, for where 푎/푝푒 > 푡 approximates 푡 from above. the applications to the result on symbolic powers, the mul- Although this definition looks a bit technical, it turns tiplier ideal satisfies the following “subadditivity” property out that it satisfies many properties (including the subaddi- [DEL00]: tivity property (†)) similar to the multiplier ideal 풥(퐴, 픞푡) in characteristic 0. This allows us to prove Theorem 21. (†) 풥(퐴 , 픞푡푛) ⊆ 풥(퐴 , 픞푡)푛 for all positive integers 푛. 픪 픪 Moreover, the crucial reason that the subadditivity holds 푡 This essentially follows from the Kawamata–Viehweg type for 휏(퐴, 픞 ) is because 퐴∞,∞ is almost big Cohen– vanishing result that accompanies the multiplier ideals Macaulay. We refer the interested reader to [MS18a, Sec- [Laz04], which can be stated dually as: tion 4] for details. ℍ푖 (푅휋 풪 (⌊푡퐺⌋)) = 0 푖 < 푑. 픪 ∗ 푌 for ACKNOWLEDGMENTS. The authors thank the refer- Philosophy. We examine the definition again. Roughly ees as well as Yves André, Bhargav Bhatt, Ray Heitmann, speaking, the multiplier ideals associated to the pair (퐴, 픞푡) Mel Hochster, and Srikanth Iyengar for numerous help- are elements of 퐴픪 that annihilate all elements in the top ful comments on this article. 푑 local cohomology module 퐻픪(퐴) whose image in 푑 References 퐻픪(푅휋∗풪푌) 푡 [And18a] AndréY. La conjecture du facteur direct, Publ. Math. is “almost annihilated” by 픞 (this is made precise in the Inst. Hautes Etudes´ Sci. (127):71–93, 2018. MR3814651 above definition as 픞 in 풪푌 is just 풪푌(−퐺)). [And18b] André Y. Le lemme d’Abhyankar perfectoide, Therefore, in order to extend the definition to mixed Publ. Math. Inst. Hautes Etudes´ Sci. (127):1–70, 2018. characteristic and still have the nice properties such as the MR3814650 subadditivity, one needs an object 퐵 like 푅휋∗풪푌, which [And18c] AndréY. Weak functoriality of Cohen–Macaulay al- has good vanishing properties and such that one can make gebras, 2018. arXiv:1801.10010. sense of, or at least approximate, 픞푡 in 퐵. It turns out that a [AIN18] Avramov LL, Iyengar SB, Neeman A. Big Cohen– Macaulay modules, morphisms of perfect complexes, sufficiently large integral perfectoid big Cohen–Macaulay and intersection theorems in local algebra, Doc. Math. (or almost big Cohen–Macaulay) algebra will do the job! (23):1601–1619, 2018. MR3890961 Below we give a definition of perfectoid multiplier ideal [Bha12] Bhatt B. Derived splinters in positive character- J K [MS18a] for 퐴 = ℤ푝 푥2, … , 푥푑 . istic, Compos. Math., no. 6 (148):1757–1786, 2012. Using the fact that ideals are made up of principal ideals, MR2999303 one can essentially reduce the definition to the case where [Bha14a] Bhatt B. Almost direct summands, Nagoya Math. J. 픞 = (푔) is principal (we are absolutely hiding subtleties (214):195–204, 2014. MR3211823 … here to keep the definitions cleaner). Let 퐴∞ denote the [Bha14b] Bhatt B. What is a perfectoid space?, No- ∞ 1/푝∞ 1/푝∞ tices Amer. Math. Soc., no. 9 (61):1082–1084, 2014. 푝 퐴[푝1/푝 , 푥 , … , 푥 ] -adic completion of 2 푑 ; this is MR3241564 the same as the definition of 퐴∞ we used in the proof of [Bha18] Bhatt B. On the direct summand conjecture and its Theorem 1 in the second section. We then form 퐴∞,∞ us- derived variant, Invent. Math., no. 2 (212):297–317, 2018. ing Theorem 16 for our fixed element 푔 (or in the non- MR3787829 principal case, to the generators of 픞). Then for a fixed [BMS18] Bhatt B, Morrow M, Scholze P. Integral 푝-adic real number 푡 > 0, we can approximate 푡 (from above) by Hodge theory, Publ. Math. Inst. Hautes Etudes´ Sci. rational numbers of the form 푎/푝푒, and observe we can (128):219–397, 2018, DOI 10.1007/s10240-019-00102-z. 