Kähler differentials: Difference between revisions
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imported>Giovanni Antonio DiMatteo m (Kähler differential moved to Kähler differentials: because the module is made up of many differentials, we almost never care about a single differential...) |
imported>Giovanni Antonio DiMatteo (→Definition: mathmode) |
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==Definition== | ==Definition== | ||
Let <math>A\to B</math> be an [[Module|algebra]]. An ''A'' differential of ''B'' into an <math>A</math>-module <math>M</math> is a map | Let <math>A\to B</math> be an [[Module|algebra]]. An ''A'' differential of ''B'' into an <math>A</math>-module <math>M</math> is a map <math>D:B\to M</math> such that | ||
#<math>D(a)=0</math> for all <math>a\in A</math> | #<math>D(a)=0</math> for all <math>a\in A</math> | ||
#<math>D(b+b')=D(b)+D(b')</math> for <math>b,b'\in B</math> | #<math>D(b+b')=D(b)+D(b')</math> for <math>b,b'\in B</math> | ||
#<math>D(bb')=b'D(b)+bD(b')</math> | #<math>D(bb')=b'D(b)+bD(b')</math> | ||
Observe that the set of all such maps <math>Der_A(B,M)</math> is a <math>B</math>-module. Moreover, <math>Der_A(B,-)</math> is a [[Category of functors|representable functor]]; we call the representative <math>\Omega_{B/A}</math> the module of Kähler differentials. That is, <math>\Omega_{B/A}</math> satisfies the following universal property: | Observe that the set of all such maps <math>Der_A(B,M)</math> is a <math>B</math>-module. Moreover, <math>Der_A(B,-)</math> is a [[Category of functors|representable functor]]; we call the representative <math>\Omega_{B/A}</math> the module of Kähler differentials. That is, <math>\Omega_{B/A}</math> satisfies the following universal property: | ||
Latest revision as of 10:45, 1 January 2008
Definition
Let be an algebra. An A differential of B into an -module is a map such that
- for all
- for
Observe that the set of all such maps is a -module. Moreover, is a representable functor; we call the representative the module of Kähler differentials. That is, satisfies the following universal property:
