In this section, we focus on Section 2 in [Sch], following [Hu], [Hu1], and [Hu2]. Moreover, we need to compare Huber's adic spaces with Berkovich's analytic spaces and Tate's rigid analytic spaces. Hence, we will briefly introduce the notion of Berkovich's analytic spaces in §1.3 and the notion of rigid analytic varieties in §1.4.§1. Adic SpacesDefinition 1.1.  A morphism $f:X\rightarrow Y$ of adic spaces is adic if, for every $x\in X$, there exist open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that the ring homomorphism $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ of $f$-adic rings is adic.§1.1. Morphisms of finite type. The material can be seen in [SP] and [Hu1].First, we review the definition of morphisms of schemes of finite type/presentation (see [SP], Definition 29.15.1, Lemma 29.15.2, and Definition 29.21.1, and Lemma 29.21.2).Definition 1.2. Let $f:X\rightarrow Y$ be a morphism of schemes.We say that $f$ is locally of finite type if, for all affine opens $U,V$ of $X,Y$ with $f(U)\subset V$, the ring map $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ is of finite type.We say that $f$ is of finite type if it is quasi-compact and locally of finite type.We say that $f$ is locally of finite presentation if, for all affine opens $U,V$ of $X,Y$ with $f(U)\subset V$, the ring map $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ is of finite presentation.We say that $f$ is of finite presentation if it is quasi-compact, quasi-separated, and locally of finite presentation. Compared with the above definition, we reach to the case of adic spaces.Definition 1.3 ([Hu1, Definition 1.2.1]). Let $f:X\rightarrow Y$ be a morphism of adic spaces.We say that $f$ is locally of finite type if, for every $x\in X$, there exists open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that the ring homomorphism $(\mathscr{O}_{Y}(V),\mathscr{O}^{+}_{Y}(V))\rightarrow(\mathscr{O}_{X}(U),\mathscr{O}^{+}_{X}(U))$ of affinoid rings is topologically of finite type.We say that $f$ is of finite type if it is quasi-compact and locally of finite type.We say that $f$ is locally of finite presentation if, for every $x\in X$, there exists open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that the ring homomorphism $(\mathscr{O}_{Y}(V),\mathscr{O}^{+}_{Y}(V))\rightarrow(\mathscr{O}_{X}(U),\mathscr{O}^{+}_{X}(U))$ of affinoid rings is topologically of finite type and, if the topology of $\mathscr{O}_{Y}(V)$ is discrete, the ring map $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ is of finite presentation.Then $\{\textrm{morphisms locally of finite presentation}\}\subset\{\textrm{morphisms locally of finite type}\}\subset\{\textrm{adic}\newline\textrm{morphisms}\}$.§1.2. Unramified, smooth, and étale morphisms.For definitions of morphisms of finite type and finite presentation, see §1.1.First, we review the notions of unramified, smooth, and étale ring maps (see [SP], 10.138, 10.148, and 10.150, and 10.151).Definition 1.4. Let $R\rightarrow S$ be a ring map. We say $R\rightarrow S$ is formally smooth/formally unramified/formally étale or $S$ is formally smooth/formally unramified/formally étale over $R$ if for every solid commutative diagramwhere $I\subset A$ is a square zero ideal, there exists at least one/at most one/a unique dotted map $S\rightarrow A$ making the diagram commute.The definitions of smooth and étale ring maps make use of the naive cotangent complex, but we will simplify this.Definition 1.5. Let $R\rightarrow S$ be a ring map.We say $R\rightarrow S$ is smooth/étale or $S$ is smooth/étale over $R$ if $R\rightarrow S$ is of finite presentation and formally smooth/formally étale.We say $R\rightarrow S$ is unramified or $S$ is unramified over $R$ if $R\rightarrow S$ is of finite type and formally unramified.Compared with the definitions above, we reach to the case of adic spaces via changing some arrows.Definition 1.6 ([Hu1, Definition 1.6.5]). A morphism $f:X\rightarrow Y$ of adic spaces is unramified/smooth/étale if $f$ is locally of finite type/locally of finite presentation/locally of finite presentation and if, for any affinoid ring $A$, any ideal $I\subset A^{\vartriangleright}$ with $I^{2}=0$, and any morphism ${\rm{Spa}}(A)\rightarrow Y$, the map ${\rm{Hom}}_{Y}({\rm{Spa}}(A),X)\rightarrow{\rm{Hom}}_{Y}({\rm{Spa}}(A/I),X)$ is injective/surjective/bijective.A morphism $f:X\rightarrow Y$ of adic spaces is unramified/smooth/étale at a point $x\in X$ if there exist open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that $f|_{U}:U\rightarrow V$ is unramified/smooth/étale.Note that the second statement of (i) above can be described as follows. For every solid commutative diagram in the following, there exist at most one/at least one/a unique one dotted map making the diagram commute.§1.3. Berkovich’s analytic spaces.We will introduce the notion of Berkovich's analytic spaces following [Ber] and [Ber1]. Berkovich's analytic spaces is one of the non-archimedean analogues of complex analytic spaces. The definition of analytic spaces in [Ber1] is more general than the definition in [Ber] (the analytic spaces in [Ber] corresponds to the good analytic spaces in [Ber1]). So we will make use of the definition in [Ber1].§1.3.1 Underlying topological spaces.First, we introduce some structures on topological spaces for further use (see [Ber1, §1, 1.1]). All compact, locally compact, and paracompact spaces are assumed to be Hausdorff.Definition 1.7.A topological space is paracompact if it is Hausdorff and every open cover of it admits a locally finite refinement.A topological space $X$ is locally Hausdorff if every point $x\in X$ admits an open Hausdorff neighborhood.Remark 1.8. Note that in [Tam], a paracompact space also requires that the locally finite refinement in (i) above is an open cover.Let $X$ be a topological space and let $\tau$ be a collection of subsets of $X$ provided with the induced topology. We put $\tau|_{Y}:=\{V\in\tau;V\subset Y\}$ for any subset $Y\subset X$.Definition 1.9. We say that the collection $\tau$ above is a quasi-net on $X$ if, for every point $x\in X$, there exist $V_{1},...,V_{n}\in\tau$ such that $x\in V_{1}\cap\cdot\cdot\cdot\cap V_{n}$ and the set $V_{1}\cup\cdot\cdot\cdot\cup V_{n}$ is a neighborhood of $x$, i.e. $V_{1}\cup\cdot\cdot\cdot\cup V_{n}$ contains an open set $U\subset X$ with $x\in U$. Furthermore, $\tau$ is said to be a {\rm{net on $X$}} if it is a quasi-net and, for any $U,V\in\tau$, $\tau|_{U\cap V}$ is a quasi-net on $U\cap V$.Definition 1.10 ([Dug, p255]). Let $X$ be a topological space and $S\subset X$ be a subset. $S$ is said to be locally closed if every point $s\in S$ has a neighborhood $U$ such that $S\cap U$ is closed in $U$.