푎/푝푒 MR3905467 identify elements 푔 ∈ 퐴∞,∞ by construction. These 푒 [DEL00] Demailly J-P, Ein L, Lazarsfeld R. A subadditivity 푔푎/푝 푔푡 approximate . property of multiplier ideals, Michigan Math. J. (48):137– 156, 2000. Dedicated to William Fulton on the occasion 13 This means 휋 ∶ 푌 ⟶ Spec 퐴 is proper birational, 푌 is regular, and 픞 ⋅ 풪푌 defines a of his 60th birthday. MR1786484 (2002a:14016) SNC divisor. [Dut87] Dutta SP. On the canonical element conjecture, 14This is not the usual definition of multiplier ideals asin [Laz04], but it is equivalent to the usual definition via local and Grothendieck’s duality [Har66], see [MS18b, Section 2] Trans. Amer. Math. Soc., no. 2 (299):803–811, 1987. for a detailed explanation. MR869233

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 829 [ELS01] Ein L, Lazarsfeld R, Smith KE. Uniform bounds and ference Series in Mathematics, No. 24. MR0371879 (51 symbolic powers on smooth varieties, Invent. Math., no. 2 #8096) (144):241–252, 2001. MR1826369 (2002b:13001) [Hoc83] Hochster M. Canonical elements in local cohomol- [EV92] Esnault H, Viehweg E. Lectures on vanishing theorems, ogy modules and the direct summand conjecture, J. Alge- DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. bra, no. 2 (84):503–553, 1983. MR723406 MR1193913 (94a:14017) [Hoc94] Hochster M. Solid closure. Commutative algebra: syzy- [EG81] Evans E, Griffith P. The syzygy problem, Ann. of Math. gies, multiplicities, and birational algebra (South Hadley, MA, (2), no. 2 (114):323–333, 1981. MR632842 1992): Amer. Math. Soc., Providence, RI; 1994:103–172. [Fal88] Faltings G. 푝-adic Hodge theory, J. Amer. Math. Soc., MR1266182 no. 1 (1):255–299, 1988. MR924705 [Hoc97] Hochster M. Nonnegativity of intersection [Fal02] Faltings G. Almost étale extensions, Astérisque multiplicities in ramified regular local rings follow- (279):185–270, 2002. Cohomologies 푝-adiques et ing Gabber/De Jong/Berthelot, 1997. available at: applications arithmétiques, II. MR1922831 http://www.math.lsa.umich.edu/∼hochster [Gab18] Gabber O. Observations made after the /mult.ps, unpublished notes. MSRI workshop on homological conjectures, 2018. [Hoc02] Hochster M. Big Cohen–Macaulay algebras in di- https://docs.google.com/viewer?url= mension three via Heitmann’s theorem, J. Algebra, no. 2 https://www.msri.org/workshops/842 (254):395–408, 2002. /schedules/23854/documents/3322/assets [Hoc04] Hochster M. Current state of the homo- /31362. logical conjectures: Five talks of a VIGRE-funded [GR03] Gabber O, Ramero L. Almost ring theory, Lecture Notes minicourse at the University of Utah, 2004. avail- in Mathematics, vol. 1800, Springer–Verlag, Berlin, 2003. able at: http://www.math.utah.edu/vigre [GR04] Gabber O, Ramero L. Foundations for almost ring /minicourses/algebra/hochster.pdf. theory, 2004. arXiv:0409584. [Hoc17] Hochster M. Homological conjectures and lim [GR70] Grauert H, Riemenschneider O. Verschwin- Cohen–Macaulay sequences. Homological and computa- dungssätze für analytische Kohomologiegruppen auf tional methods in commutative algebra: Springer, Cham; komplexen Räumen, Invent. Math. (11):263–292, 1970. 