---- - [ ] general categorical description > [!definition] Definition. (Of Modules) > A **short exact sequence** of [[linear map|modules]] is an [[exact sequence|exact]] [[chain complex of modules|complex]] of the form $0 \xrightarrow{ \ \ \ \ \ } L \xrightarrow{ \ \alpha \ } M \xrightarrow{ \ \beta \ } N \xrightarrow{ \ \ \ \ \ }0.$ > Equivalently, since $\im \beta=\text{ker }(N \to 0)=N$ ($\beta$ is a [[surjection]]) and exactness ensures $\im \alpha=\ker \beta$, the [[first isomorphism theorem for modules]] provides us with [[module isomorphism|an identificiation]] $N \cong \frac{M}{\im \alpha}$. So the form $0 \xrightarrow{ \ \ \ \ \ } L \xrightarrow{ \ \alpha \ }_{\subset} M \xrightarrow{ \ \beta \ } \frac{M}{\im \alpha} \xrightarrow{ \ \ \ \ \ }0$ > where the fact $\alpha$ is an [[injection]] allows one to identify $L$ with a [[submodule]] $L \subset M$, may be equivalently used for defining a short exact sequence. > > There is a sense in which every [[exact sequence]] breaks up into many [[short exact sequence|short exact sequences]]. Behold: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoAmAXVJADcBDAGwFcYkQBZAfWCwGpyAXxCDS6TLnyEUARlIBmanSat2AHTU4YADxy8AtgAJBhqD35DDAXkMatu4AGsYAJ2NnefGcNHjseAiJyCiUGFjZETnMvHzEQDH8pIgAWEJow1UjuLBE4hMlAlAA2NOVw9m5eAFpvXL8C6WRg5NCVCJBiOviJAMayFvS29k7fbsTC5HlSGVbyyLsdPSwjEw9+b2tbTUWnV3dzWPrelNJiWcyOrvzjlGCzwbnL0euklABWBXP2hYdl41MDpsfnpnG5BGsaocxg0TgMyhcRnkeq9kCU4Rl2iMlDAoABzeBEUAAMxcEH0SDIIBwECQchAAAsYPQoOwcAB3CCM5kIUYksm0mjUpDBKn0LCMdj0iAQRxdPnkxCUoWIEUY9hrARQ+VIEpUmmIOk4MUSyJSmVy0kKw36j4MpksyLszn2nlxbUqwX6qbw9oa2o0Rj0ABGMEYAAVkYUQC4sLj6TgLfyPXqkAB2Ghch1UjmZ13Ey1pz1IAAcGftrJzLsTCoAnEXEOnReLJdLZbyCw3696jc3Ta3q0hu-rUj71Vwcu2k0PaZTMxXndyBwalfqZCK547K4vJ1brTPBcaW+ad-uU4gRz2TQz+4JKIIgA > \begin{tikzcd} > & & 0 \arrow[rd, hook] & & 0 & & \\ > & & & \text{im } d_{i+1} = \text{ker }d_{i} \arrow[ru, two heads] \arrow[rd, hook] & & & \\ > M_{i+2} \arrow[rd, two heads] \arrow[rr, "d_{i+2}"] & & M_{i+1} \arrow[rr, "d_{i+1}"'] \arrow[ru, two heads] & & M_i \arrow[rr, "d_i"] \arrow[rd, two heads] & & M_{i-1} \\ > & \text{im } d_{i+2} = \text{ker }d_{i+1} \arrow[ru, hook] \arrow[rd, two heads] & & & & \text{im } d_{i} = \text{ker }d_{i-1} \arrow[rd, two heads] \arrow[ru, hook] & \\ > 0 \arrow[ru, hook] & & 0 & & 0 \arrow[ru, hook] & & 0 > \end{tikzcd} > \end{document} > ``` > The diagonal sequences are short exact sequences. > [!basicexample] > - A single [[linear map]] $\varphi:M \to M'$ gives rise to a short exact sequence $0 \xrightarrow{} \ker \varphi \xrightarrow{\iota} M \xrightarrow{\varphi} \im \varphi \xrightarrow{}0.$ ^basic-example > [!definition] (Of [[chain complex of modules|chain complexes]]) > [[chain map|Chain maps]] $i_{\bullet}:A_{\bullet} \to B_{\bullet}$ and $j_{\bullet}:B_{\bullet} \to C_{\bullet}$ form a **short exact sequence of chain complexes** if for each $n$ $0 \to A_{n} \xrightarrow{i_{n}} B_{n} \xrightarrow{j_{n}} C_{n} \to 0$ is a short exact sequence of [[abelian group|abelian groups]]. > [!definition] (Of [[sheaf|sheaves]]) > Things behave how one hopes. See the discussion in [[characterizing short exact sequences of sheaves]]. ^definition > [!definition] Definition. (Of general [[group|groups]]) > A **short exact sequence** is an [[exact sequence]] of the form $1 \to H \xrightarrow{\varphi} G \xrightarrow{\psi}K \to 1.