2026 Theses Doctoral

The Miracle of Flatness in Algebraic Geometry

Zelich, Ivan

Algebraic Geometry was revolutionized in the 1960s by many incredible techniques introduced by Grothendieck and his collaborators. Arguably the most fundamental machinery was the introduction of sheaf cohomology, which in turn relied on the developments of homotopical algebra. Here, the notion of flatness becomes a crucial technical condition, especially when considering the interaction between base-change and cohomology, where the flatness condition is imposed on a morphism of schemes 𝑓: 𝑋 → 𝑌 to rid 𝑋 of any pathological behavior for the resulting family of fibers 𝑋_𝑦 for 𝑦 in 𝑌. It would be unreasonable, however, to expect that the flatness condition on 𝑓 is generally sufficient to guarantee good behavior of the family of fibers 𝑋_y, because flatness is a module-theoretic condition that can generally exhibit quite strange behavior without additional finiteness hypotheses. Broadly speaking, the pursuit of understanding this dichotomy is an underlying theme for this thesis.

Semirings, studied in Chapter 1, provided new aspects of this dichotomy to wrestle with. One of our main results is that the most natural notion of homotopy theory for semirings is not enough to capture the properties of flatness required to guarantee good base-change properties.

𝐓𝐡𝐞𝐨𝐫𝐞𝐦 𝟎.𝟎.𝟏 (See Corollary 1.2.21). 𝘍𝘰𝘳 𝘢𝘯𝘺 𝘤𝘰𝘮𝘮𝘶𝘵𝘢𝘵𝘪𝘷𝘦 𝘮𝘰𝘯𝘰𝘪𝘥 𝘔 ∈ 𝘕−𝘮𝘰𝘥, 𝘔 ⊗𝘓 𝑁 𝘡 𝘪𝘴 𝘢 𝘥𝘪𝘴𝘤𝘳𝘦𝘵𝘦 𝘮𝘰𝘥𝘶𝘭𝘦. 𝘖𝘯 𝘵𝘩𝘦 𝘰𝘵𝘩𝘦𝘳 𝘩𝘢𝘯𝘥, 𝘵𝘩𝘦 𝘴𝘦𝘮𝘪𝘳𝘪𝘯𝘨 𝘮𝘢𝘱 𝘕 → 𝘡 𝘪𝘴 𝘯𝘰𝘵 𝘧𝘭𝘢𝘵, 𝘪𝘯 𝘵𝘩𝘦 𝘴𝘦𝘯𝘴𝘦 𝘵𝘩𝘢𝘵 𝘵𝘩𝘦 𝘧𝘶𝘯𝘤𝘵𝘰𝘳 −⊗𝘕 → 𝘡 : 𝘕−𝘮𝘰𝘥 →𝘕−𝘮𝘰𝘥 𝘥𝘰𝘦𝘴 𝘯𝘰𝘵 𝘱𝘳𝘦𝘴𝘦𝘳𝘷𝘦 𝘧𝘪𝘯𝘪𝘵𝘦 𝘭𝘪𝘮𝘪𝘵𝘴 (𝘴𝘦𝘦 𝘊𝘰𝘳𝘰𝘭𝘭𝘢𝘳𝘺 1.1.3).

While this result is indeed in stark contrast to what happens in classical commutative algebra, where flatness of a ring map 𝑅 → 𝑆 can be detected via the derived functors of −⊗L/𝑅 𝑆, the semiring situation does bear some similarities to the classical situation.

𝐓𝐡𝐞𝐨𝐫𝐞𝐦 𝟎.𝟎.𝟐 (See Theorem 1.3.10). 𝘈 𝘧𝘭𝘢𝘵, 𝘧𝘪𝘯𝘪𝘵𝘦𝘭𝘺 𝘱𝘳𝘦𝘴𝘦𝘯𝘵𝘦𝘥 𝘦𝘱𝘪𝘮𝘰𝘳𝘱𝘩𝘪𝘴𝘮 𝘰𝘧 𝘴𝘦𝘮𝘪𝘳𝘪𝘯𝘨𝘴 𝑓: 𝐴 → 𝐵 𝘪𝘯𝘥𝘶𝘤𝘦𝘴 𝘢𝘯 𝘪𝘴𝘰𝘮𝘰𝘳𝘱𝘩𝘪𝘴𝘮 𝘰𝘧 𝘚𝘱𝘦𝘤(𝘉) 𝘰𝘯𝘵𝘰 𝘢 𝘲𝘶𝘢𝘴𝘪-𝘤𝘰𝘮𝘱𝘢𝘤𝘵 𝘰𝘱𝘦𝘯 𝘰𝘧 𝘚𝘱𝘦𝘤(𝐴) 𝘪𝘯 𝘵𝘩𝘦 𝘡𝘢𝘳𝘪𝘴𝘬𝘪-𝘵𝘰𝘱𝘰𝘭𝘰𝘨𝘺.

