---
date: 2023-01-06 19:56
aliases:
  - Galois theory for schemes
annotation-target: https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf
status: ✅ Started
page_current: 21
page_total: 113
flashcard: 
created: 2023-01-06T19:56
updated: 2024-01-29T17:51
---

# Review

- [x] What is an $A\dash$algebra $B$? ✅ 2023-01-07
	- [ ] A ring $B$ equipped with a ring morphism $A\to B$.
- [x] What is the [[trace]] of a module? ✅ 2023-01-07
	- [ ] Realized as a morphism $$\Tr: \mods{A}(M, M) \to A$$ of $A\dash$modules for $M\in \mods{A}^\fg$. For $M\in \algs{A} \intersect \mods{B}^{\fg, \free}$, define $$\Tr_{B/A}(b) \da \Tr(x\mapsto bx) \in \mods{A}$$ which satisfies $\Tr(a) = \rank_A(B)\cdot a$ for $a\in A$.
- [x] What is a [[separable]] algebra? ✅ 2023-01-07
	- [ ] When the following natural map is an iso: $$\begin{align}\phi: B &\to \mods{A}(B, A) \\ x &\mapsto \Tr(x\cdot)\end{align}$$
- [x] What are the free separable $k\dash$algebras $L$ over $k$ a field? ✅ 2023-01-07
	- [ ] $L = \prod_{i=1}^t B_i$ where each $B_i/k$ is a finite separable extension and $t>0$.
- [x] What is a finite etale morphism $f:Y\to X$? ✅ 2023-01-07
	- [ ] $\exists \spec A_i \covers Y$ where each $f\inv(\spec A_i) = \spec B_i$ is affine and $B_i/A_i$ is a free separable $A_i\dash$algebra.
	- [ ] In particular, $f$ is a finite morphism, so each $B_i\in \mods{A_i}^{\fg}$ which need not be free, but turns out to always be projective.
- [x] What are the finite etale covers $Y\to \spec \ZZ_K$, the ring of integers of an algebraic number field $K$? ✅ 2023-01-07
	- [ ] $Y\to X$ where $Y \cong \disjoint_{i=1}^t \spec A_i$ where $t\geq 0$  and each $A_i = \ZZ_{K_i}$ where $K_i/K$ is a finite extension unramified at all $\mfp \in \spec(A_i)\smz$.
- [x] How is the projective limit as a set? ✅ 2023-01-07
	- [ ] $$\varprojlim S_i=\left\{\left(x_i\right)_{i \in I} \in \prod_{i \in I} S_i \st f_{i j}\left(x_i\right)=x_j \text { for all } i, j \in I \text { with } i \geq j\right\} \text {. }$$
- [x] Discuss topological properties of profinite groups. ✅ 2023-01-07
	- [ ] Always compact and totally disconnected; conversely every compact totally disconnected topological group is profinite.
- [x] What is the profinite completion of a group? ✅ 2023-01-07
	- [ ] Take the poset $N\geq M \iff N \subseteq M$ for $N, M \normal G$ of finite index and the directed system of quotients $G/N$.
- [x] Give examples of profinite groups. ✅ 2023-01-07
	- [ ] $\hat\ZZ = \varprojlim_{n>0} \ZZ/n\ZZ$ the profinite completion of $\ZZ$ 
	- [ ] $\ZZpadic \da \varprojlim \ZZ/p^n\ZZ$ the $p\dash$completion
- [x] What is the fundamental theorem of Galois theory for schemes? ✅ 2023-01-07
	- [ ] Let $X$ be a connected scheme. Then there exists a profinite group $\pi$, uniquely determined up to isomorphism, such that the category $\mathbf{F E t}_X$ of finite étale coverings of $X$ is equivalent to the category $\pi$-sets of finite sets on which $\pi$ acts continuously.
- [x] What is $\pi_1^\et \spec \ZZ$? ✅ 2023-01-07
	- [ ] Trivial: every finite etale cover is the free separable $\ZZ\dash$algebra $\ZZ^n$ (and there are not others).
- [x] What is $\pi_1^\et \spec k$ for $k=\kbar$? ✅ 2023-01-07
	- [ ] Trivial: the only covers are $k^n$.
- [x] What is $\pi_1^\et \spec k$ in general? ✅ 2023-01-07
	- [ ] $\Gal(k^\sep/k)$.
- [x] What is $\pi_1^\et \spec \FF_q$? ✅ 2023-01-07
	- [ ] $\hat \ZZ$
- [x] What is $\pi_1^\et \spec \ZZ_K$ for $K$ an algebraic number field? ✅ 2023-01-07
	- [ ] $\Gal(M/K)$ where $M$ is the maximal algebraic extension unramified over $\spec(\ZZ_K)\smz$. 
- [x] What is $\pi_1^\et \spec {\ZZ_K} \invert{a}$ for $a\in \ZZ_K\smz$ for $K$ a number field? ✅ 2023-01-07
	- [ ] $\Gal(M_a/K)$ where $M_a$ is the maximal extension unramified at all primes not dividing $a$.
- [x] What is $\pi_1^\et \spec \ZZpadic$? ✅ 2023-01-07
	- [ ] $\hat \ZZ$.
- [x] What is $\pi_1^\et \PP^1\slice k$ for $k$ a field? ✅ 2023-01-07
	- [ ] $\pi_1^\et \spec k = \Gal(k^\sep/k)$.


