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
is exact . (See tensor product of modules .)
Equivalent characterizations are:
- for all modules . (See Tor (math) .)
- For every ideal of is injective.
- for all ideals of .
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
- an exact sequence . Then there is the sequence
- exact if or is flat. This corresponds to the symmetry of the functor gate.
- Are and flat modules, so too .
- In the ring of dual numbers , flat is equivalent to free .
- Be . Then it is flat exactly when it is flat for everyone .
Examples
- is a flat but not projective module.
- For each ring , the module is flat.
- Let be a commutative ring with one element and a multiplicatively closed set, then the module is flat.
- This makes a flat module in particular
- is a flat algebra.
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
- ^ Hideyuki Matsumura, Commutative ring theory . Cambridge University Press, Cambridge 1989, Theorem 7.7 and Theorem 7.8, pp. 51f.
- ^ 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
- ^ David Eisenbud, Commutative algebra with a view toward algebraic geometry . Springer-Verlag, New York 1995, Corollary 6.3, p. 164
- ^ Qing Liu, Algebraic Geometry and Arithmetic Curves . Oxford University Press, Oxford 2006, Corollary 1.2.14, p. 11
- ^ Qing Liu, Algebraic Geometry and Arithmetic Curves . Oxford University Press, Oxford 2006, Proposition 2.6, p. 9