Formally smooth map

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> and <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a>, a <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> 
 
 
 
 f
 :
 A
 →
 B
 
 
 {\displaystyle f:A\to B}
 
 is called formally smooth if it satisfies the following infinitesimal <a href="/facts/Lift_(mathematics)/WCqXTXVI">lifting</a> property: 
Suppose B is given the structure of an A-algebra via the map f. Given a <a href="/facts/Commutative_algebra/mKtUdPCT">commutative</a> A-algebra, C, and a <a href="/facts/Nilpotent_ideal/pqzqWEge">nilpotent ideal</a> 
 
 
 
 N
 ⊆
 C
 
 
 {\displaystyle N\subseteq C}
 
, any A-algebra homomorphism 
 
 
 
 B
 →
 C
 
 /
 
 N
 
 
 {\displaystyle B\to C/N}
 
 may be lifted to an A-algebra map 
 
 
 
 B
 →
 C
 
 
 {\displaystyle B\to C}
 
. If moreover any such lifting is unique, then f is said to be <a href="/facts/Formally_%25C3%25A9tale_morphism/TLdIqRJe">formally étale</a>.
Formally smooth maps were defined by <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> in <a href="/facts/%25C3%2589l%25C3%25A9ments_de_g%25C3%25A9om%25C3%25A9trie_alg%25C3%25A9brique/SDtmhwWH">Éléments de géométrie algébrique</a> IV.
For finitely presented morphisms, formal smoothness is equivalent to <a href="/facts/Smooth_morphism/GNxTBHTG">usual notion of smoothness</a>.

Formally smooth map open-in-new

Formally smooth map