§1.3.2 The category of analytic spaces.Throughout, we fix a nonarchimedean field $k$ whose valuation can be trivial. The category of $k$-affinoid spaces is dual to the category of $k$-affinoid algebras (see [Ber, §2.1]). The $k$-affinoid spaces associated with a $k$-affinoid algebra $\mathscr{A}$ is denoted by $X:=\mathscr{M}(\mathscr{A})$.If for each nonarchimedean field $K$ over $k$, we are given a class $\Phi_{K}$ of $K$-affinoid spaces, the system $\Phi=\{\Phi_{K}\}$ is assumed to satisfy the following conditions:(i) $\mathscr{M}(K)\in\Phi_{K}$.(ii) $\Phi_{K}$ is stable under isomorphisms and direct products. In other words, for $X\in\Phi_{K}$, if $X'$ is a $K$-affinoid space with $X\cong X'$, then we have $X'\in\Phi_{K}$, and for $X,Y\in\Phi_{K}$, we have $X\times Y\in\Phi_{K}$.(iii) If $\varphi:Y\rightarrow X$ is a finite morphism of $K$-affinoid spaces with $X\in\Phi_{K}$, then $Y\in\Phi_{K}$.(iv) If $(V_{i})_{i\in I}$ is a finite affinoid covering of a $K$-affinoid space $X$ with $V_{i}\in\Phi_{K}$, then $X\in\Phi_{K}$.(v) If $K\hookrightarrow L$ is an isometric embedding of nonarchimedean fields over $k$, then for any $X\in\Phi_{K}$, one has $X{\widehat{\otimes}_{K}L}\in\Phi_{L}$.Definition 1.11. The class $\Phi_{K}$ is said to be dense if each point of each $X\in\Phi_{K}$ admits a fundamental system of affinoid neighborhoods $V\in\Phi_{K}$. The system $\Phi$ is said to be dense if all $\Phi_{K}$ are dense.The affinoid spaces from $\Phi_{K}$ (resp. $\Phi$) and their affinoid algebras will be called $\Phi_{K}$-affinoid (resp. $\Phi$-affinoid).From (ii) and (iii) above, we deduce that $\Phi_{K}$ is stable under fiber products. In other words, for $X,Y,Z\in\Phi_{K}$ with morphisms $X\rightarrow Z$ and $Y\rightarrow Z$, we have $X\times_{Z}Y\in\Phi_{K}$.Let $X$ be a locally Hausdorff space and let $\tau$ be a net of compact subsets on $X$.Definition 1.12. A $\Phi_{K}$-atlas $\mathscr{A}$ on $X$ with the net $\tau$ is a map that assigns, to each $U\in\tau$, a $\Phi_{K}$-affinoid algebra $\mathscr{A}_{U}$ together with a homeomorphism $U\xrightarrow{\sim}\mathscr{M}(\mathscr{A}_{U})$ and, to each pair $U,V\in\tau$ with $U\subset V$, a bounded homomorphism $\mathscr{A}_{V}\rightarrow\mathscr{A}_{U}$ of $\Phi_{K}$-affinoid algebras that identifies $(U,\mathscr{A}_{U})$ with an affinoid domain in $(V,\mathscr{A}_{V})$.Definition 1.13. A triple $(X,\mathscr{A},\tau)$ of the above form is said to be a $\Phi_{K}$-analytic space.§1.4. Rigid analytic varieties.The notion of rigid analytic variety is also one of the nonarchimedean analogues of complex analytic space. It originated in John Tate's thesis, [Tat]. In this subsection, we briefly introduce it following [BGR] and [BS].§1.4.1 $G$-topological spaces. As a technical trick, we generalize the usual topology to the so-called Grothendieck topology, [SGA4]. Roughly speaking, a $G$-topological space is a set that admits a Grothendieck topology. We will first introduce Grothendieck topology following the definition in [BS], where the "Grothendieck topology" means the "Grothendieck pretopology" in [SGA4].Definition 1.14. Let $\mathscr{C}$ be a (small) category. A Grothendieck topology $T$ consists of the category ${\rm{Cat}}(T)=\mathscr{C}$ and a set ${\rm{Cov}}(T)$ of families $(U_{i}\rightarrow U)_{i\in I}$ of morphisms in $\mathscr{C}$, called open coverings, such that the following axioms are satisfied:If $U'\rightarrow U$ is an isomorphism in $\mathscr{C}$, then the one-element family $(U'\rightarrow U)\in{\rm{Cov}}(T)$.If $(U_{i}\rightarrow U)_{i\in I}$ and $(V_{ij}\rightarrow U_{i})_{j\in I}$ are open coverings, then $(V_{ij}\rightarrow U)_{i,j\in I}\in{\rm{Cov}}(T)$.If $(U_{i}\rightarrow U)_{i\in I}$ is an open covering and $V\rightarrow U$ is a morphism in $\mathscr{C}$, then the fiber products $V\times_{U}U_{i}$ exist in $\mathscr{C}$ and $(V\times_{U}U_{i}\rightarrow V)_{i\in I}\in{\rm{Cov}}(T)$.Remark 1.15. Note that this is slightly different to the definition in [Poon], which requires that a Grothendieck topology consists of the set ${\rm{Cov}}(T)$ only. Moreover, the pair $(\mathscr{C},T)$ is usually called a site. However, to suite our needs in rigid geometry, we stick with the terminology in [BS].We specialize the definition above to the case that is more suited to our needs. And from now on, we will exclusively consider the Grothendieck topology of such a special type, unless explicitly stated otherwise.Definition 1.16. Let $X$ be a set. A Grothendieck topology (also called $G$-topology) $\mathfrak{T}$ on $X$ consists ofa category of subsets of $X$, called admissible open subsets or $\mathfrak{T}$-open subsets of $X$, with inclusions as morphisms, anda set ${\rm{Cov}}(\mathfrak{T})$ of families $(U_{i}\rightarrow U)_{i\in I}$ of inclusions with $\bigcup_{i\in I}U_{i}=U$, called admissible coverings or $\mathfrak{T}$-coverings.Remark 1.17. Note that in this case, the fiber products will come as intersections of sets.We call $X$ a $G$-topological space and write more explicitly as $X_{\mathfrak{T}}$ when $\mathfrak{T}$ is needed to be specified.§1.4.2 Presheaves and sheaves on $G$-topological spaces. The notion of Grothendieck topology defined in § 1.4.1 enables us to adapt presheaf or sheaf to such a general situation.Definition 1.18 ([BS, 5.1, Definition 2]). Let $\mathfrak{C}$ be a category and let $\mathfrak{T}$ be a Grothendieck topology in the sense of Definition 1.14. A presheaf $\mathscr{F}$ on $\mathfrak{T}$ with values in $\mathscr{C}$ is a functor $$\mathscr{F}:{\rm{Cat}}(\mathfrak{T})^{opp}\longrightarrow\mathfrak{C}.$$If $\mathfrak{C}$ is a category admitting products, then the presheaf $\mathscr{F}$ is said to be a sheaf if the sequence $$\mathscr{F}(U)\rightarrow\prod_{i\in I}\mathscr{F}(U_{i})\mathrel{\mathop{\rightrightarrows}} \prod_{i,j\in I}\mathscr{F}(U_{i}\times_{U}U_{j})$$ is exact for any open covering $(U_{i}\rightarrow U)_{i\in I}$ in ${\rm{Cov}}(\mathfrak{T})$.Remark 1.19. Note that the definition of Grothendieck topology assures the existence of the fiber products $U_{i}\times_{U}U_{j}$ in $\textrm{Cat}(\mathfrak{T})$.Morphisms of presheaves or sheaves are just natural transformations of functors.Definition 1.20. A morphism of presheaves $f:\mathscr{F}\rightarrow\mathscr{G}$ is a morphism of functors from $\mathscr{F}$ to $\mathscr{G}$. A morphism of sheaves $f:\mathscr{F}\rightarrow\mathscr{G}$ is a morphism of presheaves $f:\mathscr{F}\rightarrow\mathscr{G}$.Hence, we can define presheaves and sheaves on a $G$-topological space.Definition 1.21 ([BGR, 9.2.1, Definition 1]). A presheaf $\mathscr{F}$ with values in a category $\mathscr{C}$ on a $G$-topological space $X$ is a contravariant functor $$\mathscr{F}:{\rm{Cat}}(\mathfrak{T})\longrightarrow\mathscr{C},$$ where $\mathfrak{T}$ is a Grothendieck topology on $X$. If $\mathscr{C}$ is a category admitting products, then $\mathscr{F}$ is a sheaf on the $G$-topological space $X$ if it is a sheaf in the sense of Definition 1.18.The following kind of Grothendieck topology is of special interest to us.Definition/Proposition 1.22 ([BGR, §5.1, Proposition 5]). Let $K$ be a field and let $X$ be an affinoid $K$-space. Then the strong Grothendieck topology on $X$ is a Grothendieck topology on $X$ that satisfies the following conditions:$(G_{0})$ $\varnothing$ and $X$ are admissible open subsets of $X$.