Flatness (algebra)

Flatness of modules is a generalization of the term " free module ".

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

definition

A module over a ring is called flat if the functor ${\displaystyle M}$ ${\displaystyle A}$

${\displaystyle N\mapsto M\otimes _{A}N}$

is exact . (See tensor product of modules .)

Equivalent characterizations are:

• ${\displaystyle \operatorname {Tor} _{1}(N,M)=0}$for all modules . (See Tor (math) .) ${\displaystyle A}$ ${\displaystyle N}$
• For every ideal of is injective. ${\displaystyle I}$ ${\displaystyle A}$ ${\displaystyle I\otimes _{A}M\to IM}$
• ${\displaystyle \operatorname {Tor} _{1}(A/I,M)=0}$for all ideals of . ${\displaystyle I}$ ${\displaystyle A}$

properties

• All projective and therefore all free modules are flat. Conversely, every finally presented flat module is projective.
• Flat modules are torsion-free . The terms “flat” and “torsion-free” even coincide with Dedekind rings (especially with main ideal rings ).
• Be it

${\displaystyle 0\to N'\to N\to N''\to 0}$

an exact sequence . Then there is the sequence

${\displaystyle 0\to M\otimes N'\to M\otimes N\to M\otimes N''\to 0}$

exact if or is flat. This corresponds to the symmetry of the functor gate. ${\displaystyle M}$ ${\displaystyle N''}$

• Are and flat modules, so too . ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle M\otimes _{R}N}$
• In the ring of dual numbers , flat is equivalent to free .
• Be . Then it is flat exactly when it is flat for everyone . ${\displaystyle M=\bigoplus _{i\in I}M_{i}}$ ${\displaystyle M}$ ${\displaystyle M_{i}}$ ${\displaystyle i\in I}$

Examples

• ${\displaystyle \mathbb {Q} }$is a flat but not projective module. ${\displaystyle \mathbb {Z} }$
• For each ring , the module is flat. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$
• Let be a commutative ring with one element and a multiplicatively closed set, then the module is flat. ${\displaystyle R}$ ${\displaystyle S\subseteq R}$ ${\displaystyle R}$ ${\displaystyle S^{-1}R}$

This makes a flat module in particular ${\displaystyle k(t_{1},\ldots ,t_{n})}$ ${\displaystyle k[t_{1},\ldots ,t_{n}]}$

• ${\displaystyle R[X]}$is a flat algebra. ${\displaystyle R}$

literature

• David Eisenbud, Commutative algebra with a view toward algebraic geometry . Springer-Verlag, New York 1995, ISBN 0-387-94269-6 .
• Hideyuki Matsumura, Commutative ring theory . Cambridge University Press, Cambridge 1989, ISBN 0-521-36764-6 .
• Qing Liu, Algebraic Geometry and Arithmetic Curves . Oxford University Press, Oxford 2006, ISBN 0-19-920249-4 .

Individual evidence

1. ^ Hideyuki Matsumura, Commutative ring theory . Cambridge University Press, Cambridge 1989, Theorem 7.7 and Theorem 7.8, pp. 51f.
2. ^ David Eisenbud, Commutative algebra with a view toward algebraic geometry . Springer-Verlag, New York 1995, Corollary 6.6, p. 166; Hideyuki Matsumura, Commutative ring theory . Cambridge University Press, Cambridge 1989, Corollary 7.12, p. 53
3. ^ David Eisenbud, Commutative algebra with a view toward algebraic geometry . Springer-Verlag, New York 1995, Corollary 6.3, p. 164
4. ^ Qing Liu, Algebraic Geometry and Arithmetic Curves . Oxford University Press, Oxford 2006, Corollary 1.2.14, p. 11
5. ^ Qing Liu, Algebraic Geometry and Arithmetic Curves . Oxford University Press, Oxford 2006, Proposition 2.6, p. 9