2017:173–197. MR3751886 MR0302938 (46 #2081) [HH90] Hochster M, Huneke C. Tight closure, invariant the- [Har98] Hara N. A characterization of rational singularities ory, and the Briançon–Skoda theorem, J. Amer. Math. Soc., in terms of injectivity of Frobenius maps, Amer. J. Math., no. 1 (3):31–116, 1990. MR1017784 (91g:13010) no. 5 (120):981–996, 1998. MR1646049 (99h:13005) [HH92] Hochster M, Huneke C. Infinite integral extensions [HW02] Hara N, Watanabe K-I. F-regular and F-pure rings and big Cohen–Macaulay algebras, Ann. of Math. (2), no. 1 vs. log terminal and log canonical singularities, J. Al- (135):53–89, 1992. MR1147957 (92m:13023) gebraic Geom., no. 2 (11):363–392, 2002. MR1874118 [HH94] Hochster M, Huneke C. 퐹-regularity, test elements, (2002k:13009) and smooth base change, Trans. Amer. Math. Soc., no. 1 [Har66] Hartshorne R. Residues and duality, Lecture notes of (346):1–62, 1994. MR1273534 (95d:13007) a seminar on the work of A. Grothendieck, given at Har- [HH95] Hochster M, Huneke C. Applications of the exis- vard 1963/64. With an appendix by P. Deligne. Lecture tence of big Cohen–Macaulay algebras, Adv. Math., no. 1 Notes in Mathematics, No. 20, Springer–Verlag, Berlin, (113):45–117, 1995. MR1332808 1966. MR0222093 (36 #5145) [HH02] Hochster M, Huneke C. Comparison of symbolic [Har79] Hartshorne R. Algebraic vector bundles on projec- and ordinary powers of ideals, Invent. Math., no. 2 tive spaces: a problem list, Topology, no. 2 (18):117–128, (147):349–369, 2002. MR1881923 (2002m:13002) 1979. MR544153 [HKV09] Huneke C, Katz D, Validashti J. Uniform equiva- [HM18] Heitmann R, Ma L. Big Cohen–Macaulay algebras lence of symbolic and adic topologies, Illinois J. Math., and the vanishing conjecture for maps of Tor in mixed char- no. 1 (53):325–338, 2009. MR2584949 acteristic, Algebra Number Theory, no. 7 (12):1659–1674, [Kaw82] Kawamata Y. A generalization of Kodaira- 2018. MR3871506 Ramanujam’s vanishing theorem, Math. Ann., no. 1 [Hei02] Heitmann RC. The direct summand conjecture in di- (261):43–46, 1982. MR675204 (84i:14022) mension three, Ann. of Math. (2), no. 2 (156):695–712, [KL15] Kedlaya KS, Liu R. Relative 푝-adic Hodge theory: foun- 2002. MR1933722 (2003m:13008) dations, Astérisque (371):239, 2015. MR3379653 [Hoc73] Hochster M. Contracted ideals from integral exten- [Kov00] Kovács SJ. A characterization of rational singu- sions of regular rings, Nagoya Math. J. (51):25–43, 1973. larities, Duke Math. J., no. 2 (102):187–191, 2000. MR0349656 (50 #2149) MR1749436 (2002b:14005) [Hoc75] Hochster M. Topics in the homological theory of mod- [Laz04] Lazarsfeld R. Positivity in algebraic geometry. II, Ergeb- ules over commutative rings, Published for the Conference nisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Board of the Mathematical Sciences by the American Math- Series of Modern Surveys in Mathematics [Results in Math- ematical Society, Providence, R.I., 1975. Expository lec- ematics and Related Areas. 3rd Series. A Series of Modern tures from the CBMS Regional Conference held at the Uni- Surveys in Mathematics], vol. 49, Springer–Verlag, Berlin, versity of Nebraska, Lincoln, Neb., June 24–28, 1974, Con- 2004. Positivity for vector bundles, and multiplier ideals. ference Board of the Mathematical Sciences Regional Con- MR2095472 (2005k:14001b)

830 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 [MS18a] Ma L, Schwede K. Perfectoid multiplier/test ideals Credits in regular rings and bounds on symbolic powers, Invent. Photo of Melvin Hochster is courtesy of Melvin Hochster. Math., no. 2 (214):913–955, 2018. MR3867632 Photo of Yves Andréis courtesy of Yves André. [MS18b] Ma L, Schwede K. Singularities in mixed char- Photo of Ofer Gabber is courtesy of Renate Schmid, Copyright acteristic via big Cohen–Macaulay algebras, 2018. is with MFO [CC BY-SA 2.0 de (https:// arXiv:1806.09567. creativecommons.org/licenses/by-sa/2.0 [MS97] Mehta