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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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