User:Jan A. Sanders/An Introduction to Leibniz Algebra Cohomology/Lecture 6
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The Serre-Hochschild spectral sequence
Let \(\mathfrak{h}\) be a subalgebra or an ideal in \(\mathfrak{g}\ .\) Define a filtration on \(C^n(\mathfrak{g},\mathfrak{a})\) by \[F^pC^n(\mathfrak{g},\mathfrak{a})=\{a^n\in C^n(\mathfrak{g},\mathfrak{a})| a^n(x_1,\cdots,x_n)=0 \ \mathrm{if}\ n-p+1 \ \mathrm{of\ its \ variables\ are\ in\ } \mathfrak{h}\}.\] Then \[C^n(\mathfrak{g},\mathfrak{a})=F^0 C^n(\mathfrak{g},\mathfrak{a})\supset\cdots\supset F^nC^n(\mathfrak{g},\mathfrak{a})\supset F^{n+1}C^n(\mathfrak{g},\mathfrak{a})=0\]
remark
Since, when \(\mathfrak{h}\) is an ideal, \[ F^n C^n (\mathfrak{g},\mathfrak{a})\simeq C^n (\mathfrak{g}/\mathfrak{h},\mathfrak{a})\] one can see this as an approximation scheme to go from \(C^n (\mathfrak{g},\mathfrak{a})\) to \(C^n (\mathfrak{g}/\mathfrak{h},\mathfrak{a})\ .\)
lemma
\[ d^{(n)} (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^n(\mathfrak{g},\mathfrak{a})\ .\]
proof
Let \(a^n\in F^pC^n(\mathfrak{g},\mathfrak{a})\ .\) That means that \( a^n\) will be zero if \(n-p+1\) of its variables are in \(\mathfrak{h}\ .\) Since \[ (d^{(n)}(y)a^n)(x_1,\cdots,x_n)=d_+^{(0)}(y)a^n(x_1,\cdots,x_n)-\sum_{i=1}^n a^n(x_1,\cdots, [y,x_i],\cdots,x_n),\] it is clear that \( (d^{(n)}(y)a^n)(x_1,\cdots,x_n)=0\) if \(n-p+2\) of its variables are in \(\mathfrak{h}\ ,\) that is, \(d^{(n)}(y)a^n\in F^{p-1}C^n(\mathfrak{g},\mathfrak{a})\ .\)
lemma
For \(x\in \mathfrak{g}\) that \[ \iota^n (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^{n-1}(\mathfrak{g},\mathfrak{a})\ .\]
lemma
\[ d^n F^p C^n(\mathfrak{g},\mathfrak{a})\subset F^{p} C^{n+1}(\mathfrak{g},\mathfrak{a})\]
proof
For \(n=0\) this is clear, since \( d^0 C^0(\mathfrak{g},\mathfrak{a})\subset C^1(\mathfrak{g},\mathfrak{a})\ .\) Suppose the statement holds for all \(k< n\ .\) Then, since \[\iota^{n+1}(x)d^n+d^{n-1}\iota^n(x)=d^{(n)}(x)\ ,\] the statement holds by induction for all \(n\in\N\ .\) Indeed, \[ d^{(n)}(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1} C^n(\mathfrak{g},\mathfrak{a})\] and, using the induction hypothesis, \[d^{n-1}\iota^n(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset d^{n-1}F^{p-1} C^{n-1}(\mathfrak{g},\mathfrak{a}) \subset F^{p-1} C^{n}(\mathfrak{g},\mathfrak{a})\ .\] This implies that for \(a^n\in F^p C^n(\mathfrak{g},\mathfrak{a})\ ,\) \[ d^n a^n (x, x_1,\cdots, x_{n})\] will be zero if \(n+2-p\) of its arguments are in \(\mathfrak{h}\ .\) But this implies that \( d^n a^n \in F^{p} C^{n+1}(\mathfrak{g},\mathfrak{a})\ .\)
definition
Let \(K^{p,n}=F^p C^n(\mathfrak{g},\mathfrak{a})\ .\) With \(d^n K^{p,n}\subset K^{p,n+1}\) one is now in the right setting to define a spectral sequence.
