# Wednesday, January 18


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Some review:

- $M\in \kalg \iff M\in \kmod \intersect \Ring$ and $\exists m: M\tensor_k M\to M$ a multiplication map.
- $M\in \liealg \iff \exists [\wait,\wait]: M\tensor_k M\to M$ satisfying the usual identities.
  - E.g. $\Endo_k(V) \in\liealg$ when $V\in \kmod$.
- $\Der_k(M)$ is *not* closed under composition, but is a Lie algebra under $[\delta_1 \delta_2] \da \delta_1 \circ \delta_2 - \delta_2 \circ \delta_1$.
  - Counterexample: $\Der_k(k[x]) = k[x] \dd{}{x} \cong k[D]$ but $\dd{}{x}\circ \dd{}{x}$ s not a derivation.
- $\HoH(M)$ will make meaningful higher analogs of derivations, $\delta^n: A\tensorpowerk{n} \to A$.
  - 1-cocycles are derivations
  - 2-cocyles are $\delta: M\tensorpowerk{2} \to M$ such that $\delta(ab,c) - \delta(a, bc) = a\delta(b,c) - \delta(a,b)c$.
  - $n\dash$cocycles will be $\delta: M\tensorpowerk{n} \to M$ satisfying
  \[
 \sum_{i=1}^n(-1)^i \delta\left(a_1, a_2, \cdots, a_i a_{i+1}, \cdots, a_{n+1}\right)=-a_1 \delta\left(a_2, \cdots, a_{n+1}\right)+(-1)^n \delta\left(a_1, \cdots, a_n\right) a_{n+1} . 
  .\]
- Define $Z^n(M)$ to be $n\dash$cycles -- this is not a Lie algebra for $n\geq 2$ unless the bracket is trivial.
- Gerstenhaber's idea: define a new bracket $[\wait, \wait]: Z^m(M) \tensor_k Z^n(M) \to Z^{n+m-1}(M)$ with for $m=n=1$ is the commutator; this makes $Z^*(M)$ into a graded Lie algebra.
- Define boundaries $B_n(M)$ and $\HoH^n(M) \da Z^n(M)/B^n(M)$.
  -  $\HoH^1(M) = Z(M)$ is the center.
  - $\HoH^2(M) = \Der(M)$ when $M$ is commutative.

- Recall the definitions of chain complexes and their morphisms.

- Recall the different formulations of projectives $P$ in $\rmod$:

  - $\exists F \in \rmod^\free$ with $F \cong P \oplus T$ for some $T\in \rmod$ (not necessarily free).
  - Every $B\surjects P$ and $B'\to P$ lifts to $B'\to B$.
  - Every SES $A\injects B\surjects P$ splits.

- Some useful resolutions:

  - $\ZZ \mapsvia{\cdot n} \ZZ \surjectsvia{\eps} \ZZ/n\ZZ$ for $R = \ZZ$ where $\eps$ is the quotient and $\ker \eps = n\ZZ$.
  - $k[x] \injectsvia{\cdot x} k[x] \surjectsvia{\eps(x) = 0} k \in \rmod$ for $R = k[x]$, where the kernels are all $\gens{x}$.
  - $\cdots \to k[x] \mapsvia{\cdot x} k[x] \mapsvia{\cdot x} k[x]\surjectsvia{\eps(x) = 0} k$ for $R= k[x]/\gens{x^2}$ where the kernels are all $\gens{x}$.
  Note that this is an infinite periodic resolution.
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