Flatness

by iarnab_kundu, Dec 9, 2018, 10:31 PM

Proposition- Let $M$ be an $A-$ module. Given an exact sequence of $A-$ modules $N'\to N\to N''\to 0$ , then the tensored sequence $N'\otimes M\to N\otimes M\to N''\otimes M\to 0$ is exact.

We say that tensoring is "right exact". Thus we can consider its left derived functors. We shall call them the $\text{Tor}$ functors. Given an $A-$ module $M$ , we consider its free(or projective) resolution $F_1\to F_0\to M\to 0$ . Then we define $\text{Tor}^A_n(N,M)$ to be the homology of the complex $\cdots\to N\otimes F_1\to N\otimes F_0\to 0$ . As we might check, we have that $\text{Tor}_0(N,M)=N\otimes M$ .

Proposition- Given an exact sequence of $A-$ modules $0\to N'\to N\to N''\to 0$ we have a long exact sequence $\cdots\to\text{Tor}_1(N',M)\to\text{Tor}_1(N,M)\to\text{Tor}_0(N'',M)\to N'\otimes M\to N\otimes M\to N''\otimes M\to 0$ .

Some computations-
(a) Let $F$ be a free module. Then $0\to F\to F\to 0$ is a free resolution of $F$ . Thus $Tor_n(N,F)=0$ , for $n\ge 1$ . Thus free modules are flat, as one can prove otherwise.
(b) We can extend the above result for projective modules, and prove that they are flat.
(c) Let $M=M_1\times M_2$ . Then we have a

Examples-
(a) Let $f\in A$ be an (non-zero divisor?) element. Then $A_f$ is flat over $A$ .
Proof- $A_f=\text{colim}_n f^{-n}A$ . Thus it commutes with colimits, and therefore with kernels.
What I have in mind is this. Given an exact sequence $0\to M'\to M\to M''$ we have exact sequences $0\to f^{-n}M'\to f^{-n}M\to f^{-n}M''$ . Therefore we can take the colimit and preserve the kernel.
(b) Let $p$ be a prime ideal. Then $A_p$ is flat over $A$ .
Proof- Can we use similar idea as above?

Locality of Flatness-
Proposition- Let $M$ be an $A-$ module, then the following conditions are equivalent
(a) $M$ is flat over $A$
(b) $M_p$ is flat over $A_p$ for all prime ideals $p$ in $A$
(c) $M_P$ is flat over $A_m$ for all maximal ideals $p$ in $A$
Proof-
(a) $\implies$ (b)
We know $A_p$ is exact over $A$ and we know that $M_p=M\otimes A_p$ .

Proposition- Let $M$ be an $A-$ module, then the following conditions are equivalent
(a) $M$ is flat over $A$
(b) For all ideals $I$ in $A$ , the tensored map $I\otimes M\to A\otimes M$ is injective
(b) For all finitely generated ideals $I$ in $A$ , the tensored map $I\otimes M\to A\otimes M$ is injective.

Proposition- $$

This post has been edited 3 times. Last edited by iarnab_kundu, Dec 10, 2018, 1:56 PM

Commutative Algeba

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Cracking the Nut Open

Flatness

by iarnab_kundu, Dec 9, 2018, 10:31 PM

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