# Galois theory for schemes


>%%
>```annotation-json
>{"created":"2023-01-07T01:07:09.040Z","text":"An example where $g=V^3+2 V^2-15 V-4 U$ and $C \\da V(g) \\subseteq \\AA^2$ and the projection $f: C\\to \\AA^1$ onto the first coordinate: \n\n![2023-01-06](attachments/2023-01-06.png)","updated":"2023-01-07T01:07:09.040Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":7169,"end":7271},{"type":"TextQuoteSelector","exact":". To illustrate this, and to show how finite  ́etale coverings are more general, weconsider an example","prefix":" direct scheme-theoreticanalogue","suffix":".Define g ∈C[U,V ] by g = V 3 +2"}]}]}
>```
>%%
>*%%PREFIX%%direct scheme-theoreticanalogue%%HIGHLIGHT%% ==. To illustrate this, and to show how finite  ́etale coverings are more general, weconsider an example== %%POSTFIX%%.Define g ∈C[U,V ] by g = V 3 +2*
>%%LINK%%[[#^hydzz0708y6|show annotation]]
>%%COMMENT%%
>An example where $g=V^3+2 V^2-15 V-4 U$ and $C \da V(g) \subseteq \AA^2$ and the projection $f: C\to \AA^1$ onto the first coordinate: 
>
>![2023-01-06](attachments/2023-01-06.png)
>%%TAGS%%
>
^hydzz0708y6


>%%
>```annotation-json
>{"created":"2023-01-07T01:08:27.636Z","updated":"2023-01-07T01:08:27.636Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":8751,"end":8989},{"type":"TextQuoteSelector","exact":"Translating this definition in concrete terms, one findsthat the local “triviality” condition from the topological definition has been replaced by ananalogous algebraic condition, namely that a certain discriminant does not vanish locally","prefix":"s notion is given in Section 1. ","suffix":"(cf. Exercises 1.3 and 1.6). In "}]}]}
>```
>%%
>*%%PREFIX%%s notion is given in Section 1.%%HIGHLIGHT%% ==Translating this definition in concrete terms, one findsthat the local “triviality” condition from the topological definition has been replaced by ananalogous algebraic condition, namely that a certain discriminant does not vanish locally== %%POSTFIX%%(cf. Exercises 1.3 and 1.6). In*
>%%LINK%%[[#^h51t4emf9xn|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^h51t4emf9xn



>%%
>```annotation-json
>{"text":"Separable","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":15220,"end":15386},{"type":"TextQuoteSelector","exact":"If φ is an isomorphismwe call B separable over A, or a free separable A-algebra if we wish to stress the conditionthat B be finitely generated and free as an A-module","prefix":"(φ(x))(y) = Tr(xy), for x,y ∈B.","suffix":". See Exercise 1.3 for a reformu"}]}],"created":"2023-01-07T01:21:19.542Z","updated":"2023-01-07T01:21:19.542Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}
>```
>%%
>*%%PREFIX%%(φ(x))(y) = Tr(xy), for x,y ∈B.%%HIGHLIGHT%% ==If φ is an isomorphismwe call B separable over A, or a free separable A-algebra if we wish to stress the conditionthat B be finitely generated and free as an A-module== %%POSTFIX%%. See Exercise 1.3 for a reformu*
>%%LINK%%[[#^a7dba02d1p8|show annotation]]
>%%COMMENT%%
>Separable
>%%TAGS%%
>
^a7dba02d1p8


