Eilenberg–Moore, the Bar, and the Koszul Spectral Sequences¶
Job: develop the EMSS (Tor / loop space), the bar complex / bar SS, the Koszul / Koszul duality, and the Rothenberg–Steenrod — with the EMSS for \(\Omega S^n\) as the worked example.
Core¶
The Eilenberg–Moore spectral sequence (EMSS)¶
| EMSS / Tor / loop space | Description |
|---|---|
| EMSS | \(\mathrm{Tor}_{H^*(B)}^{*,*}(H^*(E),H^*(F))\Rightarrow H^*(\Omega B)\) (the loop space, the Tor) |
| Tor | the \(\mathrm{Tor}_{H^*(B)}\) the derived tensor (the homological algebra) |
| \(\Omega B\) | the loop space (the topology) |
The bar spectral sequence and the bar complex¶
| Bar / \(B(A)\) / Tor | Description |
|---|---|
| Bar complex \(B(A)\) | the \(A\) the algebra, the bar (the homological algebra) |
| Bar SS | the \(\mathrm{Tor}_A\) the bar (the homological algebra) |
The Koszul and the duality¶
| Koszul / \(K\) / duality / \(A^!\) | Description |
|---|---|
| Koszul complex \(K(x_1,\dots,x_n)\) | the regular sequence, the Koszul (the homological algebra) |
| Koszul duality | the Koszul dual \(A^!\) (the PBW, the homological algebra) |
The Rothenberg–Steenrod and the path / loop¶
| Rothenberg–Steenrod / \(H^*(G)\) / \(BG\) | Description |
|---|---|
| Rothenberg–Steenrod | \(H^*(G)\cong\mathrm{Tor}_{H^*(BG)}\) (the group / classifying, the topology) |
| Path / loop | \(\Omega BG\) the loop, \(H^*(G)\) the group (the topology) |
See it¶
EMSS for \(\Omega S^n\) and the bar / Tor. The EMSS: \(\mathrm{Tor}_{H^*(S^n)}(k,k)\Rightarrow H^*(\Omega S^n)\) (the \(S^n\) the base, the \(\Omega S^n\) the loop). The \(\mathrm{Tor}_{k[x]}(k,k)=\Lambda(e)\) (the exterior, \(H^*(\Omega S^n)=\Lambda(e_{n-1})\) for the odd \(n\) the free). The bar: the \(\mathrm{Tor}_A\) the bar complex \(B(A)\) (the homological algebra). The key point: the EMSS the \(\Omega S^n\) the Tor, the bar the \(\mathrm{Tor}\) — the loop space the Eilenberg–Moore.
Check¶
Show that \(\mathrm{Tor}_{k[x]}(k,k)=\Lambda(e)\) and explain the EMSS for \(\Omega S^n\).
The \(\mathrm{Tor}_{k[x]}(k,k)\): the \(k[x]\) the polynomial, the resolution \(0\to k[x]\xrightarrow{x}k[x]\to k\to0\), the \(\mathrm{Tor}_0=k\), \(\mathrm{Tor}_1=k\cdot e\) (the \(e\) the degree 1, the \(x\) the degree \(n\)), the \(\mathrm{Tor}=\Lambda(e_{n-1})\) the exterior. The EMSS: the \(\mathrm{Tor}_{H^*(S^n)}(k,k)\Rightarrow H^*(\Omega S^n)\), the \(H^*(S^n)=\Lambda(x_n)\) (the odd \(n\) the exterior), the \(\mathrm{Tor}_{\Lambda(x_n)}(k,k)=k[e_{n-1}]\) the polynomial (the \(e\) the degree \(n-1\), the \(H^*(\Omega S^n)=k[e_{n-1}]\) the free for the odd \(n\)). So the \(\mathrm{Tor}\) the exterior / the polynomial, the EMSS the \(\Omega S^n\).
Explain the Koszul duality \(A^!\) and the PBW.
The Koszul: \(K(x_1,\dots,x_n)\) the regular sequence (the \(x_i\) the \(R\)-regular, the Koszul the exact, the homological algebra). The Koszul duality: the \(A\) the quadratic algebra, the \(A^!\) the Koszul dual (the quadratic dual, the PBW the \(A\otimes A^!\cong\mathrm{Clifford}\), the Priddy). The PBW: the \(A\) the Koszul \(\Rightarrow\) the PBW the \(A\) the PBW (the Poincaré–Birkhoff–Witt, the homological algebra). So the Koszul the duality, the \(A^!\) the dual.
Pitfalls¶
- The EMSS the Tor / the loop (the \(\mathrm{Tor}_{H^*(B)}\), the \(\Omega B\)) — the EMSS the Tor, the loop the \(\Omega\).
- The bar the \(B(A)\) (the \(A\) the algebra, the \(\mathrm{Tor}\)) — the bar the \(\mathrm{Tor}\), the homological algebra.
- The Koszul the regular (the \(K(x_1,\dots,x_n)\), the \(R\)-regular) — the Koszul the exact, the duality the \(A^!\).
- The Rothenberg–Steenrod (the \(H^*(G)\cong\mathrm{Tor}_{H^*(BG)}\), the group / classifying) — the Rothenberg–Steenrod the group, the topology.
Refs¶
- Eilenberg–Moore, "Limits and colimits" / the EMSS; Smith, "The Eilenberg–Moore spectral sequence"
- McCleary, A User's Guide to Spectral Sequences; the bar / the Koszul
- Priddy, "Koszul resolutions" (TAMS 1970); the Koszul duality / the PBW
- Rothenberg–Steenrod, "Independent cohomology operations" (1965); the group / classifying
Next¶
SS index · Foundations · Adams & Adams–Novikov · Motivic & Slice · Bockstein & Descent · Homological algebra · Topology