Bockstein, Mod-\(p\), and the Descent Spectral Sequences¶
Job: develop the Bockstein SS (mod-\(p\) torsion), the Cartan–Leray / Galois descent, the Lyndon–Hochschild–Serre (group extension), and the change-of-rings / base change — with the Bockstein for \(H^*(\mathbb{RP}^\infty;\mathbb{Z})\) / the Cartan–Leray as the worked example.
Core¶
The Bockstein spectral sequence¶
| Bockstein / \(\beta\) / SS / \(p\)-torsion | Description |
|---|---|
| Bockstein \(\beta\) | \(H^n(X;\mathbb{Z}/p)\to H^{n+1}(X;\mathbb{Z})\) (the short exact \(0\to\mathbb{Z}\xrightarrow{p}\mathbb{Z}\to\mathbb{Z}/p\to0\)) |
| Bockstein SS | \(E_1=H^*(X;\mathbb{Z}/p)\), \(d_1=\beta\), \(E_\infty=H^*(X;\mathbb{Z})/p\)-torsion |
The descent and the Cartan–Leray¶
| Cartan–Leray / \(G\) / descent / Galois | Description |
|---|---|
| Cartan–Leray | \(H^*(G;H^*(X))\Rightarrow H^*(X/G)\) (the quotient, the topology) |
| Descent / Galois | \(H^*(\mathrm{Gal};H^*)\Rightarrow H^*\) (the Galois, the arithmetic geometry / étale) |
The Lyndon–Hochschild–Serre (LHS)¶
| LHS / \(N\to G\to Q\) / \(E_2\) / group | Description |
|---|---|
| LHS | \(E_2=H^p(Q;H^q(N;M))\Rightarrow H^{p+q}(G;M)\) for \(1\to N\to G\to Q\to1\) (the homological algebra) |
The change-of-rings and the base change¶
| Change-of-rings / \(\mathrm{Ext}\) / base change / AG | Description |
|---|---|
| Change-of-rings | \(\mathrm{Ext}_S\cong\mathrm{Ext}_R\) (the homological algebra) |
| Base change / flat | \(H^*(X\times_S Y)\) the base change (the algebraic geometry / cohomology) |
See it¶
Bockstein for \(H^*(\mathbb{RP}^\infty;\mathbb{Z})\) and the Cartan–Leray for \(S^\infty\to\mathbb{RP}^\infty\). The Bockstein: \(H^*(\mathbb{RP}^\infty;\mathbb{Z}/2)=\mathbb{Z}/2[x]\) (\(|x|=1\)), the \(\beta(x)=y\) the \(H^2\) the \(\mathbb{Z}/2\)-torsion, the \(H^*(\mathbb{RP}^\infty;\mathbb{Z})=\mathbb{Z}[y]/(2y)\) (\(|y|=2\), the \(2\)-torsion). The Cartan–Leray: \(S^\infty\to\mathbb{RP}^\infty\) the \(G=\mathbb{Z}/2\) the free action, the \(H^*(\mathbb{Z}/2;H^*(S^\infty))\Rightarrow H^*(\mathbb{RP}^\infty)\) (the \(S^\infty\) the contractible, the \(H^*(\mathbb{Z}/2;k)=k[x]\)). The key point: the Bockstein the \(p\)-torsion, the Cartan–Leray the quotient — the descent the spectral sequence.
Check¶
Show that the Bockstein \(\beta(x)=y\) for \(H^*(\mathbb{RP}^\infty;\mathbb{Z})\), and explain the \(2\)-torsion.
The \(H^*(\mathbb{RP}^\infty;\mathbb{Z}/2)=\mathbb{Z}/2[x]\) (\(|x|=1\), the \(\mathbb{Z}/2\)-coefficients). The Bockstein: the \(0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to0\), the \(\beta(x)\in H^2(\mathbb{RP}^\infty;\mathbb{Z})\) the connecting. The \(H^*(\mathbb{RP}^\infty;\mathbb{Z})\): the \(\mathbb{Z}[y]/(2y)\) (\(|y|=2\), the \(y=\beta(x)\), the \(2\)-torsion the \(2y=0\)). So the \(\beta\) the \(y\), the \(H^*(\mathbb{RP}^\infty;\mathbb{Z})\) the \(2\)-torsion.
Explain the LHS for \(1\to N\to G\to Q\to1\) and the Cartan–Leray for the quotient.
The LHS: the \(1\to N\to G\to Q\to1\) the group extension, the \(E_2=H^p(Q;H^q(N;M))\Rightarrow H^{p+q}(G;M)\) (the \(Q\) the quotient, the \(N\) the kernel, the homological algebra). The Cartan–Leray: the \(G\) the group acting freely on \(X\), the \(H^*(G;H^*(X))\Rightarrow H^*(X/G)\) (the quotient, the topology). The LHS the group, the Cartan–Leray the space: the \(X=EG\) the contractible the free \(G\)-space, the \(H^*(G;H^*(EG))=H^*(G)\Rightarrow H^*(BG)\) (the \(BG=EG/G\), the classifying). So the LHS the group extension, the Cartan–Leray the quotient.
Pitfalls¶
- The Bockstein the \(p\)-torsion (the \(\beta\), the \(0\to\mathbb{Z}\xrightarrow{p}\mathbb{Z}\to\mathbb{Z}/p\to0\)) — the \(\beta\) the connecting, the \(E_\infty\) the \(p\)-torsion.
- The Cartan–Leray the quotient (the \(H^*(G;H^*(X))\Rightarrow H^*(X/G)\), the topology) — the \(G\) the free, the \(X/G\) the quotient.
- The LHS the group extension (\(1\to N\to G\to Q\to1\), the homological algebra) — the LHS the group, the \(E_2\) the \(H^*(Q;H^*(N))\).
- The change-of-rings (\(\mathrm{Ext}_S\cong\mathrm{Ext}_R\), the homological algebra) — the change-of-rings the \(\mathrm{Ext}\), the base change the AG.
Refs¶
- Bockstein, "Eine neue Homologiegruppe" (1958); the \(\beta\)
- Cartan–Leray, "Structures et théorèmes d'homologie" (1947); the Cartan–Leray
- Hochschild–Serre, "Cohomology of group extensions" (1953); the LHS
- Cartan–Eilenberg, Homological Algebra (1956); the change-of-rings
Next¶
SS index · Foundations · Adams & Adams–Novikov · Eilenberg–Moore & Bar · Motivic & Slice · Topology · Homological algebra · AG cohomology