>%%
>```annotation-json
>{"created":"2023-01-07T01:23:05.734Z","text":"Generally, if $K$ is a field, then the free separable $K$-algebras are precisely the $K$-algebras of the form $\\prod_{i=1}^t B_i$, where each $B_i$ is a finite separable field extension of $K$ in the sense of Galois theory,","updated":"2023-01-07T01:23:05.734Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":15799,"end":15996},{"type":"TextQuoteSelector","exact":"Generally,if K is a field, then the free separable K-algebras are precisely the K-algebras of the form∏ti=1 Bi, where each Bi is a finite separable field extension of K in the sense of Galois theor","prefix":"closed field (see Theorem 2.7). ","suffix":"y,and t ≥ 0, see Theorem 2.7. (N"}]}]}
>```
>%%
>*%%PREFIX%%closed field (see Theorem 2.7).%%HIGHLIGHT%% ==Generally,if K is a field, then the free separable K-algebras are precisely the K-algebras of the form∏ti=1 Bi, where each Bi is a finite separable field extension of K in the sense of Galois theor== %%POSTFIX%%y,and t ≥ 0, see Theorem 2.7. (N*
>%%LINK%%[[#^8ssgk95xcqm|show annotation]]
>%%COMMENT%%
>Generally, if $K$ is a field, then the free separable $K$-algebras are precisely the $K$-algebras of the form $\prod_{i=1}^t B_i$, where each $B_i$ is a finite separable field extension of $K$ in the sense of Galois theory,
>%%TAGS%%
>
^8ssgk95xcqm


>%%
>```annotation-json
>{"created":"2023-01-07T01:23:56.153Z","text":"What is a finite etale morphism?\n\nA morphism $f: Y \\rightarrow X$ of schemes is finite étale if there exists a covering of $X$ by open affine subsets $U_i=\\operatorname{Spec} A_i$, such that for each $i$ the open subscheme $f^{-1}\\left(U_i\\right)$ of $Y$ is affine, and equal to $\\operatorname{Spec} B_i$, where $B_i$ is a free separable $A_i$ algebra. In this situation we also say that $f: Y \\rightarrow X$ is a finite étale covering of $X$.","updated":"2023-01-07T01:23:56.153Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":16117,"end":16145},{"type":"TextQuoteSelector","exact":".4 Finite  ́etale morphisms.","prefix":"found in Exercises 1.5 and 1.6.1","suffix":" A morphism f : Y → X of schemes"}]}]}
>```
>%%
>*%%PREFIX%%found in Exercises 1.5 and 1.6.1%%HIGHLIGHT%% ==.4 Finite  ́etale morphisms.== %%POSTFIX%%A morphism f : Y → X of schemes*
>%%LINK%%[[#^27ii61x13q|show annotation]]
>%%COMMENT%%
>What is a finite etale morphism?
>
>A morphism $f: Y \rightarrow X$ of schemes is finite étale if there exists a covering of $X$ by open affine subsets $U_i=\operatorname{Spec} A_i$, such that for each $i$ the open subscheme $f^{-1}\left(U_i\right)$ of $Y$ is affine, and equal to $\operatorname{Spec} B_i$, where $B_i$ is a free separable $A_i$ algebra. In this situation we also say that $f: Y \rightarrow X$ is a finite étale covering of $X$.
>%%TAGS%%
>
^27ii61x13q


