# Ext函子

## 定義

${\displaystyle {\mathcal {C}}}$ 為有充足內射元的阿貝爾範疇，例如一個 ${\displaystyle R}$ 上的左範疇 ${\displaystyle R-\mathbf {Mod} }$。固定一對象 ${\displaystyle A}$，定義函子 ${\displaystyle T_{A}(-):=\mathrm {Hom} _{\mathcal {C}}(A,-)}$，此為左正合函子，故存在右導函子 ${\displaystyle R^{\bullet }T_{A}(-)}$，記為 ${\displaystyle \mathrm {Ext} _{\mathcal {C}}^{\bullet }(A,-)}$。當 ${\displaystyle {\mathcal {C}}=R-\mathbf {Mod} }$ 時，常記之為 ${\displaystyle \mathrm {Ext} _{R}^{\bullet }(A,-)}$

${\displaystyle J(B)\longleftarrow B\longleftarrow 0}$

${\displaystyle \mathrm {Hom} _{\mathcal {C}}(A,J(B))\longleftarrow \mathrm {Hom} _{\mathcal {C}}(A,B)\longleftarrow 0}$

${\displaystyle P(A)\longrightarrow A\longrightarrow 0}$

${\displaystyle \mathrm {Hom} _{\mathcal {C}}(P(A),B)\longrightarrow \mathrm {Hom} _{\mathcal {C}}(A,B)\longrightarrow 0}$

## 基本性質

• ${\displaystyle A}$射影對象${\displaystyle B}$內射對象，則對所有 ${\displaystyle i>0}$${\displaystyle \mathrm {Ext} _{\mathcal {C}}^{i}(A,B)=0}$
• 反之，若 ${\displaystyle \mathrm {Ext} _{\mathcal {C}}^{1}(A,-)=0}$，則 ${\displaystyle A}$射影對象。若 ${\displaystyle \mathrm {Ext} _{\mathcal {C}}^{1}(-,B)=0}$，則 ${\displaystyle B}$內射對象
• ${\displaystyle \mathrm {Ext} _{\mathcal {C}}^{\bullet }(\bigoplus _{i}A_{i},B)=\coprod _{i}\mathrm {Ext} _{\mathcal {C}}^{\bullet }(A_{i},B)}$
• ${\displaystyle \mathrm {Ext} _{\mathcal {C}}^{\bullet }(A,\prod _{j}B_{j})=\prod _{j}\mathrm {Ext} _{\mathcal {C}}^{\bullet }(A,B_{j})}$
• 根據導函子性質，對每個短正合序列 ${\displaystyle 0\to B'\to B\to B''\to 0}$，有長正合序列
${\displaystyle \cdots \to \mathrm {Ext} _{\mathcal {C}}^{n-1}(A,B'')\to \mathrm {Ext} _{\mathcal {C}}^{n}(A,B')\to \mathrm {Ext} _{\mathcal {C}}^{n}(A,B)\to \mathrm {Ext} _{\mathcal {C}}^{n}(A,B'')\to \mathrm {Ext} _{\mathcal {C}}^{n+1}(A,B'')\to \cdots }$
• 承上，若 ${\displaystyle {\mathcal {C}}}$ 有充足的射影元，則對第一個變數也有長正合序列；換言之，對每個短正合序列 ${\displaystyle 0\to A'\to A\to A''\to 0}$，有長正合序列
${\displaystyle \cdots \to \mathrm {Ext} _{\mathcal {C}}^{n-1}(A',B)\to \mathrm {Ext} _{\mathcal {C}}^{n}(A'',B)\to \mathrm {Ext} _{\mathcal {C}}^{n}(A,B)\to \mathrm {Ext} _{\mathcal {C}}^{n}(A',B)\to \mathrm {Ext} _{\mathcal {C}}^{n+1}(A'',B)\to \cdots }$

## 譜序列

${\displaystyle \mathrm {Hom} _{B}(-,\mathrm {Hom} _{A}(B,-))\simeq \mathrm {Hom} _{A}(-,-)}$

${\displaystyle E_{2}^{pq}=\mathrm {Ext} _{B}^{p}(M,\mathrm {Ext} _{A}^{q}(B,N))\Rightarrow \mathrm {Ext} _{A}^{p+q}(M,N)}$

## Ext函子與擴張

Ext 函子得名於它與群擴張的聯繫。抽象地說，給定兩個對象 ${\displaystyle A,B\in {\mathcal {C}}}$，在擴張

${\displaystyle 0\rightarrow B\rightarrow C\rightarrow A\rightarrow 0}$

${\displaystyle 0\rightarrow B\rightarrow C\rightarrow A\rightarrow 0}$
${\displaystyle 0\rightarrow B\rightarrow C'\rightarrow A\rightarrow 0}$

${\displaystyle 0\rightarrow B\rightarrow X_{n}\rightarrow \cdots \rightarrow X_{1}\rightarrow A\rightarrow 0}$
${\displaystyle 0\rightarrow B\rightarrow X'_{n}\rightarrow \cdots \rightarrow X'_{1}\rightarrow A\rightarrow 0}$

${\displaystyle 0\rightarrow B\rightarrow Y_{n}\rightarrow X_{n-1}\oplus X'_{n-1}\rightarrow \cdots \rightarrow X_{2}\oplus X'_{2}\rightarrow X''_{1}\rightarrow A\rightarrow 0}$

## 重要例子

• ${\displaystyle G}$ 為群，取環 ${\displaystyle R:=\mathbb {Z} G}$，可以得到群上同調${\displaystyle \mathrm {Ext} _{\mathbb {Z} G}^{\bullet }(\mathbb {Z} ,M)=H^{\bullet }(G,M)}$
• ${\displaystyle {\mathcal {C}}}$局部賦環空間 ${\displaystyle X}$ 上的 ${\displaystyle {\mathcal {O}}_{X}}$-模範疇，可以得到層上同調${\displaystyle \mathrm {Ext} _{\mathcal {C}}^{\bullet }({\mathcal {O}}_{X},{\mathcal {F}})=H^{\bullet }(X,{\mathcal {F}})}$
• ${\displaystyle {\mathfrak {g}}}$李代數，取環 ${\displaystyle R:=U({\mathfrak {g}})}$ 為其泛包絡代數，可以得到李代數上同調${\displaystyle \mathrm {Ext} _{R}^{\bullet }(R,M)=H^{\bullet }({\mathfrak {g}},M)}$
• ${\displaystyle k}$ 為域，${\displaystyle A}$${\displaystyle k}$-代數，取環 ${\displaystyle R:=A\times A^{\mathrm {op} }}$${\displaystyle A}$ 帶有自然的 ${\displaystyle R}$-模結構，此時得到 Hochschild 上同調：${\displaystyle \mathrm {Ext} _{R}^{\bullet }(A,M)=HH^{\bullet }(A,M)}$

## 文獻

• Charles A. Weibel, An introduction to homological algebra, Cambridge University Press. ISBN 0-521-55987-1