$(G_{1})$ Let $U\subset X$ be an admissible open subset with an admissible covering $(U_{i})_{i\in I}$ and let $V\subset U$ a subset. If $U_{i}\cap V$ is admissible open in $X$ for each $i\in I$, then $V$ is admissible open in $X$.$(G_{2})$ If $\mathfrak{U}=(U_{i})_{i\in I}$ is a covering of an admissible open $U\subset X$ with an admissible refinement such that each $U_{i}$ is admissible open in $X$, then $\mathfrak{U}$ is an admissible covering of $U$.§1.4.3 Locally $G$-ringed spaces and analytic varieties.The definition of rigid analytic varieties makes use of the notion of locally $G$-ringed spaces. The so-called $G$-ringed spaces are analogous to our familiar ringed spaces.Definition 1.23 ([BGR, §9.1.1]). A $G$-ringed space is a pair $(X,\mathscr{O}_{X})$ consisting of a $G$-topological space $X$ and a sheaf $\mathscr{O}_{X}$ of rings on $X$, called the structure sheaf of $X$. A locally $G$-ringed space is a $G$-ringed space $(X,\mathscr{O}_{X})$ such that all stalks $\mathscr{O}_{X,x},x\in X$, are local rings. If the structure sheaf $\mathscr{O}_{X}$ is a sheaf of algebras over a fixed ring $R$, then such a $G$-ringed space $(X,\mathscr{O}_{X})$ is said to be over $R$.Definition 1.24 ([BGR, §9.1.1]). A map $f:X\rightarrow Y$ between $G$-topological spaces is said to be continuous if the following conditions are satisfied:(i) If $V\subset Y$ is an admissible subsets, then $f^{-1}(V)$ is an admissible subsets of $X$.(ii) If $(V_{i})_{i\in I}$ is an admissible covering of an admissible subset $V\subset Y$, then $(f^{-1}(V_{i}))_{i\in I}$ is an admissible covering of the admissible subset $f^{-1}(V)$.We need appropriate morphisms for $G$-ringed spaces. In fact, we have the following definitions analogous to that of morphisms of ringed spaces and locally ringed spaces.Definition 1.25 ([BGR, 9.3.1]). A morphism of $G$-ringed spaces $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ is a pair $(f,f^{*})$ where $f:X\rightarrow Y$ is a continuous map of $G$-topological spaces and $f^{*}$ is a collection $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(f^{-1}(V))$ of ring maps for any admissible open subset $V\subset Y$ that are compatible with restriction maps.A morphism of locally $G$-ringed spaces $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ is a morphism of $G$-ringed space $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ such that all induced ring maps $f^{*}_{x}:\mathscr{O}_{Y,f(x)}\rightarrow\mathscr{O}_{X,x}$ for $x\in X$ are local.Let $R$ be a fixed ring. An $R$-morphism $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ of $G$-ringed spaces over $R$ is a morphism of $G$-ringed spaces $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ such that, in addition, $f^{*}$ is a collection $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(f^{-1}(V))$ of $R$-algebra homomorphisms for all admissible open subsets $V\subset Y$.Remark 1.26. We follow the convention of ringed spaces that we denote a $G$-ringed space $(X,\mathscr{O}_{X})$ simply by $X$ and we denote a morphism of $G$-ringed spaces by suppressing the morphism of structure sheaves.In the following, let $k$ be a fixed complete nonarchimedean field. Next, we are in a position to introduce global analytic varieties.Definition 1.27 ([BGR, 9.3.1, Definition 4]). A rigid analytic variety over $k$ (also called a $k$-analytic variety) is a locally $G$-ringed space $(X,\mathscr{O}_{X})$ over $k$ such that the following axioms are verified:(i) The Grothendieck topology of $X$ satisfies properties $G_{0}$, $G_{1}$, and $G_{2}$ described in Proposition 1.22.(ii) There exists an admissible covering $(X_{i})_{i\in I}$ of $X$ with $(X_{i},\mathscr{O}_{X}|_{X_{i}})$ being a $k$-affinoid variety for each $i\in I$.§2. Almost mathematicsIn this section, we focus on Faltings' almost mathematics which first arose in his paper [Hodg], which is the first of a series works on the subject of $p$-adic Hodge theory, ending with [Falt]. The motivating point of $p$-adic Hodge theory can be traced back to Tate's classical paper [Tat1]. We will use Gabber's book [Gab] as a basic reference. The content will be useful in understanding Section 4 in Scholze's paper [Sch].References [BGR] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis. A systematic approach to rigid analyticgeometry, Grundlehren der Mathematischen Wissenschaften, Bd. 261, Springer, Berlin-Heidelberg-New York, 1984. [BS] Siegfried Bosch, Lectures on Formal and Rigid Geometry, Lect.Notes Mathematics vol. 2105, Springer, Cham, 2014. [Poon] Bjorn Poonen, Rational Points on Varieties, Graduate Studies in Mathematics Volume: 186, American Mathematical Society, 2017. [SGA4] M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Math. 269, 270, 305, Berlin-Heidelberg-New York, Springer. 1972-1973. [Gab] O. Gabber and L. Ramero, Almost ring theory, volume 1800 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003. [Hodg] G.Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255-299. [Falt] G.Faltings, Almost étale extensions, Astérisque 279 (2002), 185-270. [Tat] J. Tate, Rigid analytic spaces, Invent. Math. 12 (1971), 257-289. [Tat1] J. Tate, p-divisible groups, Proc. conf. local fields (1967), 158-183. [Dug] James Dugundji, Topology, Allyn and Bacon, Inc., 470 Atlantic Avenue, Boston, 1966. [Tam] Tammo Tom Dieck, Algebraic Topology, European Mathematical Society, 2008. [Ber] V.G. Berkovich, Spectral Theory and analytic Geometry over NonArchimedean fields, Math. Surv. Monogr. vol. 33, Am. Math. Soc., Providence, RI, 1990. [Ber1] V.G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Etud. Sci. 78 (1993). [SP] The Stacks Project Authors, Stacks Project. Available at http://math.columbia.edu/algebraic_geometry/stacks-git/. [Sch] Peter Scholze, Perfectoid Spaces, IHES Publ. math. 116 (2012), 245-313. [Hu] R. Huber, Continuous valuations, Math. Z. 212 (1993), 455-477. [Hu1] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30., Friedr. Vieweg & Sohn, Braunschweig, Springer Fachmedien Wiesbaden, 1996. [Hu2] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), 513-551.

