# Useful Facts

## Category Theory

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\envlist

- Products: a collection of maps into factors $Y\to X_i$ is the same as a map $Y\to \prod X_i$.
  Products are easy to map *into*.
  Products have projections $\prod X_i \to X_i$.

  - Products are limits.

- Coproducts: a collection of out of factors $X_i\to Y$ is the same as a map $\coprod X_i \to Y$.
  Coproducts are easy to map *out* of.
  Coproducts have injections $X_i \to \coprod X_i$.

  - Coproducts are colimits.

- If $\cat{C}$ has a zero object, there is a canonical map $\coprod_{i\in I} X_i \to \prod_{j\in I} X_j$ given by assembling maps $\delta_{ij}$.
- $\colim_{i\in I}(\wait)$ is generally not exact, but is exact if the colimit is filtered.
  - In any case, the functor of taking stalks $(\wait)_x: \Sh(X; \Ab\Grp) \to \Ab\Grp$ is always exact.
- Left adjoints/colimits are characterized by morphisms on $F(x)$, and right adjoints/limits by morphisms *into* it.
- Why RAPL and LAPC:
\[
[\colim_i L(x_i), \wait] \cong \cocolim_i [L(x_i), \wait] \cong \cocolim_i [x_i, R(\wait)] \cong [\colim_i x_i, R(\wait)] = [L(\colim_i x_i), \wait]
.\]

- Right-derived functors are left Kan extensions.
- Colimits are quotients of coproducts and **receive** maps from objects (i.e. they are cocones).
  Taking colims is right exact.
  Limits send maps.

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## Tor and Ext

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![](figures/2022-03-30_22-08-05.png)

Tor:

- $\Tor$ commutes with arbitrary direct sums, colimits (direct limits), localization.
- If $M$ is flat over $R, \operatorname{Tor}_{i}^{R}(M \otimes A, B) \cong M \otimes_{R} \operatorname{Tor}_{i}^{R}(A, B)$.
- If $S$ is a flat $R$-algebra, $S \otimes_{R} \operatorname{Tor}_{i}^{R}(A, B) \cong \operatorname{Tor}_{i}^{S}\left(S \otimes_{R} A, S \otimes_{R} B\right)$.
- $\Tor_1^R(R/I, R/J) \cong {I \intersect J \over IJ}$
- If $I$ is an $R\dash$regular sequence $I = \gens{x_1,\cdots, x_n}$, then $\Tor_n^R(R/I, M) = (0 :_M I)$ is a colon ideal.
- If $A\in \rmod^\flat$ then $\Tor_{\geq 1}^R(A, B) = 0$.
- $\Tor(A, B) \cong \Tor(B, A)$.

Ext:

- $\tau_{\geq 1}\cocomplex{\Ext}_R(A, B) = 0$ if either $A$ is injective or $B$ is projective.
- The Koszul complex for $k[x,y]$: $K_x \tensor K_y = 0\to k[x,y]\to k[x,y]\cartpower{2}\to k[x,y] \to k \to 0$ where $K_x = 0\to k[x,y] \mapsvia{\cdot x} k[x,y] \to 0$. 
- $\cocomplex{\Ext}_{k[x,y]}(k) = k \oplus \Sigma k\cartpower{2} \oplus \Sigma^2 k$.

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