Exact Sequences
The sequence \(A \xrightarrow{f_1} B \xrightarrow{f_2} C\) is exact if and only if \(\operatorname{im}f_i = \ker f_{i+1}\) and thus \(f_2 \circ f_1 = 0\).
Some useful results:
- \(0 \to A \hookrightarrow_{f} B\) is exact iff \(f\) is injective
- \(B\twoheadrightarrow_{f} C \to 0\) is exact iff \(f\) is surjective
- \(0\to A \to B \to 0\) is exact iff \(A \cong B\).
- \(A \hookrightarrow B \to C \to D \twoheadrightarrow E\) iff \(C = 0\)
-
\(0\to A \to B \xrightarrow{\cong} C \to D\to 0\) iff \(A = D = 0\).
- Todo: Proof
- \(0\to A\to B \to C \to 0\) splits iff \(C\) is free.
- Can think of \(C \cong \frac{B}{\operatorname{im}f_1}\).
The sequences splits when a morphism \(f_2^{-1}: C \to B\) exists. In \(\textbf{Ab}\), this means \(B \cong A \oplus C\), in \(\mathbf{Grp}\) it’s \(B \cong A \rtimes_\phi C\).
- \(0 \to{\mathbf{Z}}\xrightarrow{\times 2} {\mathbf{Z}}\xrightarrow{\text{mod}~2} \frac{{\mathbf{Z}}}{2{\mathbf{Z}}} \to 0\)
-
\(1 \to N \xrightarrow{\iota} G \xrightarrow{p} \frac{G}{N} \to 1\)
- Groups and normal subgroups
-
\(1 \to\frac{{\mathbf{Z}}}{n{\mathbf{Z}}} \xrightarrow{\iota} D_{2n} \xrightarrow{?} \frac{{\mathbf{Z}}}{2{\mathbf{Z}}} \to 1\)
- Dihedral group and cyclic groups
-
\(0 \to I \cap J \xrightarrow{\Delta: x\mapsto(x,x)} I \oplus J \xrightarrow{f:(x,y) \mapsto x-y} I + J \to 0\)
- \(R\)-Modules
- \(0 \to\frac{R}{I \cap J} \xrightarrow{\Delta: x\mapsto(x,x)} \frac{R}{I} \oplus \frac{R}{J} \xrightarrow{f:(x,y) \mapsto x-y} \frac{R}{I + J} \to 0\)
-
\(0 \to\mathbb{H}_1 \xrightarrow{\nabla} \mathbb{H}_\text{curl} \xrightarrow{\nabla \times} \mathbb{H}_\text{div} \xrightarrow{\nabla \cdot} \mathbb{L}_2 \to 0\)
- Since \(\nabla \times\nabla F = \nabla \cdot\nabla\times\overline{v} = 0\) in Hilbert spaces
Is \(f_1\circ f_2 = 0\) equivalent to exactness..? Answer: yes, every exact sequence is a chain complex with trivial homology. Therefore homology measures the failure of exactness.
Alternatively stated: Exact sequences are chain complexes with no cycles.
Any LES \(A_1 \to\cdots \to A_6\) decomposes into a twisted collection of SES’s; define \(C_k = \ker (A_k \to A_{k+1}) \cong \operatorname{im}(A_{k-1} \to A_k)) \cong \operatorname{coker}(A_{k-2} \to A_{k-1})\), then all diagonals here are exact:
Five Lemma
If \(m, p\) are isomorphisms, \(l\) is an surjection, and \(q\) is an injection, then \(n\) is an isomorphism.
Proof: diagram chase two “four lemmas”, one on each side. Full proof here.
Free Resolutions
The canonical example: \begin{align*} 0 \to {\mathbf{Z}}\xrightarrow{\times m} {\mathbf{Z}}\xrightarrow{\operatorname{mod}m} {\mathbf{Z}}_m \to 0 \end{align*}
Or more generally for a finitely generated group \(G = \left\langle{g_1, g_2, \cdots, g_n}\right\rangle\), \begin{align*} \cdots \to \ker(f) \to F[g_1, g_2, \cdots, g_n] \xrightarrow{f} G \to 0 \end{align*} where \(F\) denotes taking the free group.
Every abelian groups has a resolution of this form and length 2.