1. Introduction代数几何是数学的核心领域，也是如今国际数学界的主流。代数几何与许多数学分支都存在广泛的联系，比如数论、微分几何、代数拓扑、复几何、表示论、同调代数、交换代数、偏微分方程等等，这些分支的发展同时也对代数几何起到促进作用。数学史上的许多重大的事件，比如，费马大定理、莫德尔猜想、韦伊猜想的证明都跟代数几何有关。同时，代数几何存在广泛的应用，比如密码学、弦理论、大数据、统计学习理论等等。代数几何之下有众多分支，比如复代数几何，热带几何，算术几何，远阿贝尔几何，$p$进霍奇理论（complex algebraic geometry, tropical geometry, arithmetic geometry, anabelian geometry, p-adic hodge theory），每个分支代表代数几何研究的一个大方向，而在每个大方向下，又有各种以不同的问题为导向的子方向。在这篇文章中，我们将会对代数几何，包括它的分支算术代数几何，做一个简短的介绍。2. An Introduction to Arithmetic Geometry算术几何是算术代数几何的简称，它是代数几何的一个分支，主要研究与数论有关的问题，比如丢番图方程。著名的费马大定理其实就是丢番图方程的一种。Definition 2.1. Diophantine equations are equations whose solutions are required to be integers.Example 2.2. The equations in Fermat's Last Theorem : $x^{n} + y^{n} = z^{n}$ for all integers $n\geq 2$ are Diophantine equations.Example 2.3. The equations $ax + by = c$ are called linear Diophantine equations.Example 2.4. The equations $x^{2} + y^{2} = z^{2}$ are called Pythagorean equations.从上可以看出椭圆曲线与丢番图方程之间存在某种联系，因此数论上的问题就可以转移到几何上的椭圆曲线进行研究。接下来，我们将给出椭圆曲线的定义，但是在此之前我们先做一些约定。我们记$K$为一个任意的域，$f(x)\in K[x]$ 为$K$上的一个三次多项式，假设这个多项式有不同的根，由于这个域并不一定是代数闭域，因此有些不同的根存在于这个域的代数闭包 $\overline{K}$上。同时，我们假设域$K$不是特征2的。Definition 2.5. The solutions to the equation $y^{2} = f(x)$ , where $x$ and $y$ are in some extension $K'$ of $K$, are called the $K'$-points of the elliptic curve defined by the equation.Example 2.6. The locus of the equations $y^{2} = x^{3} - n^{2}x$ is a special case of elliptic curve.Figure 1. Elliptic curves从上面的定义和这个例子，我们可以看出椭圆曲线的方程形式上像一个丢番图方程。事实上，当我们限定椭圆曲线方程的解为整数解时，方程就成为了丢番图方程。既然说到了椭圆曲线，我们不得不提及一下跟椭圆曲线有关联的椭圆函数。椭圆函数是19 世纪数学最光辉的成就之一，它当初是由求椭圆弧长诱导出来的，与椭圆积分也有很密切的联系，毕竟椭圆积分就是用来求椭圆弧长的。顺带一提，椭圆周长目前没有办法求精确值，其周长表达式没法表达成初等函数的形式，它只有椭圆积分表达式以及级数展开式。在定义椭圆函数之前，我们需要先定义复数域$\mathbb{C}$上的lattice。Definition 2.7. A lattice $L$ in the complex plane is the set of all integral linear combinations of two given complex numbers $\omega_{1}$ and $\omega_{2}$, where $\omega_{1}$ and $\omega_{2}$ are linear independent.Example 2.8. If we take $\omega_{1}$ = 1 and $\omega_{2}$ = $i$, we will get a lattice of Gaussian integers $\{mi+n| m , n\in \mathbb{Z}\}$.Definition 2.9. A meromorphic function on $\mathbb{C}$ is said to be an elliptic function relative to a given lattice $L$, if $f(z+l)=f(z)$ for all $l\in L$.从定义可以看出，椭圆函数是一个双周期的函数。这使人联想到实数情况的单周期函数。一个$\mathbb{R}$上的周期函数，可以看成一个圆上的函数，而一个$L$的椭圆函数则可以看成一个圆环上的函数。我们可以证得关于一个lattice 的所有椭圆函数的集合构成一个域$\mathcal{E}_{L}$，它是所有亚纯函数的域的子域，因为任意两个椭圆函数的和差积商都是椭圆函数。接下来，我们继续讨论椭圆曲线。椭圆曲线与模形式有紧密的关联，而它们之间的联系成为了证明费马大定理的关键。由于作者并不能完全看懂费马大定理的证明，因此这里不做过多阐述。我们知道当年最后完成费马大定理证明的数学家是Wiles，而Wiles在他的paper 中证明了所有有理数集上的半稳定的椭圆曲线都是modular的，从而使费马大定理成为一个推论被证明。值得一提的是，Wiles在十岁的时候在一本叫做《最后定理》的书中了解到了费马大定理，他很受震撼并打算成为第一个解决费马大定理的人，最后正如他自己所说，很多数学家用自己的一生尝试解决费马大定理都没有成功，最后只有他成功了。关于椭圆曲线、椭圆函数、模形式、费马大定理的证明，想了解更多的读者可以参考[1], [11]。讲完费马大定理，接下来我们来讲讲费马大定理背后的故事，即费马大定理之所以最后能够被Wiles证明，主要是归功于某些数学家的关键性工作。其中两位即是日本数学家Shimura 和Taniyama，他们提出的谷山—志村猜想成为了证明费马大定理的关键一步。还有一位数学大师，在讲他之前我们需要先做一些铺垫。上个世纪，算术几何中不仅仅只有费马大定理，还有韦伊猜想（有限域上的黎曼猜想）、莫德尔猜想。韦伊猜想被Deligne所证明，而莫德尔猜想被Faltings所证明。Deligne和Faltings都是如今数学界的泰斗级人物，不论是Wiles、Deligne还是Faltings ，他们的证明都离不开一个人的奠基性工作，他就是被很多人认为是20世纪最伟大的数学家Grothendieck。Grothendieck被称作代数几何的教皇，有一句很经典的描述他的话就是：“20世纪代数几何涌现了很多天才和菲尔兹奖，但是上帝只有Grothendieck一个。”Grothendieck的工作使代数几何这门古老的学科重新焕发出新的生命力，这也使代数几何进入如今的黄金时期。Grothendieck的哲学直接被数学所吸收，以至于现在数学的新人根本无法想象Grothendieck时代前这个领域的模样。从二十世纪中叶开始，整个代数几何领域越来越抽象和普遍的研究倾向，大部分都得归功于Grothendieck的影响。Grothendieck 的影响之大，几乎所有数学分支都能感受到。如今的代数几何已经是后Grothendieck时代了，代数几何涌现出了很多后起之秀，比如说日本数学家Shinichi Mochizuki、德国数学家Peter Scholze。接下来，我们继续介绍算术几何的有关内容。上文中我们提到了可以通过研究椭圆曲线和模形式，进而研究数论问题。而椭圆曲线其实只是代数曲线中的一种特殊情况，代数曲线是算术几何的一个重要研究课题。别看名字很高大上，它其实很常见，比如说在欧几里得平面上的代数曲线，就是我们用多项式方程$f(x,y) = 0$所定义的平面曲线。而想要定义一般的代数曲线就不那么简单了，这需要用到Grothendieck发展的概形的理论。在定义一般的曲线之前，我们需要不少的预备知识，因此在这里我们只做简单的描述，想要了解更多细节的读者可以参考[2]。首先，在定义概形之前，我们需要定义层的概念。我们有阿贝尔群层、环层、模层等等，取决于层所取的范畴。关于范畴论的概念不熟悉的读者可以参考[7]。Definition 2.10 ([2], [16]). Let $X$ be a topological space. A presheaf $\mathcal{F}$ of abelian group on $X$ is a contravariant functor $$ \mathcal{F}:\textbf{Top}^{\textrm{opp}}\rightarrow \textbf{Ab}$$ from the category of open sets of $X$ to the category of abelian groups.If $\mathcal{F}$ is a presheaf on $X$, the set $\mathcal{F}(U)$ consists of the sections of $\mathcal{F}$ over the open set $U$. If $s\in \mathcal{F}(U)$, we write $s|_{V}$ for an element of $\mathcal{F}(V)$ corresponding to $s$.Definition 2.11. A presheaf $\mathcal{F}$ on a topological space $X$ is a sheaf, if it satisfies the following conditions:(Uniqueness) if $U$ is an open set of $X$, and $\{V_{i}\}$ is an open covering of $U$, then for an element $s\in \mathcal{F}(U)$ such that $s|_{V_{i}}$ = 0 for all $i$, we have $s = 0$.if $U$ is an open set of $X$, and $\{V_{i}\}$ is an open covering of $U$. If we have elements $s_{i}\in \mathcal{F}(V_{i})$ for each $i$, such that for each $i, j$, $s_{i}|_{V_{i} \cap V_{j}} = s_{j}|_{V_{i}\cap V_{j}}$, then there is an element $s \in \mathcal{F}(U)$ such that $s|_{V_{i}} = s_{i}$ for each $i$.Definition 2.12. Let $\mathcal{F}$ be a presheaf on $X$, if $P$ is a point of $X$, we define the stalk $\mathcal{F}_{P}$ of $\mathcal{F}$ at $P$ to be direct limit of the groups $\mathcal{F}(U)$ $$\lim\limits_{\longrightarrow}\mathcal{F}(U)$$ for all open sets $U$ containing $P$.一个预层上某个点的茎$\mathcal{F}_{P}$，其实就是一个等价类的集合，我们可以记茎中任意一个元素为$\langle U,s\rangle$，并称它为$\mathcal{F}$截面的芽。其中$U$为$P$ 点的开邻域，$s\in\mathcal{F}(U)$。接下来，我们记$A$为一个环，$Spec(A)$为该环所有素理想的集合，称为谱。