# Tuesday, March 01

:::{.remark}
A property is *local* on $A$ if it can be checked on an affine open cover of $\spec A$.
Note that e.g. $\AA^2\slice k$ is affine but $\spec k[x,y]/\gens{x,y} \cong\AA^2\slice k\smz$ is not an affine open subset.
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:::{.exercise title="?"}
Show that the property of being zero is local.

> Hint: $M_{f_i} = 0 \implies$ for every $m\in M$ there is an $n_i$ with $f_i^{n_i} m = 0$, write $\sum a_i f_i = 1$, and consider $(\sum a_i f_i)^N m$ for $N\gg 1$.

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:::{.exercise title="?"}
Show that being injective/surjective/bijective is local.

> Hint: localization is exact, so take the SES $0\to \ker g \to M \mapsvia{g} N\to \coker g\to 0$.

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:::{.exercise title="?"}
Show that being finitely generated is local.

> Hint: tensor a presentation. For the other direction, take generators $M_{f_i} = \gens{{x_{i1} \over a_{i1}}, {x_{i2} \over a_{i2}},\cdots, {x_{in} \over a_{in}} }$ for $\ts{f_1,\cdots, f_m}$, where without loss of generality $a_{ij} = 1$, and take $\ts{x_{ij}}$ and check surjectivity locally.

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:::{.exercise title="?"}
Show that flatness is local, i.e. if $M\in \mods{A}^\flat$ then $M_f\in \mods{A_f}^\flat$.

> Hint: to show $M_f$ is flat assuming $M$ is flat, show that $\Tor^{A_f}_i(A_f/a, M_f) = 0$ by taking $\cocomplex{P}\surjects M$ and $\cocomplex{P}_f \surjects M_f$.
  Then compute
\[
\Tor_1(M_f, A_f/a)
&= H_1(\cocomplex{P}_f \tensor A_f/a) \\
&= H_1(\cocomplex{P} \tensor A/a' \tensor A_f) \\
&= H_1(\cocomplex{P} \tensor A/a') \tensor A_f \\
&= 0 \tensor A_f = 0
,\]
using that localization is exact and thus commutes with taking homology.
In the other direction, show that for $0\to A \mapsvia{f} B\to C\to 0$ leads to $A\tensor M \mapsvia{\tilde f} B\tensor M$, and injectivity can be checked locally.

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:::{.remark}
Define the **support** of $M$ as $\supp(M) \da \ts{p\in \spec A\st M_p\neq 0}$, thought of as points where "functions on $A$" defined by $M$ do not vanish.
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:::{.exercise title="?"}
Show that if $M\in \mods{A}^\fg$ then $\supp M = V(\Ann_A(M))$ where $\Ann_A(M) \da \ts{a\in A\st am=0 \, \forall m\in M}$.
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:::{.exercise title="?"}
Show that for $M\in \mods{A}^\fg$ with $A$ Noetherian, $M=0 \iff \supp M = 0$.
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:::{.remark}
Modules give sheaves over $\spec A$, and the following theorem is a special case of faithfully flat descent:
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:::{.theorem title="Serre"}
If $M\in \mods{A}^\fg$ and $\ts{f_i}$ is a finite generating set, then the following sequence is exact:
\[
0 \to M \to \bigoplus_{f_i} M_{f_i} &\to \bigoplus M_{f_i f_i} \\
x\in M_{f_i} &\mapsto \tv{\cdots, {x\over f_i f_j}, \cdots, -{x\over f_j f_i}, \cdots }
.\]
with the positive sign in the $i$th component and the negative in the $j$th.
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:::{.exercise title="?"}
Prove this: check injectivity locally, and use that localization commutes with direct sums.
Note that essentially the same proof goes through for faithfully flat descent.
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:::{.theorem title="Classification of flat finitely-generated modules over a Noetherian ring"}
If $M\in \mods{A}^{\flat, \fg}$ for $A$ Noetherian, then $M$ is locally free, i.e. there exist $f_i$ generating the unit ideal with $M_{f_i}$ free for all $i$.
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:::{.proof title="?"}
Philosophy: reduce to local case.

1. For $A$ local: finitely-generated flat flat modules over a Noetherian *local* ring is free.
2. Hence $M_p$ is free over $A_p$ for all $p\in \spec A$.
3. Spreading out: by the HW, there exist $f_i^p$ not in $p$ such that $M_{f_1^p \cdots f_\ell^p}$ is free and equals $M_{f^p}$.
3. The finite collection $\ts{P^*}$ generated the unit ideal.

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:::{.remark}
Next: Artinian, local, complete local rings, DVRs, etc -- the building blocks of the theory of local rings!
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