 
Summary: EXTENSIONS OF A DUALIZING COMPLEX BY ITS RING:
COMMUTATIVE VERSIONS OF
A CONJECTURE OF TACHIKAWA
LUCHEZAR L. AVRAMOV, RAGNAROLAF BUCHWEITZ, AND LIANA M. S¸EGA
Dedicated to Wolmer Vasconcelos on the occasion of his 65th birthday
Abstract. Let (R, m, k) be a commutative noetherian local ring with dual
izing complex DR, normalized by Ext
depth(R)
R (k, DR) = k. Partly motivated
by a long standing conjecture of Tachikawa on (not necessarily commutative)
kalgebras of finite rank, we conjecture that if Extn
R(DR, R) = 0 for all n > 0,
then R is Gorenstein, and prove this in several significant cases.
Introduction
Let (R, m, k) be a local ring, that is, a commutative noetherian ring R with
unique maximal ideal m and residue field k = R/m. We write edim R for the
minimal number of generators of m and set codepth R = edim R  depth R.
We consider only rings with dualizing complexes, see Section 1 for details. Let
DR
denote a dualizing complex of R, positioned so that Hi(DR