如果$\alpha$是环$A$的任意一个理想，我们记$V(\alpha)\subseteq Spec(A)$为所有包含理想$\alpha$ 的素理想的集合。我们令$V(\alpha)$为$Spec(A)$中的闭集，从而在$Spec(A)$上定义了一个Zariski拓扑。接着，我们再定义拓扑空间$Spec(A)$上的环层$\mathcal{O}$。 这样下来，$(Spec(A),\mathcal{O})$成为一个局部赋环空间。接下来我们给出赋环空间的定义。回顾一下，一个环$A$被称为局部环，如果它只有唯一一个极大理想$\mathfrak{m}_{A}$。Definition 2.13. A ringed space is a pair $(X,\mathcal{O}_{X})$, where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf of rings on $X$ called the structure sheaf. A ringed space is a locally ringed space, if for each $P\in X$, the stalk $\mathcal{O}_{X,P}$ is a local ring.有了上面这些储备，我们终于可以定义概形。首先我们定义仿射概形，之后就是一般的概形。Definition 2.14. An affine scheme is a locally ringed space $(X,\mathcal{O}_{X})$, which is isomorphic to a spectrum $\textrm{Spec }A$ of some ring $A$. A scheme is a locally ringed space $(X,\mathcal{O}_{X})$ in which every point $p$ of $X$ has an open neighborhood $U$ such that $(U,\mathcal{O}_{X}|_{U})$ is an affine scheme.从以上的定义，我们可以看出概形跟流形有异曲同工之妙。对于一个流形来说，它局部上都是一个欧几里得空间。而对于一个概形来说，它局部上都是一个仿射概形，同时因为同构关系，概形局部上的仿射概形可以看成某个环的谱。这样下来，流形由一个个欧几里得空间拼起来，而概形由一个个环的谱拼起来。而事实上，微分几何里的流形是可以用局部赋环空间表示的（更多细节请参考[10], [15]）。现在我们有了概形，就可以定义一般意义上的代数曲线了。在此之前，我们先定义概形的一些基本性质。Definition 2.15. Let $X$ be a scheme. We say that $X$ is integral if for each open affine set $U\subset X$, $\mathcal{O}_{X}(U)$ is an integral domain.Definition 2.16. Let $f:X\rightarrow Y$ be a morphism of schemes. The diagonal morphism of $X$ is a morphism $\triangle:X\rightarrow X\times_{Y}X$ such that $\textrm{pr}_{1}\circ\triangle=\textrm{pr}_{2}\circ\triangle=\textrm{id}_{X}$. We say that $f$ is separated or that $X$ is separated over $Y$ if the diagonal morphism of $X$ is a closed immersion.Definition 2.17. Let $f:X\rightarrow Y$ be a morphism of schemes. We say that $f$ is proper or that $X$ is proper over $Y$ if $f$ is separated, of finite type, and universally closed.Definition 2.18. Let $X$ be a scheme. The dimension of $X$ is the dimension of its underlying topological space $\textrm{sp}(X)$, which we will denote by $\textrm{dim }X$.Definition 2.19. An algebraic curve is an integral scheme of dimension 1, proper over a field $K$, all of whose local rings are regular.因此，一个代数曲线其实就是一个一维的概形。流形也如此，一维的流形也叫做曲线。以上我们完成了对代数曲线的定义，通过代数曲线我们可以研究数论问题。但是，研究代数曲线是需要工具的。在这些工具中，就有algebraic stack和moduli theory。Algebraic stack是stack的特殊情况，stack是对概形的进一步推广。而stack可以看成某种群胚纤维化范畴(category fibred in groupoid)，可以运用Descent à la Grothendieck来定义。而moduli theory就是研究某一类数学对象的参数空间，比如曲线的模空间、椭圆曲线的模空间。由于目前这些理论不是作者的研究方向，作者不作过多阐述。2.1 The $p$-adic numbers field $\mathbb{Q}_{p}$ and the $p$-adic integers ring $\mathbb{Z}_{p}$接下来，我们来简单说明一下$p$进数域$\mathbb{Q}_{p}$是如何构造出来的。首先，我们以有理数域$\mathbb{Q}$为例，粗略解释一下完备化（completion）的过程：我们取有理数域所有柯西序列构成的集合，定义逐项加法和乘法后可以证明它构成一个交换环，接着模掉所有零序列构成的理想，我们就得到一个完备的域，它是有理数域的域扩张。一个域的完备化不是唯一的，对应不同定义于域上的绝对值，我们可以定义不同的柯西序列，进而构造出不同的完备化。在这里，我们给出任意域上的绝对值与完备域的定义。Definition 2.20. Let $K$ be a field. An absolute value on $K$ is a map $\left|\cdot\right|:K\rightarrow\mathbb{R}_{\geq0}$ such that $\left|x\right|=0\Leftrightarrow x=0$, $\left|xy\right|=\left|x\right|\left|y\right|$, and $\left|x+y\right|\leq\left|x\right|+\left|y\right|$. We say that $K$ is complete if it is complete with respect to the distance $d(x,y)=\left|x-y\right|$ induced by the absolute value $\left|\cdot\right|$ on it.接下来我们先定义有理数域上的$p$进序数。Definition 2.21. Let $p$ be any prime number. We define the $p$-adic ordinal ord$_{p}a$ of an non-zero integer $a$ to be the highest power of $p$ which divides $a$, i.e. the greatest $m$ such that $p^{m}|a$ or $a\equiv0(\textrm{mod }p^{m})$.我们约定当整数$a=0$时，ord$_{p}a=\infty$。接着对于任意$x=a/b\in\mathbb{Q}$，我们定义$\textrm{ord}_{p}x=\textrm{ord}_{p}a-\textrm{ord}_{p}b$。如果将ord看成一个函数，那么它是良定义的，因为如果将$x$写成$x=ac/bc$，我们有$\textrm{ord}_{p}x=\textrm{ord}_{p}ac-\textrm{ord}_{p}bc=\textrm{ord}_{p}a-\textrm{ord}_{p}b$。接着我们定义$p$进绝对值：$$\left| x \right|_{p} = \begin{cases} \frac{1}{p^{\textrm{ord}_{p}x}}, & \textrm{if} \ x\neq 0\\ 0,  & \textrm{if} \ x = 0. \end{cases}$$我们先阐述复数域$\mathbb{C}$的构造过程，首先我们作有理数域$\mathbb{Q}$的完备化（关于通常的绝对值$\left|\cdot\right|$）$\widehat{\mathbb{Q}}$得到实数域$\mathbb{R}$，然后取实数域的代数闭包$\overline{\mathbb{R}}$ 得到复数域。$p$进数域$\mathbb{Q}_{p}$其实就是有理数域$\mathbb{Q}$的$p$进完备化（关于$p$进绝对值 $\left|\cdot\right|_{p}$）$\widehat{\mathbb{Q}}$。然而当我们取$p$进数域的代数闭包$\overline{\mathbb{Q}}_{p}$时，发现它不是完备的，因此我们对其再作一次完备化，最后得到$\mathbb{C}_{p}$。它是最小的包含有理数域的既是代数闭的，又是完备的域。于是，我们有如下关系：$$\begin{cases} \mathbb{C}_{p}=\widehat{\overline{\mathbb{Q}}}_{p}=\widehat{\overline{\widehat{\mathbb{Q}}}}, \textrm{p-adic analog} \\ \mathbb{C}=\overline{\mathbb{R}}=\overline{\widehat{\mathbb{Q}}}, \textrm{usual case} \end{cases}$$接着$p$进整数环$\mathbb{Z}_{p}$即是$p$进数域$\mathbb{Q}_{p}$的离散赋值环：$$\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}\mid \left|x\right|_{p}\leq1\}.$$3. Grothendieck's Theory接下来，我们来回顾一下上世纪Grothendieck所做的工作。其实代数几何如今整体上能分成两个方向，一个是以Grothendieck发展的抽象理论为基础的方向，另一个是与微分几何结合主要研究复几何的方向（参考[14]）。Grothendieck所做的工作当然远远不止上文所说的概形，还有étale cohomology（平展上同调）, crystalline cohomology（晶体上同调）, $l$-adic cohomology（$l$进上同调）, topos（拓扑范）, motives, Grothendieck topology, Grothendieck universe等等。除此之外，Grothendieck 还有三本被誉为代数几何圣经的著作，分别是EGA(Éléments de géométrie algébrique),SGA(Séminaire de géométrie algébrique)和FGA(Fondements de la Géometrie Algébrique)，翻译成中文就是《代数几何原理》、《代数几何讨论班》和《代数几何基础》。首先我们来说说Grothendieck著名的motives理论，该理论的哲学即是将所有的性质相似的上同调，诸如奇异上同调、德拉姆上同调、平展上同调和晶体上同调，统一起来。下面我们给出上同调的定义，该定义涉及到阿贝尔范畴。所谓的阿贝尔范畴，它的原型是阿贝尔群范畴，上世纪Grothendieck将其重要的性质抽象出来，只剩下足够计算同调代数的东西。