George M. Bergman and Warren Dicks,
Universal derivations and universal ring constructions.
Pacific J. Math. 79 (1978), 293-337.


July 12, 2005.

In Section 5,  we gave a long proof  of
                  (95):  for any ring R,  and any universal localization S of R
                           Tor1R and  Tor1S are naturally isomorphic as bi-functors of S-modules,
                           and hence   Tor1R(S,S) = 0.
and deduced
                  Theorem 5.3:  if R is right (or left) hereditary then so is S.  

Dicks and Schofield,  by developing an argument of Dlab and Ringel,  
gave a short proof of these facts, and some other results.
These proofs appeared  in pages 57-58 of

                  A. H. Schofield,  Representations of rings over skew fields
                  LMS Lecture Notes 92, Cambridge University Press,  1985,

where credit is very carefully given for everything except (95).

In Example 4.1, we presented a ring of right (and left) global dimension two,
and a (nonfaithful) universal localization thereof of infinite right (and left) global dimension.
We remarked that "no example is known where the global dimension shows a finite increment".

Twenty-six years later,  Theorem 2.1 of

                  Amnon Neeman, Andrew Ranicki and Aidan Schofield,  
                  Representations of algebras as universal localizations

                  Math. Proc. Cambridge Philos. Soc. 136 (2004), 105-117. arxiv

provided many examples where the global dimension of a (faithful!) universal localization shows
a finite increment.

Incidentally, the authors were unaware of both our example and our remark.  

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