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).