Definition 3.1. A cochain complex $\mathcal{C}= \{\mathcal{C}^{n},d^{n}\}$ in an abelian category $\mathfrak{U}$ is a collection of objects $C^{i},i\in \mathbb{Z}$ , and morphisms $d^{i} : C^{i} \rightarrow C^{i+1}$, such that $d^{i}\circ d^{i+1} = 0$. The morphisms $d=\{d^{i}\}$ are called the differential (or coboundary operator).The $i$th cohomology object of the complex $\mathcal{C}$ is defined to be $H^{i}(\mathcal{C}) = \textrm{Ker }d^{i}/\textrm{Im }d^{i-1}$.根据范畴的不同，我们可以定义上同调群、上同调模，接着就可以定义singular cohomology（奇异上同调）、de Rham cohomology（德拉姆上同调）、Galois cohomology（伽罗华上同调）、Čech cohomology （切赫上同调）等等。在集合论中，我们有类与集合的概念。所谓的类由所有享有共同性质的数学对象构成，但是它不一定是一个集合，如果它不是一个集合，我们称这个类是真类。接下来，我们给出Grothendieck universe 的定义，它是在上世纪由Grothendieck提出来的，用来避免不构成集合的真类。如果读者想要了解更多相关内容，可以参考[5], [6]。Definition 3.2. A Grothendieck universe is a non-empty set $\mathcal{U}$ that satisfied the following conditions:if $x\in \mathcal{U}$ and $y\in x$, then $y\in \mathcal{U}$.if $x,y\in \mathcal{U}$, then $\{ x,y\}\in \mathcal{U}$.if $x \in \mathcal{U}$, then $\mathcal{P}(x) \in \mathcal{U}$, where $\mathcal{P}(x)$ denotes the set of all subsets of $x$.if $(x_{i},i\in I)$ is a family of elements of $\mathcal{U}$ and $I \in \mathcal{U}$, then $\bigcup_{i\in I}x_{i} \in \mathcal{U}$.4. Modern Mathematics以上内容其实都已经是以前发展的理论了，基本上都是20世纪的内容，已经有点旧了。接下来，我们讲一下21世纪比较新的内容：Shinichi Mochizuki和Peter Scholze的工作。Shinichi Mochizuki(望月新一)就是那位声称证明了abc猜想的数学家，我们习惯叫他为望月大神。他刚开始主要是做hyperbolic curve相关的研究的，后来他开始通过运用自己以前的研究成果来研究远阿贝尔几何（anabelian geometry）。远阿贝尔几何最初是Grothendieck提出来的一个宏伟的理论，如今它被望月新一进一步发展，构建了一个名叫宇宙际理论（Inter-universal Teichmüller Theory）的东西，用于证明abc猜想，可惜世界上没有多少数学家能够看得懂他的证明，因此关于他的证明主流数学界仍不认可。不同的是，Peter Scholze的工作则更为主流数学界所接受，很多人都更愿意做Peter Scholze的方向。Peter Scholze就是那个国际奥林匹克数学竞赛拿金牌，高中开始学习研究生数学的数学家，很年轻。在他的博士论文中，他发展出了一个叫状似完备空间（perfectoid spaces）的新东西，成为了当代算术几何最具影响力的数学家之一。4.1. Rigid GeometryPeter Scholze 所做的perfectoid spaces与刚性几何（Rigid Geometry）有关，接下来我们将对刚性几何的部分内容做介绍。想要了解更多的读者请参考[3], [4]。首先我们需要研究非阿基米德的绝对值。对于与绝对值相关的valuation，在本文中我们将不予讨论。我们着重讨论非阿基米德的绝对值的特别之处。Definition 4.1. A (non-archimedean) absolute value $\upsilon$ on a field $K$ is a map $\left| \cdot \right|$ : K $\rightarrow$ $\mathbb{R}_{\geq0}$, such that for all $x,y\in K$ the following conditions verified:$\left| x \right|$ = 0 $\Leftrightarrow$ $x=0$.$\left| xy \right|$ = $\left| x \right|$$\left| y \right|$$\left| x+y \right| \leq \max\{\left| x \right|, \left| y \right|\}$Proposition 4.2. Let $x,y\in K$, we have $\left| x+y \right|$ = $\max\{\left| x \right|, \left| y \right|\}$, if $\left| x \right| \neq \left| y \right|$.Proof. Without loss of generality, we assume $\left| x \right| < \left| y \right|$. Then $\left| x+y \right|$ $<$ $\max\{\left| x \right|, \left| y \right|\}$ =$ \left| y \right|$ implies$$\ \left| y \right| = \left| (y+x)-x \right| \leq \max\{\left| x+y \right|, \left| x \right|\} < \left| y \right|$$which is contradictory. So we must have $\left| y \right| = \left| y+x \right| = \max\{\left| x \right|, \left| y \right|\}$ as claimed.通过绝对值，我们定义任意域$K$上的距离为$d(x,y) = \left| x-y \right|$，然后该距离诱导出$K$上的一个拓扑。有了$K$中任意两点的距离，根据非阿基米德的三角不等式，对于所有$x,y,z \in K$，我们可以得出：$$d(y,z) \leq \max\{d(x,y),d(x,z)\}$$根据命题4.2，该不等式两边相等，如果不等式右边的两个距离不相等。这意味着：在域$K$中的任意三角形，都是等腰三角形。更进一步，我们可以证出：域$K$中任意一个圆盘中的点都可以作为该圆盘的中心。因此，如果$K$中的两个圆盘有非空交集，那么它们就是共心的。下面我们给出证明。Definition 4.3. For a centre $a\in K$ and a radius $r\in \mathbb{R}_{> 0}$, we define the disk without boundary to be the set $$D^{-}(a,r) = \{ x \in K\mid d(x,a)<r \}$$ And we define the disk with boundary to be the set $$D^{+}(a,r) = \{ x \in K\mid d(x,a)\leq r\}$$Proposition 4.4. Each point of disk without boundary in K is the centre of the disk.Proof. Assume that $a$ is the centre of a disk, $b$ is a point different from $a$. For any $x\in D^{-}(a,r)$, we have $$ d(x,b) = \left| x-b \right| = \left| (x-a)+(a-b) \right| \leq \max\{\left| x-a \right|,\left| a-b \right|\} < r $$类似的，我们可以证明对于有边界的圆盘，其中的任意一点都可以是它的中心。4.2 Perfectoid Geometry接下来我们粗略地说一下，Perfectoid spaces, [4]，这篇文章里面的一些内容，鉴于作者水平有限，不能一一详述。首先，perfectoid是perfect+oid，意思就是more or less perfect，类完美。首先，我们回顾一下什么是完美域（perfect fields）。Definition 4.5. Let $K$ be a field. We say that $K$ is perfect if either $K$ has characteristic $0$, or if $K$ has characteristic $p>0$, the Frobenius $$ \Phi:K\rightarrow K, x\mapsto x^{p}$$ is an isomorphism.Perfectoid spaces这篇文章的动机源于以下Fontaine-Wintenberger的一个定理：Theorem 4.6. The absolute Galois groups of $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and $\mathbb{F}_{p}((t))$ are canonically isomorphic.Remark 4.7. $$\mathbb{Q}_{p}(p^{1/p^{\infty}})=\lim_{\substack{\longrightarrow \\ n>0}}\mathbb{Q}_{p}(p^{1/p^{n}})=\bigcup_{n>0}\mathbb{Q}_{p}(p^{1/p^{n}}).$$$\mathbb{Q}_{p}(p^{1/p^{\infty}})$是一个特征0的域，它的剩余类域$\mathbb{F}_{p}$是特征$p$，这种域被称为混合特征的（mixed characteristic）。而$\mathbb{F}_{p}((t))$ 是一个特征$p$的域。意思是如果将所有$X^{p^{n}}-p\in\mathbb{Q}_{p}[X]$的根加到$\mathbb{Q}_{p}$里面，它会看起来像一个特征$p$的域$\mathbb{F}_{p}((t))$。想要更好地理解$\mathbb{Q}_{p}(p^{1/p^{n}})$是什么意思，可以参考$\mathbb{C}\cong\mathbb{R}(i)\cong\mathbb{R}[X]/(X^{2}+1)$这个例子。同时，我们有这样一个tower:$$\mathbb{Q}_{p}\subseteq \mathbb{Q}_{p}(p^{1/p})\subseteq \mathbb{Q}_{p}(p^{1/p^{2}})\subseteq \cdot\cdot\cdot \subseteq \mathbb{Q}_{p}(p^{1/p^{n}})\subseteq \cdot\cdot\cdot \subseteq \mathbb{Q}_{p}(p^{1/p^{\infty}}).$$定理4.6可以在更加一般的框架下研究，这就引申出了perfectoid fields。 首先，我们给出非阿基米德域的定义，它其实就是一个拓扑由一个非阿基米德绝对值生成的拓扑域。