>%%
>```annotation-json
>{"created":"2023-01-07T01:25:13.349Z","text":"If $X=$ Spec $A$, where $A$ is the ring of algebraic integers in an algebraic number field $K$, then the finite étale coverings $Y \\rightarrow X$ are precisely given by $Y=\\coprod_{i=1}^t \\operatorname{Spec} A_i$, where $t \\geq 0$ and where for each $i$ the ring $A_i$ is the ring of algebraic integers in a finite extension $K_i$ of $K$ that is unramified at all non-zero prime ideals of $A$, see $6.18$.","updated":"2023-01-07T01:25:13.349Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":17294,"end":17621},{"type":"TextQuoteSelector","exact":" If X = Spec A, where A is the ring ofalgebraic integers in an algebraic number field K, then the finite  ́etale coverings Y →X areprecisely given by Y = ∐ti=1 Spec Ai, where t ≥0 and where for each i the ring Ai is the ringof algebraic integers in a finite extension Ki of K that is unramified at all non-zero primeideals of A","prefix":"pec Bi, with Bi and t as in 1.3.","suffix":", see 6.18.1.6 Morphisms of cove"}]}]}
>```
>%%
>*%%PREFIX%%pec Bi, with Bi and t as in 1.3.%%HIGHLIGHT%% ==If X = Spec A, where A is the ring ofalgebraic integers in an algebraic number field K, then the finite  ́etale coverings Y →X areprecisely given by Y = ∐ti=1 Spec Ai, where t ≥0 and where for each i the ring Ai is the ringof algebraic integers in a finite extension Ki of K that is unramified at all non-zero primeideals of A== %%POSTFIX%%, see 6.18.1.6 Morphisms of cove*
>%%LINK%%[[#^6t4y5w53hkh|show annotation]]
>%%COMMENT%%
>If $X=$ Spec $A$, where $A$ is the ring of algebraic integers in an algebraic number field $K$, then the finite étale coverings $Y \rightarrow X$ are precisely given by $Y=\coprod_{i=1}^t \operatorname{Spec} A_i$, where $t \geq 0$ and where for each $i$ the ring $A_i$ is the ring of algebraic integers in a finite extension $K_i$ of $K$ that is unramified at all non-zero prime ideals of $A$, see $6.18$.
>%%TAGS%%
>
^6t4y5w53hkh


>%%
>```annotation-json
>{"text":"Trace","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":14361,"end":14458},{"type":"TextQuoteSelector","exact":"Let M be a finitely generated free A-module with basis w1,w2,...,wn and let f : M →Mbe A-linear. ","prefix":"e rank is finite (Exercise 1.1).","suffix":"Thenf(wi) =n∑j=1aijwj (1 ≤i ≤n)f"}]}],"created":"2023-01-07T01:27:08.059Z","updated":"2023-01-07T01:27:08.059Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}
>```
>%%
>*%%PREFIX%%e rank is finite (Exercise 1.1).%%HIGHLIGHT%% ==Let M be a finitely generated free A-module with basis w1,w2,...,wn and let f : M →Mbe A-linear.== %%POSTFIX%%Thenf(wi) =n∑j=1aijwj (1 ≤i ≤n)f*
>%%LINK%%[[#^9duiogag2c|show annotation]]
>%%COMMENT%%
>Trace
>%%TAGS%%
>
^9duiogag2c


