The finitistic dimension conjecture via DG-rings

A BSTRACT . Given an associative ring A , we present a new approach for establishing the finiteness of the big finitistic projective dimension FPD( A ) . The idea is to find a sufficiently nice non-positively graded differential graded ring B such that H 0 ( B ) = A and such that FPD( B ) < ∞ . We show that one can always find such a B provided that A is noetherian and has a noncommutative dualizing complex. We then use the intimate relation between D ( B ) and D (H 0 ( B )) to deduce results about FPD( A ) . As an application, we generalize a recent sufficient condition of Rickard, for FPD( A ) < ∞ in terms of generation of D ( A ) from finite dimensional algebras over a field to all noetherian rings which admit a dualizing complex.

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