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Almost smooth algebras. (English) Zbl 0777.16003
The “almost smooth algebras” (which generalize the formally smooth algebras of Grothendieck) are introduced. An \(A\)-algebra \(B\) is almost smooth if for every singular \(A\)-extension of \(B\) by a \(B\)-module \(M\) the second fundamental exact sequence of \(B\)-modules \[ 0\to M\to\Omega_{E\setminus A} \otimes_ E B\to\Omega_{B\setminus A}\to 0 \] is short exact. Several characterizations of almost smooth algebras are given. It is proved that an \(A\)-algebra \(B\) is almost smooth iff for any \(A\)-algebra \(C\), any ideal \(I\) of \(C\) satisfying \(I^ 2=0\), and any \(A\)-algebra homomorphism \(g:B\to C/I\) such that \(I\) is an injective \(B\)- module via \(g\), there exists a lifting \(f:B\to C\) of \(g\). In the end of the paper an example of an almost smooth algebra which is not formally smooth is given.

MSC:
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16S80 Deformations of associative rings
16D50 Injective modules, self-injective associative rings
16S70 Extensions of associative rings by ideals
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