>%%
>```annotation-json
>{"created":"2023-01-07T01:27:48.553Z","text":"A particularly wide class of spaces $X$ can be treated if one wishes to classify only the finite coverings of $X$. For this it suffices that $X$ be connected, i.e., have exactly one connected component. (In these notes the empty space is not considered to be connected.) For any connected space $X$ there is a topological group $\\hat{\\pi}(X)$ such that the category of finite coverings of $X$ is equivalent to the category of finite discrete sets provided with a continuous action of $\\hat{\\pi}(X)$.","updated":"2023-01-07T01:27:48.553Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":6081,"end":6545},{"type":"TextQuoteSelector","exact":" particularly wide class of spaces X can be treated if one wishes to classify only thefinite coverings of X. For this it suffices that X be connected, i.e., have exactly one connectedcomponent. (In these notes the empty space is not considered to be connected.) For anyconnected space X there is a topological group ˆπ(X) such that the category of finite coveringsof X is equivalent to the category of finite discrete sets provided with a continuous actionof ˆπ(X)","prefix":" be the fundamental group of X.A","suffix":". This result, which is difficul"}]}]}
>```
>%%
>*%%PREFIX%%be the fundamental group of X.A%%HIGHLIGHT%% ==particularly wide class of spaces X can be treated if one wishes to classify only thefinite coverings of X. For this it suffices that X be connected, i.e., have exactly one connectedcomponent. (In these notes the empty space is not considered to be connected.) For anyconnected space X there is a topological group ˆπ(X) such that the category of finite coveringsof X is equivalent to the category of finite discrete sets provided with a continuous actionof ˆπ(X)== %%POSTFIX%%. This result, which is difficul*
>%%LINK%%[[#^v5jpjz8htyo|show annotation]]
>%%COMMENT%%
>A particularly wide class of spaces $X$ can be treated if one wishes to classify only the finite coverings of $X$. For this it suffices that $X$ be connected, i.e., have exactly one connected component. (In these notes the empty space is not considered to be connected.) For any connected space $X$ there is a topological group $\hat{\pi}(X)$ such that the category of finite coverings of $X$ is equivalent to the category of finite discrete sets provided with a continuous action of $\hat{\pi}(X)$.
>%%TAGS%%
>
^v5jpjz8htyo


>%%
>```annotation-json
>{"created":"2023-01-07T22:43:36.403Z","updated":"2023-01-07T22:43:36.403Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":12893,"end":12974},{"type":"TextQuoteSelector","exact":" If A is a ring, an A-algebra is a ring B equipped with a ring homomorphismA → B.","prefix":"its of a ring A isdenoted by A∗.","suffix":" Everything we need from commuta"}]}]}
>```
>%%
>*%%PREFIX%%its of a ring A isdenoted by A∗.%%HIGHLIGHT%% ==If A is a ring, an A-algebra is a ring B equipped with a ring homomorphismA → B.== %%POSTFIX%%Everything we need from commuta*
>%%LINK%%[[#^41ahzvyfj7n|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^41ahzvyfj7n


>%%
>```annotation-json
>{"created":"2023-01-07T22:58:54.753Z","updated":"2023-01-07T22:58:54.753Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":19502,"end":19672},{"type":"TextQuoteSelector","exact":"Profinite groups are compact (Exercise 1.9(a)) and totally disconnected; it can beproved that conversely every compact totally disconnected topological group is profinite","prefix":"way is called a profinitegroup. ","suffix":" (see[5, Chapter V, Theorem 1])."}]}]}
>```
>%%
>*%%PREFIX%%way is called a profinitegroup.%%HIGHLIGHT%% ==Profinite groups are compact (Exercise 1.9(a)) and totally disconnected; it can beproved that conversely every compact totally disconnected topological group is profinite== %%POSTFIX%%(see[5, Chapter V, Theorem 1]).*
>%%LINK%%[[#^6ff8ttn1p0x|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^6ff8ttn1p0x


>%%
>```annotation-json
>{"created":"2023-01-07T22:59:09.236Z","updated":"2023-01-07T22:59:09.236Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":19862,"end":20000},{"type":"TextQuoteSelector","exact":"Since each continuous bijection from a compact space to a Hausdorff spaceis a homeomorphism, each bijective homomorphism is an isomorphism","prefix":"se that is againa homomorphism. ","suffix":".1.9 Examples. Let G be an arbit"}]}]}
>```
>%%
>*%%PREFIX%%se that is againa homomorphism.%%HIGHLIGHT%% ==Since each continuous bijection from a compact space to a Hausdorff spaceis a homeomorphism, each bijective homomorphism is an isomorphism== %%POSTFIX%%.1.9 Examples. Let G be an arbit*
>%%LINK%%[[#^8gm98gzksd9|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^8gm98gzksd9


>%%
>```annotation-json
>{"created":"2023-01-07T23:00:44.508Z","updated":"2023-01-07T23:00:44.508Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":20319,"end":20408},{"type":"TextQuoteSelector","exact":". Hence ˆG = lim←−G/N is aprofinite group, and it is called the profinite completion of G","prefix":"eing the canonical homomorphisms","suffix":". In particular we haveˆZ= lim←−"}]}]}
>```
>%%
>*%%PREFIX%%eing the canonical homomorphisms%%HIGHLIGHT%% ==. Hence ˆG = lim←−G/N is aprofinite group, and it is called the profinite completion of G== %%POSTFIX%%. In particular we haveˆZ= lim←−*
>%%LINK%%[[#^qax42meowl|show annotation]]
>%%COMMENT%%
>
>%%TAGS%%
>
^qax42meowl