Definition 4.8. A non-archimedean field is a topological field $K$ whose topology is induced by a non-trivial valuation of rank 1.Definition 4.9. A perfectoid field is a complete non-archimedean field $K$ with residue characteristic $p>0$ whose associated rank-1-valuation is non-discrete and the Frobenius $\Phi:K^{\circ}/p\rightarrow K^{\circ}/p,x\mapsto x^{p}$ is surjective.Example 4.10. The $p$-adic completion $\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}$ of $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and the $t$-adic completion $\widehat{\mathbb{F}_{p}((t))(t^{1/p^{\infty}})}:=\mathbb{F}_{p}((t))((t^{1/p^{\infty}}))$ of $\mathbb{F}_{p}((t))(t^{1/p^{\infty}})$ are perfectoid fields.$$\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}=\widehat{\mathbb{Z}_{p}[p^{1/p^{\infty}}]}[\frac{1}{p}]=(\lim_{\longleftarrow} \mathbb{Z}_{p}[p^{1/p^{\infty}}]/p^{n})[\frac{1}{p}],$$$$\widehat{\mathbb{F}_{p}((t))(t^{1/p^{\infty}})}=\widehat{\mathbb{F}_{p}[t^{1/p^{\infty}}]}[\frac{1}{t}]=(\lim_{\longleftarrow} \mathbb{F}_{p}[t^{1/p^{\infty}}]/t^{n})[\frac{1}{t}].$$Perfectoid field叫做类完美域，当它为特征$p$时，它是一个完美域。同时，这里有一个tilt的过程，它可以看成一个函子叫做tilt funtor:$$K\mapsto K^{\flat}$$将一个任意特征的perfectoid field打到一个特征$p$的perfectoid field。同时，我们有$$K^{\flat}=\lim_{\substack{\longleftarrow \\ x\mapsto x^{p}}}K.$$接着我们有了更加一般的定理，它推广了定理4.6。Theorem 4.11. The absolute Galois groups of $K$ and $K^{\flat}$ are canonically isomorphic.总之，这篇文章中，Peter Scholze提出一种框架，它能将任意特征的问题简化为特征$p$的问题，因为特征$p$往往更好研究，同时也有很多好的性质和结论。References Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd ed., Springer-Verlag New York, Inc., 1993. Robin Hartshorne, Algebraic Geometry, Springer, New York, NY, Springer Science+Business Media New York, 1977. Siegfried Bosch, Lectures on formal and rigid geometry, volume 2105 of Lecture Notes in Mathematics. Springer, Cham, 2014. Peter Scholze, Perfectoid Spaces, IHES Publ. math. 116 (2012), pp. 245–313. Grothendieck with Artin, M. and Verdier, J. L. Théorie des Topos et Cohomologie Étale des Schémas. Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Springer-Verlag Berlin Heidelberg, 1973. Pierre Deligne, Cohomologie Étale, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 1/2, Springer-Verlag Berlin Heidelberg, 1977. Peter J. Hilton and Urs Stammbach, A Course in Homological Algebra, Springer-Verlag New York, 1997. Fredrik Meyer, Notes on algebraic stacks, https://blog.fredrikmeyer.net/uio-math, 2013. G. Everest and Thomas Ward, An Introduction to Number Theory, Springer-Verlag London, 2005. Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, New York, NY, 2011. Andrew John Wiles, Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics, 141 (1995), 443-552. Michael Artin, Allyn Jackson, David Mumford, and John Tate, Coordinating Editors, Alexandre Grothendieck, Notices of the AMS 51, 2016. Joe Harris and Ian Morrison, Moduli of Curves, Springer-Verlag New York, Springer Science+Business Media New York, 1998. Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, Wiley-Interscience; 1st edition (August 16, 1994), 1978. J.S. Milne, Algebraic Geometry (v6.02), www.jmilne.org/math/ , 2017. Glen E. Bredon, Sheaf Theory, Springer-Verlag New York, Springer Science+Business Media New York, 1997.

以下短文为我2022年3月3日朋友圈的感想，如今分享出来lyh说得有理！就比如我一周十几节课，天天早八，结果就是我虽然很累，但是脑子根本就没转几下。本科的课堂对我而言根本没有什么思维强度，以至于上课我总是打瞌睡😢。以前我可能会很不喜欢某些课，但现在我的心态就是能学多点也不亏，随缘。做研究是一件其具挑战性的事情，你得站在无法预知的交叉路口，凭借自己的眼光、胆识，去艰难开辟一条属于自己的道路。这完全不是学习强度能够简单衡量的。我觉得只有我读文献的时候，才感到自己脑袋真的动起来了。为了深刻理解文献的内容，我需要做一个又一个的计算，我不知道其它分支是不是也是这样的，但似乎学本科的数学，读定义根本不需要做什么计算。同时，每接触一个全新的东西，你都得熟悉它的background，了解它的motivation，然后找一些例子，做一些计算等等，这些过程往往会耗费大量的时间和精力，最重要的是会产生巨大的压力。读文献，你每天都会接触很多全新的概念，新到你学之前完全无法想象得到它的存在。因此，我个人觉得学数学感到痛苦是一件很正常的事情，我相信每个学数学的人无论水平如何都会感到痛苦，尤其是读证明的时候。😣看了lyh的话，我似乎也开始明白为什么自己学数学、做研究的方法这么难传授给那些同样想学数学的人。因为这些东西本身就没有固定的流程，并不是标准化的流水线工程。很多时候，有的东西就是一种感觉，我只不过是努力将自己的感觉通过语言表达出来，希望别人能够理解。最后我想拿Peter Scholze说过的话来共勉：在一个采访中，问Scholze觉得看数学痛苦吗？有没有办法让数学论文更加简单更容易理解？Scholze说痛苦，但是他觉得这些痛苦是有价值的，虽然他不知道别的数学领域，但是他自己这块这些痛苦是必须的。

Let $A,B$ be commutative rings in characteristic $p$. Let $\phi_{A}:A\rightarrow A,\phi_{B}:B\rightarrow B$ be the Frobenius morphisms, i.e. the $p$-power maps. If we have ${\rm{colim}}_{n\in\mathbb{N}}A\cong {\rm{colim}}_{n\in\mathbb{N}}B$, where the transition maps are Frobenius morphisms, can we show that $A\cong B$ ?Answer: No. Recall that an $\mathbb{F}_p$-algebra $R$ is perfect if the Frobenius map $\varphi : R \ni r \mapsto r^p \in R$ is an isomorphism. The colimit of powers of the Frobenius you describe is the perfection of an $\mathbb{F}_p$-algebra, and it is named that because it's the left adjoint of the inclusion of perfect $\mathbb{F}_p$-algebras into $\mathbb{F}_p$-algebras. This makes perfect $\mathbb{F}_p$-algebras a reflective subcategory of $\mathbb{F}_p$-algebras, which among other things implies that perfection fixes any perfect $\mathbb{F}_p$-algebra.This is all abstract context for the following more specific counterexample: take $A = \mathbb{F}_p[x]$, whose perfection is $\mathbb{F}_p[x^{\frac{1}{p^{\infty}}}]$, the ring obtained by adjoining all $p^n$-power roots of $x$. Then take $B = \mathbb{F}_p[x^{\frac{1}{p^{\infty}}}]$ to be the perfection. More generally, we could take $A$ to be any $\mathbb{F}_p$-algebra which is not perfect and $B$ to be its perfection.These notes by Bhatt have a more general claim right before Remark 1.4 about when two algebras have isomorphic perfections but I'm not familiar enough with universal homeomorphisms to say anything about it.