We were informed after having proved Theorem 1.3.10 that Florian Murty obtained the same result in greater generality via a different method.

Our main interest in studying the flatness of semirings is that here the notion seems to capture some strong positivity properties of semirings. The following is a question posed to the author by James Borger.

𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧 𝟎.𝟎.𝟑. Let 𝐴 be a finitely presented flat ℝ₊-algebra. If 𝐴 is non-trivial, does it necessarily have an ℝ-point (or even better, an ℝ₊-point)?

If we asked the above question for the groupification ℝ of ℝ₊ then the answer would be no, as there are many non-trivial flat and finitely presented ℝ-algebras that do not have any ℝ-points. In our opinion, the question seems to suggest that because ℝ₊ is a totally ordered semiring with no elements admitting additive inverses, the flatness condition on 𝐴 is strong enough to guarantee the existence of a ℝ₊-point. We attempted to address this question by understanding when quotients of the form ℝ₊[𝓍₁,...,𝓍ₙ]/(𝑓 ∼ 𝑔) are flat over ℝ₊, but we were not able to obtain much progress. In private communication, Johan de Jong informed the author of the following result.

𝐓𝐡𝐞𝐨𝐫𝐞𝐦 𝟎.𝟎.𝟒 [de Jong] 𝘛𝘩𝘦 𝘴𝘦𝘮𝘪𝘳𝘪𝘯𝘨 ℝ}₊[𝓍]/(𝑎𝓍 ∼ 𝓍² + 1) 𝘪𝘴 𝘧𝘭𝘢𝘵 𝘰𝘷𝘦𝘳 ℝ₊ 𝘪𝘧 𝘢𝘯𝘥 𝘰𝘯𝘭𝘺 𝘪𝘧 𝑎 ≥ 2.

On the other hand, in Chapter 2, we will show that from the perspective of 𝘥𝘦𝘴𝘤𝘦𝘯𝘥𝘢𝘣𝘪𝘭𝘪𝘵𝘺, flat ring maps between classical rings can behave arbitrarily badly. The notion of a descendable morphism is a technical condition introduced by Akhil Mathew building upon the work of Paul Balmer. Informally, if a map 𝑓: 𝐴 → 𝐵 is descendable, then 𝐴-linear stable ∞-categories can be understood via 𝐵-linear stable ∞-categories together with `descent' data. Our main result is that faithfully flat ring maps are not necessarily descendable; see Corollary 2.2.8, Theorem 2.3.10, and
the work of Aoki [3]. This answers a question by Bhatt and Scholze ([4, 11.24]), Akhil Mathew ([20, Prop. 3.32]), and Jacob Lurie ([5, D.3.3.4]). Our strategy was to first understand the relationship between non-vanishing cup-products and cardinality in the module-theoretic setting.

𝐓𝐡𝐞𝐨𝐫𝐞𝐦 𝟎.𝟎.𝟓 (See Theorem 2.1.4). 𝘍𝘰𝘳 𝘢 𝘤𝘰𝘮𝘮𝘶𝘵𝘢𝘵𝘪𝘷𝘦 𝘳𝘪𝘯𝘨 𝘙, 𝘪𝘧 𝘙 𝘤𝘰𝘯𝘵𝘢𝘪𝘯𝘴 𝘢𝘯 𝘯-𝘪𝘯𝘥𝘪𝘷𝘪𝘴𝘪𝘣𝘭𝘦 𝘴𝘦𝘲𝘶𝘦𝘯𝘤𝘦 (𝘴𝘦𝘦 𝘋𝘦𝘧𝘪𝘯𝘪𝘵𝘪𝘰𝘯 2.1.1), 𝘵𝘩𝘦𝘯 𝘵𝘩𝘦𝘳𝘦 𝘦𝘹𝘪𝘴𝘵𝘴 𝘢 𝘧𝘭𝘢𝘵 𝘮𝘰𝘥𝘶𝘭𝘦 𝘔 𝘰𝘷𝘦𝘳 𝘙 𝘢𝘯𝘥 𝘢 𝘤𝘭𝘢𝘴𝘴 η∈𝘌𝘹𝘵1 𝘙(𝘔,𝘙′) 𝘴𝘶𝘤𝘩 𝘵𝘩𝘢𝘵 𝜂⊗𝐿 𝑅𝑛 ∈𝐸𝑥𝑡𝑛 𝑅(𝑀⊗𝐿 𝑅𝑛,𝑅′⊗𝐿 𝑅𝑛) ̸= 0 𝘸𝘩𝘦𝘳𝘦 𝑅 ′ 𝘪𝘴 𝘢 𝘧𝘳𝘦𝘦 𝑅 -𝘮𝘰𝘥𝘶𝘭𝘦.