$ > Equivalently (up to [[group isomorphism]]) it is an [[exact sequence]] of the form $1 \to H \xrightarrow{\varphi} G \xrightarrow{\psi} G / H \to 1.$ > [!equivalence] >- Exactness at $H$ means $\im \{ 1\}=1=\ker \varphi$, [[group homomorphism is injective iff kernel is trivial iff is a monomorphism|i.e.,]] $\varphi$ is an [[injection]]. > > - Hence $\textcolor{Thistle}{H \cong \varphi(H)}$. >- Exactness at $K$ means $\im \psi = \ker (K \to 1) = K$, i.e., $\psi$ is a [[surjection]]. >- Exactness at $G$ means $\im \varphi = \textcolor{Thistle}{\varphi(H) \cong H} = \ker \psi \trianglelefteq G$. ([[kernel iff normal subgroup|see.]]) > >Hence [[first isomorphism theorem|FIT]] implies $G / \ker \psi \cong \im \psi$, i.e., $G / H \cong K$. > Thus **short exact sequences** look like $1 \to H \xrightarrow{\varphi} G \xrightarrow{\psi} G / H \to 1.$ > [!note] Remark. > Given $H$ and $K$, what can $G$ look like? An obvious choice is $G=H \times K$: $1 \to H \to H \times K \to K \to 1,$where >- The map $H \hookrightarrow H \times K$ is $h \mapsto (h,1)$; >- The map $H \times K \twoheadrightarrow K$ is $(h,k) \to k$; >- By [[internal direct product of subgroups|direct product characterizations]] $H$ is isomorphic to a [[normal subgroup]] of $H \times K$. \ More generally, given any $\phi: K \to \text{Aut } H$ we can take $G = H \rtimes_{\phi} K$ and all the same things as above happen. \ But these are not the only choices (question: are they the only choices for [[split exact sequences]])? For example, let $H=C_{2}=K$ and ask, for $1 \to C_{2} \to G \to C_{2} \to 1,$ what can $G$ be? Using the characterization this looks like $1 \to C_{2} \to G \to G / C_{2} \to 1,$ $|G|=4$ so $G \cong C_{4}$ or $C_{2} \times C_{2}$. We showed above that $C_{2} \times C_{2}$ works. What about $C_{4}$? It works too: all we need to verify is that $C_{4}$ contains a [[normal subgroup]] [[group isomorphism|isomorphic to]] $C_{2}$; this is just $\{ e, x^{2} \}$ which [[subgroups of index 2 are normal|has index 2]]. However, $C_{4}$ is certainly not isomorphic to $C_{2} \rtimes_{\phi} C_{2}$ for any $\phi: C_{2} \to \text{Aut }C_{2}$, since $\text{Aut }C_{2} \cong C_{1}$ is trivial. Find *all* $G$ (up to [[group isomorphism|isomorphism]]) in the following cases: ## $H=C_{2}, K=C_{2}$ — $G \in \{ C_{4}, C_{2} \times C_{2} \}$ The [[short exact sequence]] looks like $1 \to C_{2} \to G \to C_{2} \to 1.$ $|G|=4$; the remark above shows $G \in \{ C_{4}, C_{2} \times C_{2}\}$ ## $H=C_{2}, K=C_{3}$ — $G \in \{ C_{2} \times C_{3} \cong C_{6}\}$ We have $1\to C_{2} \to G \to C_{3} \to 1.$ $|G|=6$ so we must examine $C_{6}$ and $D_{3}$. $C_{6} \cong C_{2} \times C_{3}$, so it works. To have $D_{3} / C_{2} \cong C_{3}$ we need that $D_{3}$ has a normal [[subgroup]] of [[order of an element in a group|order]] $2$ in the first place— it does not. ## $H=C_{3}, K=C_{2}$ — $G \in \{ C_{3} \times C_{2} \cong C_{6} , D_{3}\}$ We consider $1 \to C_{3} \to G \to C_{2} \to 1.