Properties of Tensor
- \(A\otimes B \cong B\otimes A\)
- \(({-}) \otimes_R R^n = \operatorname{id}\)
- \(\bigoplus_i A_i \otimes\bigoplus_j B_j = \bigoplus_i\bigoplus_j(A_i \otimes B_j)\)
- \({\mathbf{Z}}_m \otimes{\mathbf{Z}}_n = {\mathbf{Z}}_d\)
- \({\mathbf{Z}}_n \otimes A = A/nA\)
Properties of Hom
- \(\hom_R (\bigoplus_i A_i, \prod B_j) = \bigoplus_i \prod_j \hom(A_i, B_j)\)
- Contravariant in first slot, covariant in second
- Exact over vector spaces
Properties of Tor
- \(\operatorname{Tor}_R^0(A, B) = A \otimes_R B\)
- \(\operatorname{Tor}(\bigoplus_i A_i, \bigoplus_j B) = \bigoplus_i \bigoplus_j \operatorname{Tor}(\mathbf{T}A_i, \mathbf{T}B_j)\) where \(\mathbf{T}G\) is the torsion component of \(G\).
- \(\operatorname{Tor}(A, B) = \operatorname{Tor}(B, A)\)
- \(\operatorname{Tor}({\mathbf{Z}}_n, G) = \ker (g \mapsto ng) = \left\{{g\in G\mathrel{\Big|}ng = 0}\right\}\)
Properties of Ext
- \(\operatorname{Ext}_R^0(A, B) = \hom_R(A, B)\)
- \(\operatorname{Ext}(\bigoplus_i A_i, \prod_j B_j) = \bigoplus_i \prod_j \operatorname{Ext}(\mathbf{T}A_i, B_j)\)
- \(\operatorname{Ext}(F, G) = 0\) if \(F\) is free
- \(\operatorname{Ext}({\mathbf{Z}}_n, G) \cong G/nG\)
Computing Tor
\begin{align*} \operatorname{Tor}(A, B) = h[\cdots \to A_n \otimes B \to A_{n-1}\otimes B \to \cdots A_1\otimes B \to 0] \end{align*} where \(A_*\) is any free resolution of \(A\).
Shorthand/mnemonic: \begin{align*} \operatorname{Tor}: \mathcal{F}(A) \to ({-}\otimes B) \to H_* \end{align*}
Computing Ext
\begin{align*} \operatorname{Ext}(A, B) = h[\cdots \hom(A, B_n) \to \hom(A, B_{n-1}) \to \cdots \to \hom(A, B_1) \to 0 ] \end{align*} where \(B_*\) is a any free resolution of \(B\).
Shorthand/mnemonic: \begin{align*} \operatorname{Ext}: \mathcal{F}(B) \to \hom(A, {-}) \to H_* \end{align*}
Hom/Ext/Tor Tables
| \(\hom\) | \({\mathbf{Z}}_m\) | \({\mathbf{Z}}\) | \({\mathbf{Q}}\) |
|---|---|---|---|
| \({\mathbf{Z}}_n\) | \({\mathbf{Z}}_d\) | \(0\) | \(0\) |
| \({\mathbf{Z}}\) | \({\mathbf{Z}}_m\) | \({\mathbf{Z}}\) | \({\mathbf{Q}}\) |
| \({\mathbf{Q}}\) | \(0\) | \(0\) | \({\mathbf{Q}}\) |
| \(\operatorname{Tor}\) | \({\mathbf{Z}}_m\) | \({\mathbf{Z}}\) | \({\mathbf{Q}}\) |
|---|---|---|---|
| \({\mathbf{Z}}_n\) | \({\mathbf{Z}}_d\) | \(0\) | \(0\) |
| \({\mathbf{Z}}\) | \(0\) | \(0\) | \(0\) |
| \({\mathbf{Q}}\) | \(0\) | \(0\) | \(0\) |
| \(\operatorname{Ext}\) | \({\mathbf{Z}}_m\) | \({\mathbf{Z}}\) | \({\mathbf{Q}}\) |
|---|---|---|---|
| \({\mathbf{Z}}_n\) | \({\mathbf{Z}}_d\) | \({\mathbf{Z}}_n\) | \(0\) |
| \({\mathbf{Z}}\) | \(0\) | \(0\) | \(0\) |
| \({\mathbf{Q}}\) | \(0\) | \(\mathcal{A_p}/{\mathbf{Q}}\) | \(0\) |
Where \(d = \gcd(m, n)\) and \({\mathbf{Z}}_0 \coloneqq 0\).
Things that behave like “the zero functor”:
- \(\operatorname{Ext}({\mathbf{Z}}, {-})\)
- \(\operatorname{Tor}({-}, {\mathbf{Z}}), \operatorname{Tor}({\mathbf{Z}}, {-})\)
- \(\operatorname{Tor}({-}, {\mathbf{Q}}), \operatorname{Tor}({\mathbf{Q}}, {-})\)
Thins that behave like “the identity functor”:
- \(\hom({\mathbf{Z}}, {-})\)
- \({-}\otimes_{\mathbf{Z}}{\mathbf{Z}}\) and \({\mathbf{Z}}\otimes_{\mathbf{Z}}{-}\)
For description of \(\mathcal{A_p}\), see here. This is a certain ring of adeles.