>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%lim←−Z/pnZis a profinite group.%%HIGHLIGHT%% ==It is in fact a profinite ring, the ring of p-adic integers== %%POSTFIX%%.Other important examples of pro*
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>%%
>```annotation-json
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>```
>%%
>*%%PREFIX%%e are now able to formulate the%%HIGHLIGHT%% ==main theorem of Galois theory for schemes.== %%POSTFIX%%1.11 Main theorem. Let X be a co*
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>%%
>```annotation-json
>{"created":"2023-01-07T23:31:10.706Z","updated":"2023-01-07T23:31:10.706Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":22543,"end":22745},{"type":"TextQuoteSelector","exact":"he fact that for X = Spec Zthere are no other finite  ́etale coverings of X is thus expressed by the group π(Spec Z)being trivial. The same is true for π(Spec K), where K is an algebraically closed fiel","prefix":"nts on which π acts trivially. T","suffix":"d. More9generally, if K is an ar"}]}]}
>```
>%%
>*%%PREFIX%%nts on which π acts trivially. T%%HIGHLIGHT%% ==he fact that for X = Spec Zthere are no other finite  ́etale coverings of X is thus expressed by the group π(Spec Z)being trivial. The same is true for π(Spec K), where K is an algebraically closed fiel== %%POSTFIX%%d. More9generally, if K is an ar*
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>%%
>```annotation-json
>{"created":"2023-01-07T23:35:51.221Z","updated":"2023-01-07T23:35:51.221Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":23129,"end":23582},{"type":"TextQuoteSelector","exact":"Next let X = Spec A, where A is the ring of integers in an algebraic number field K.Then π(X) is the Galois group of M over K, where M is the maximal algebraic extensionof K that is unramified at all non-zero prime ideals of A. More generally, if a ∈ A, a 6= 0,then π(Spec A[1/a]) is the Galois group, over K , of the maximal algebraic extension of Kthat is unramified at all non-zero prime ideals of A not dividing a. These facts will be provedin 6.18.","prefix":"if K is a finitefield (see 2.5).","suffix":"If p is a prime number, then π(S"}]}]}
>```
>%%
>*%%PREFIX%%if K is a finitefield (see 2.5).%%HIGHLIGHT%% ==Next let X = Spec A, where A is the ring of integers in an algebraic number field K.Then π(X) is the Galois group of M over K, where M is the maximal algebraic extensionof K that is unramified at all non-zero prime ideals of A. More generally, if a ∈ A, a 6= 0,then π(Spec A[1/a]) is the Galois group, over K , of the maximal algebraic extension of Kthat is unramified at all non-zero prime ideals of A not dividing a. These facts will be provedin 6.18.== %%POSTFIX%%If p is a prime number, then π(S*
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>%%
>```annotation-json
>{"created":"2023-01-07T23:37:53.287Z","updated":"2023-01-07T23:37:53.287Z","document":{"title":"GSchemes.pdf","link":[{"href":"urn:x-pdf:6d092b278ec538c11ac933158399bfe0"},{"href":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf"}],"documentFingerprint":"6d092b278ec538c11ac933158399bfe0"},"uri":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","target":[{"source":"https://websites.math.leidenuniv.nl/algebra/GSchemes.pdf","selector":[{"type":"TextPositionSelector","start":26295,"end":26388},{"type":"TextQuoteSelector","exact":"If K is a field, then one has π(P1K) = π(Spec K), where P1K denotes theprojective line over K","prefix":"er I, Section 5]).1.16 Example. ","suffix":". If moreover char(K) = 0, then "}]}]}
>```
>%%
>*%%PREFIX%%er I, Section 5]).1.16 Example.%%HIGHLIGHT%% ==If K is a field, then one has π(P1K) = π(Spec K), where P1K denotes theprojective line over K== %%POSTFIX%%. If moreover char(K) = 0, then*
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