$ $|G|=6$ so again we examine $C_{6}$ and $D_{3}$. Again $C_{6}$ works because it is a [[direct product of groups|direct product]] of $C_{3}$ and $C_{2}$. This time $D_{3}$ works as well because the rotations are a [[normal subgroup]] of order $3$ and the [[quotient group|quotient]] will have order $2$ and thus must be $C_{2}$. ## $H=C_{4}, K=C_{2}$ — $G \in \{C_{8}, C_{4} \times C_{2}, D_{4}, Q_{8}\}$ We consider $1 \to C_{4} \to G \to C_{2} \to 1.$ $|G|=8$, so [[classification of small groups|we examine]] the possible cases - $C_{8}$ - $C_{4} \times C_{2}$ - $C_{2} \times C_{2} \times C_{2}$ - $D_{4}$ - $Q_{8}$ Clearly $C_{4} \times C_{2}$ works, as a [[direct product of groups|direct product]]. We are constrained by the fact $G / C_{4} \cong C_{2}$; For the other groups, we just need to find a [[normal subgroup]] isomorphic to $C_{4}$, since the [[quotient group|quotient]] will automatically then be $C_{2}$ as required. Similarly, $D_{4} \cong C_{4} \rtimes C_{2}$ works since it is a [[external semi-direct product|semi-direct product]]. $4|8$, [[uniqueness of subgroups of finite cyclic groups|so]] $C_{8}$ has a (unique) [[subgroup]] [[group isomorphism|isomorphic to]] $C_{4}$. $C_{8}$ is [[abelian group|abelian]] so this [[subgroup]] is certainly normal. Thus $C_{8}$ works. On the other hand, $C_{2} \times C_{2} \times C_{2}$ has elements of exclusively orders 1 and 2, so it does not work. The [[quaternion group]] $Q_{8}$ has 3 [[normal subgroup]]s [[group isomorphism|isomorphic]] to $C_{4}$. So it works. ## $H=C_{2}$, $K=C_{4}$ — $G \in \{ C_{8}, C_{2} \times C_{4} \}$ We consider $1 \to C_{2} \to G \to C_{4} \to 1.$ $|G|=8$, so [[classification of small groups|we examine]] the possible cases - $C_{8}$ - $C_{2} \times C_{4}$ - $C_{2} \times C_{2} \times C_{2}$ - $D_{4}$ - $Q_{8}$ We need $G$ such that $G / C_{2} \cong C_{4}$. $C_{8}$ has as a [[uniqueness of subgroups of finite cyclic groups|unique subgroup isomorphic to]] $C_{2}$ which is [[normal subgroup|normal]] because $C_{8}$ is [[abelian group|abelian]]. The [[quotient group|quotient]] $C_{8} / C_{2}$ is [[group isomorphism|isomorphic to]] $C_{4}$ (since the [[coset|element]] $x\{ e, x^{4} \}$ has order $4$). So $C_{8}$ works. $C_{2} \times C_{4}$ clearly works as a direct product. Quotienting $C_{2} \times C_{2} \times C_{2}$ by $C_{2}$ gives $C_{2} \times C_{2}$, not $C_{4}$. So it doesn't work. $D_{4}$ doesn't has a unique normal subgroup $\{ e, x^{2} \}$ of order $2$. If we look at its cosets we find $|y\{ e, x^{2} \}|=2$ and $|yx\{ e,x ^{2} \}|=2$ which is enough to conclude that the [[quotient group]] is $C_{2} \times C_{2}$. $Q_{8}$ has a unique [[normal subgroup]] of order 2, $\{ \pm \b 1 \}$. But we can see that the elements in the quotient all have order 2... e.g., $i^{2}\{ 1, -1\}=\{ 1,-1 \}$ and the pattern continues. So the quotient is the [[Klein 4-group]], not $C_{4}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```