以下短文出自2022年2月27号，为我发在朋友圈的一段感想，如今分享出来lyh前辈算是我高中时期数学的启蒙人之一，如今几年过去，回看过去，甚是感慨学数学早点把握方向很重要，直到你做研究那天，你会发现很多你曾学过的东西都派不上用场。当你有扎实的基本功以后，直接倒着学，这样效率或许比按部就班地按顺序学要高得多。看不懂文献怎么办，继续看，这样你才能发现自己需要学什么。然后再倒回去补知识，这种方法才是适合于做研究的。我对竞赛不感兴趣，学习数学的方向完全遵从内心的选择，顺利地从微分几何和代数几何中选择了代数几何。然后发现代数几何的浩瀚，似乎能囊括几乎所有重要的数学领域，至少纯数学是如此。我并不认可数学考得高分就是学得好，曾经高中被数学老师冷落的经历还历历在目。我看的第一本英文书就是泛函分析，我没系统学过点集拓扑。这些都不能说明什么。我15岁开始看EGA，这是我当时读过的最难的书。我被Grothendieck的风格所深深震撼，也就那时候，我开始学习法语就是为了读原版的EGA。我不认可大神级天赋就真的能16岁无障碍读EGA，万丈高楼平地起，天才也不例外，况且这个世界上可能都没有所谓的天才。我觉得真正的大神级天赋，就是Peter Scholze这种学数学、做数学仅仅是为了理解数学的人，他做数学不是为了成为某个领域的leading figure，而是for his own sake。这多自在、多自由，就如同逍遥游一样，也就只有这样才有可能做出永垂不朽的工作吧。

Question: Let $X$ be a scheme. Let ${\rm{Spec}}(R)$ be an affine scheme for some ring $R$. Suppose that there is a morphism of schemes $f:X\rightarrow{\rm{Spec}}(R)$. What properties should $f$ have so that $X$ is also an affine scheme? Or what conditions can make $X$ affine?Answer: If $f$ is an affine morphism, then $X$ is affine by definition.This is an "if and only if". If $X$ is affine, the so is $f$. See theorem 7.3.7 in Vakil's Foundations of Algebraic Geometry, or 29.11.3 and 29.11.4 in the Stacks Project.So in particular if $f$ is a closed immersion, then $f$ is affine such that $X$ is affine, for example.

下面我来讲一下我对Grothendieck universes粗浅的理解。首先，我们知道当年Cantor的朴素集合论是有漏洞的，这些漏洞所衍生出的悖论，比如罗素悖论，引发了第三次数学危机。后来为了解决这些问题，发展出了诸多新的理论，其中一个就是如今也经常使用的ZFC公理系统。在ZFC框架下，所有的数学对象都是集合，而所谓的所有集合的“集合”严格来说不是一个数学对象，它不能构成一个集合，那么我们称它为一个class。class的定义很明了，它的成员就是所有享有某些共同性质的数学对象，其实就是最初对集合的定义，现在区分开来，因为它不一定构成一个集合。如果一个cass不能构成一个集合，我们称它为一个proper class。接一下来，来到Grothendieck universes，它是上个世纪60年代由Grothendieck提出来的用来避免proper classes的。Grothendieck universe是一个非空集合，在这个集合里面，所有通常的集合的运算封闭，比如说取一个集合里的元素、索引并集将两个元素构成一个集合等，这样在计算中就不用担心出现不构成集合的情况。后面，Grothendieck还加了两条公理，分别是U A和U B，其中U A如今比较常用，比如在范畴论中，该公理就是每个集合都包含于某个Grothendieck universe中。其实关于class和集合，这是很重要的两样不同的东西，有必要加以区分，在写作的时候需要说明清楚。比如说，在范畴论中，我们有big category和small category,其中big category就是所有对象构成class的范畴，small category就是所有对象构成集合的范畴。但是写作的时候我们一般从头到尾都只会用到一种范畴，因此会特此说明我们的范畴是指什么什么。------------------------------------------------------本文原于2021年12月26日发布于QQ空间

The name "p-divisible group" is somewhat misleading, which in fact has another name "Barsotti-Tate group". A p-divisible group is not merely just a kind of group. lt is more general. In fact, a p-divisible group can be viewed as a tower of affine group schemes with some extra conditions. Historically, p-divisible groups were the main stimulus for p-adic Hodge theory. So l think that studying p-divisible groups is the key to study p-adic Hodge theory.However, today's p-adic Hodge theory is much more complicated. l failed to understand Fontaine rings in the past. But this won't suppress my interest in p-adic Hodge theory. For now, I'm still studying Peter Scholze's Perfectoid Space, which was published nearly ten years ago and has already obtained a lot of beautiful results. lf l have time, l will still try to study p-adic Hodge theory.----------This short articles was originally published at June 24, 2021.

原本我以为微分几何跟代数几何仅仅是数学两个有关联的分支，结果是我之前肤浅了。随着对代数几何深入的学习了解，我发现代数几何跟微分几何也有很深的联系。因此我完全可以说微分流形的理论是深入学习代数几何的necessity，或许你懂抽象代数、交换代数，甚至同调代数，但是若你不懂一些manifold的理论，你完全没有机会去学习étale cohomology、Hodge theory等代数几何更深层次的理论。当然想要学习代数几何最高深、最先进的部分，仅仅懂abstract algebra、homological algebra、manifold是完全不够的。以我自己为例，我的方向是算术几何，这意味着你还需要懂elliptic curve、modular forms、$\ell$-adic representation、algebraic topology等更深层次的知识。还没完，你觉得你所学的东西就真的在研究的过程中用得上吗？在看书的过程中，你还得不停地看文献，就像定制一台机器一样定制自己所需要学习的知识，这样才能保证自己学到有用的东西，否则就是浪费时间，人的脑容量是有限的，用不上的东西时间长了就会忘记。很多人想做数学研究，结果却把大量的时间浪费在无谓的学习上，其实我更加提倡边做边学的做法，先找到个问题，然后尝试去做它，在做的过程中不断学习自己所需要的知识，这样效率是不是高很多呢。但说这么多都没用，很多人本身没有这么强烈的motivation，动机是前提，连最基本的动机都没有，谈再多的方法都没用。————————————————————本文原发布于2020年10月14日

My question: Let $S$ be a base scheme and let $(Sch/S)_{fppf}$ be a big fppf site. Let $U$ be a scheme over $S$. Suppose that there is a surjective morphism $\Phi_{U}:U\rightarrow U$. Then can we show that the induced morphism of sheaves $h_{U}\rightarrow h_{U}$ is locally surjective? It seems that this is false.Note that $h_{U}={\rm{Hom}}(-,U)$ is a representable sheaf. A map of sheaves $F\rightarrow G$ on $(Sch/S)_{fppf}$ is locally surjective if for every scheme $U\in{\rm{Ob}}((Sch/S)_{fppf})$ and every $s\in G(U)$, there exists a covering $\{U_{i}\rightarrow U\}_{i\in I}$ such that for all $i$, $s|_{U_{i}}$ is in the image of $F(U_{i})\rightarrow G(U_{i})$.Answer: Let $S:={\rm Spec}(k)$ be a field and let $U={\rm Spec}(k[t]/t^2)$.The ring $k[t]/t^2$ is a $k$-algebra and there is a map of $k$-algebras $k[t]/t^2\to t$ sending $t$ to $0$ so we get morphisms $U\to{\rm Spec}(k)$ and ${\rm Spec}(k)\to U$.Let $a:U\to U$ be the composition of these two morphisms. This is surjective.Now if the answer to your question is positive, there is a faithfully flat morphism $b:V\to U$ and an element $\rho\in{\rm Hom}(V,U)$ such that $a\circ\rho=b$ (applying local surjectivity to ${\rm Id}_U\in{\rm Hom}(U,U)$). In other words, the morphism $a_V:U\times_U V\to V$ has a section. However, if the morphism $a_V$ has a section then so has the morphism $S\times_U V\to V$ obtained by base-change from the morphism $S={\rm Spec}(k)\to U$. The morphism is $S\times_U V\to V$ is a closed immersion, so it is an isomorphism, since it has a section. Isomorphisms descend along faithfully flat morphisms, so we deduce that ${\rm Spec}(k)\to U$ is an isomorphism, which is false. So this provides a counterexample.