User:Jan A. Sanders/An introduction to Lie algebra cohomology/Lecture 6

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

$d_1^{n} (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^n(\mathfrak{g},\mathfrak{a})\ .$

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 $(d_1^{n}(y)a_n)(x_1,\cdots,x_n)=d_1(y)a_n(x_1,\cdots,x_n)-\sum_{i=1}^n a_n(x_1,\cdots, [y,x_i],\cdots,x_n),$ it is clear that $$(d_1^{n}(y)a_n)(x_1,\cdots,x_n)=0$$ if $$n-p+2$$ of its variables are in $$\mathfrak{h}\ ,$$ that is, $$d_1^{n}(y)a_n\in F^{p-1}C^n(\mathfrak{g},\mathfrak{a})\ .$$

lemma

For $$x\in \mathfrak{g}$$ that $\iota_1^n (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^{n-1}(\mathfrak{g},\mathfrak{a})\ .$

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 $$d C^0(\mathfrak{g},\mathfrak{a})\subset C^1(\mathfrak{g},\mathfrak{a})\ .$$ Suppose the statement holds for all $$k< n\ .$$ Then, since $\iota_1^{n+1}(x)d^n+d^{n-1}\iota_1^n(x)=d_1^{n}(x)\ ,$ the statement holds by induction for all $$n\in\N\ .$$ Indeed, $d_1^{n}(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1} C^n(\mathfrak{g},\mathfrak{a})$ and, using the induction hypothesis, $d^{n-1}\iota_1^n(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset d^{n-1}F^{p-1} C^{n-1}(\mathfrak{g},\mathfrak{a}) \subset F^{p-1} C^{n}(\mathfrak{g},\mathfrak{a})\ .$ 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.