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
for any ring R, and any universal localization S of
and Tor1S are
naturally isomorphic as bi-functors of S-modules,
and hence Tor1R(S,S)
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
Schofield, Representations of rings over skew fields,
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
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
Neeman, Andrew Ranicki and Aidan Schofield,
of algebras as universal localizations,
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.
Return to Warren Dicks' publications.