Commutative Algebra, Notes 1: Flat Modules

We will assume all rings are commutative and unital.

Definition 1.  Let R be a ring, E a right R-module, and M a left R-module.  E is M-flat if for every left R-module M' and monomorphism \phi:M'\to M, the homomorphism 1_E\otimes\phi:E\otimes_RM'\to E\otimes_RM is injective.

Some basic results can be shown:

Proposition 2.  If E is a right R-module and M is a left R-module such that E is M-flat, then if N\leq M, we have that E is N-flat and M/N-flat.  Also if \{M_i\}_{i\in I} is a family of left R-modules such that E is M_i-flat for all i\in I, then E is \oplus_{i\in I}E_i-flat.

Proposition 3.  The following are equivalent for a right R-module E.

  1. The canonical map f:E\otimes I\to E\otimes R_S is injective (where R_S=S^{-1}R for some multiplicative subset S).
  2. E is M-flat for every left R-module M.
  3. For every exact sequence of left R-modules and homomorphisms

M'\stackrel{f}{\longrightarrow}M\stackrel{g}{\longrightarrow}M'',

   the sequence

E\otimes M'\stackrel{1\otimes f}{\longrightarrow}E\otimes M\stackrel{1\otimes g}{\longrightarrow}E\otimes M''

   is exact.

Definition 4.  A right R-module E satisfying the equivalent conditions above is flat.  (A dual definition exists for left modules)

Proposition 5.  Let E be a right R-module, then the following are equivalent.

1.  The sequence M'\stackrel{f}{\to}M\stackrel{g}{\to}M'' of left R-modules is exact iff

E\otimes M'\stackrel{1\otimes f}{\to}E\otimes M\stackrel{1\otimes g}{\to}E\otimes M''

is exact.

2.  E is flat and E\otimes N=0\Rightarrow N=0 for a left R-module N.

3.  E is flat and for every homomorphism f:N'\to N, 1_E\otimes f=0\Rightarrow f=0.

4.  E is flat and for every maximal left ideal \mathfrak{m} of R, E\neq E\mathfrak{m}.

Definition 6.  A right R-module E satisfying the above conditions is faithfully flat.

Recall that for a subset S of an R-module M, the annihilator of S is defined as

Ann_R(S)=\{r\in R:rs=0\mbox{~for all~}s\in S\}.

An R-module M is faithful if Ann_R(M)=0.

Definition 7.  Let E be a left R-module and suppose

L_k\stackrel{f_k}{\to}\cdots\stackrel{f_2}{\to}L_1\stackrel{f_1}{\to}L_0\stackrel{f_0}{\to}E

is an exact sequence such that each L_i is free and finitely generated.  Then E is said to be finitely presented, and the exact sequence is called a finite presentation of E of length k.

Note that if f:R\to S is a ring homomorphism, then a finite presentation of a left R-module E induces a finite presentation of E\otimes_R S where the ith term of the first sequence, L_i, becomes L_i\otimes_R S with rs:=f(r)s.  (We will consider E to be the -1 term of the original sequence)

Proposition 8.  Let E be a left R-module.

  1. If E admits a finite presentation, then E is finitely generated.
  2. If R is left Noetherian, then if E is finitely generated, it admits a finite presentation.
  3. If E is projective and finitely generated, then it admits a finite presentation.

Definition 9.  Let E be a left R-module.  E is pseudo-coherent if every finitely generated submodule of E is admits a finite presentation.  E is coherent if it is pseudo-coherent and finitely generated.

[1]  Bourbaki, N.  Commutative Algebra, Ch 1-7.  Springer-Verlag.  1989.

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