We then form a quite general method to convert module theoretic non-vanishing cup-products to non-descendable ring maps, and consequently obtain many examples of non-descendable faithfully flat ring maps (See Corollary 2.2.8). Our method in particular allows us to construct examples between p-boolean rings, which was a question posed to the author by Juan Esteban Rodríguez Camargo. By slightly refining our argument, we were also able to construct a faithfully flat cover of Spec 𝑘[𝓍₁,𝓍₂,...], where 𝑘 is an algebraically closed field, that is not descendable (see Theorem 2.3.10). This example is quite striking, as Spec 𝑘[𝓍₁,𝓍₂,...], while not Noetherian, is still a reasonably nice scheme from the perspective of algebraic geometry.

To wrap up this thesis, we homed in on a conjecture posed by Grothendieck and Dieudonné. In [6, 21.12.14],, the authors conjectured that for a locally of finite type map 𝑓: 𝑋 → 𝑌 of excellent, locally Noetherian schemes, with 𝑋 normal and 𝑌 regular, the ramification locus of 𝑓 is pure of codimension 1 (See Theorem 3.0.1). Having already shown that for any open subscheme 𝑉 ⊂ 𝑋 such that 𝑉 → 𝑋 is an affine morphism of schemes, 𝑋 \ 𝑉 is pure of codimension 1 ([6, 21.12.7]), the authors then conjectured that if 𝑉 is the maximal open subscheme 𝑉 ⊂ 𝑋 where 𝑓 is unramified (and hence étale), then 𝑉 → 𝑋 is an affine morphism of schemes (v) [6, 21.12.14]). The main goal of this chapter was to answer their conjecture in the affirmative, see Theorem 3.0.3.. We will in particular demonstrate the following:

𝐓𝐡𝐞𝐨𝐫𝐞𝐦 𝟎.𝟎.𝟔 (See Theorem 3.3.1).𝘓𝘦𝘵 (𝐴,𝖒_𝐴,𝑘) 𝘣𝘦 𝘢 𝘳𝘦𝘨𝘶𝘭𝘢𝘳 𝘭𝘰𝘤𝘢𝘭 𝘳𝘪𝘯𝘨 𝘢𝘯𝘥 𝑉 → Spec 𝐴 𝘢𝘯 é𝘵𝘢𝘭𝘦 𝘮𝘰𝘳𝘱𝘩𝘪𝘴𝘮 𝘵𝘩𝘢𝘵 𝘪𝘴 𝘤𝘰𝘩𝘰𝘮𝘰𝘭𝘰𝘨𝘪𝘤𝘢𝘭𝘭𝘺 𝘱𝘶𝘳𝘦 𝘪𝘯 𝘤𝘰𝘥𝘪𝘮𝘦𝘯𝘴𝘪𝘰𝘯 1. 𝘛𝘩𝘦𝘯 𝑉 𝘪𝘴 𝘢𝘯 𝘢𝘧𝘧𝘪𝘯𝘦 𝘴𝘤𝘩𝘦𝘮𝘦.

Our definition of ‘cohomologically pure in codimension 1’ (see Definition 3.0.8) could be seen as a cohomological analogue of the situation that 𝑉 arises as an open subscheme of Spec 𝐵, for a normal local ring (𝐵,𝖒_𝐵) that is finite over 𝐴, such that there is a factoring 𝑉 →Spec 𝐵 \{𝖒_𝐵} → Spec 𝐵 where the first map is affine morphism of schemes. We only realized later that the authors in [6, 21.12] were interested in characterizing rings 𝐵 for which such a V is then forced to be an affine scheme, and consequently formulated a conjecture (iv) [6, 21.12.14].

If their conjecture was true, then as noted in (v) [6, 21.12.14], the affineness of the maximal étale locus would be equivalent to the purity of the ramification locus. By essentially paraphrasing the proof of purity of the ramification locus given in [7, 0ECD], Theorem 3.0.3 would then follow by induction on the dimension of Y, where the case of dimension 2 would crucially use the 𝘮𝘪𝘳𝘢𝘤𝘭𝘦 𝘧𝘭𝘢𝘵𝘯𝘦𝘴𝘴 𝘵𝘩𝘦𝘰𝘳𝘦𝘮 (see [7, 00R4]), from which this thesis derives its name.