V, Srinivas V. A characterization of rational /de/deed.en)]. singularities, Asian J. Math., no. 2 (1):249–271, 1997. Photo of Paul C. Roberts is courtesy of Paul C. Roberts. MR1491985 (99e:13009) [PS73] Peskine C, Szpiro L. Dimension projective finie et Photo of Karen E. Smith is ©Eric Bronson, Michigan Photog- cohomologie locale. Applications à la démonstrationde raphy. conjectures de M. Auslander, H. Bass et A. Grothendieck, Photo of Kei-ichi Watanabe is courtesy of Ernst Kunz. Inst. Hautes Etudes´ Sci. Publ. Math. (42):47–119, 1973. Photo of Bhargav Bhatt is courtesy of Bhargav Bhatt. MR0374130 (51 #10330) Photo of Craig Huneke is courtesy of MFO [CC BY-SA 2.0 de [Rob80] Roberts P. Cohen–Macaulay complexes and an an- (https://creativecommons.org/licenses/by- alytic proof of the new intersection conjecture, J. Algebra, sa/2.0/de/deed.en)]. no. 1 (66):220–225, 1980. MR591254 Photo of Raymond C. Heitmann is ©Jürgen Herzog. [Rob87] Roberts P. Le théorèmed’intersection, C. R. Acad. Sci. Photo of Linquan Ma is courtesy of Linquan Ma. Paris Sér. I Math., no. 7 (304):177–180, 1987. MR880574 Photo of Karl Schwede is courtesy of Karl Schwede. [Rob89] Roberts P. Intersection theorems. Commutative algebra (Berkeley, CA, 1987): Springer, New York; 1989:417– 436. MR1015532 [Sch12] Scholze P. Perfectoid spaces, Publ. Math. Inst. Hautes Etudes´ Sci. (116):245–313, 2012. http://dx.doi.org/ 10.1007/s10240-012-0042-x. MR3090258 [Shi17] Shimomoto K. Integral perfectoid big Cohen– Macaulay algebras via André’s theorem, 2017. arXiv:1706.06946, to appear in Math. Ann. [Sil09] Silverman JH. The arithmetic of elliptic curves, Second, Graduate Texts in Mathematics, vol. 106, Springer, Dor- drecht, 2009. MR2514094 (2010i:11005) 퐐 [Sin99] Singh AK. -Gorenstein splinter rings of characteris- Applications are invited for:- tic 푝 are F-regular, Math. Proc. Cambridge Philos. Soc., no. 2 (127):201–205, 1999. MR1735920 (2000j:13006) Department of Mathematics [Smi94] Smith K. Tight closure of parameter ideals, Invent. Research Assistant Professor Math., no. 1 (115):41–60, 1994. MR1248078 (94k:13006) (Ref. 190000JX) (Closing date: June 30, 2019) [Smi97] Smith KE. 퐹-rational rings have rational singu- The Department of Mathematics in CUHK has developed a strong larities, Amer. J. Math., no. 1 (119):159–180, 1997. reputation in teaching and research. Many of the faculty members are internationally renowned and are recipients of prestigious awards MR1428062 (97k:13004) and honours. The graduates are successful in both academia and [Swa00] Swanson I. Linear equivalence of ideal topologies, industry. The Department is highly ranked internationally, ranking Math. Z., no. 4 (234):755–775, 2000. MR1778408 28th in the QS World University Rankings and 31st in the US [Vie82] Viehweg E. Vanishing theorems, J. Reine Angew. Math. News Rankings. (335):1–8, 1982. MR667459 (83m:14011) The Department is now inviting applications for the position of Research Assistant Professor in all areas of Mathematics. Applicants should have a relevant PhD degree and good potential for research and teaching. Appointment will initially be made on contract basis for up to three years commencing August 2019, renewable subject to mutual agreement. Applications will be considered on a continuing basis until the position is fi lled. Application Procedure The University only accepts and considers applications submitted online for the post above. For more information and to apply online, please visit http://career.cuhk.edu.hk.

JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 831