| Finite Generation Of Hochschild Homology Algebras (2000) | |||||||||||||||
Abstract | |||||||||||||||
| . We prove converses of the Hochschild-Kostant-Rosenberg Theorem, in particular: If a commutative algebra S is at and essentially of nite type over a noetherian ring |, and the Hochschild homology HH (S j|) is a nitely generated S-algebra for shue products, then S is smooth over |. Introduction Let S be a commutative algebra over a commutative noetherian ring |. Shue products on the Hochschild complex dene the Hochschild homology algebra HH (S j|), which is graded-commutative and is natural in S and |, cf. [11], [23]. Since HH 0 (S j|) is S itself, and HH 1 (S j|) is the S-module of Kahler dierentials 1 S j| , there is a canonical homomorphism of graded algebras ! S j| : V S 1 S j| ! HH (S j|) mapping dierential forms to Hochschild homology. It provides a piece of the product: ! n S j| is injective if n! is invertible in S. Little more is known in general. In a special case the story is complete. Recall that S is regular over | if it is at, and the r... | |||||||||||||||
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