Extensions of a Dualizing Complex by its Ring: Commutative Versions of a Conjecture of Tachikawa

Details Extensions of a Dualizing Complex by its Ring: Commutative Versions of a Conjecture of Tachikawa Extensions of a Dualizing Complex by its Ring: Commutative Versions of a Conjecture of Tachikawa

2002-08-22

arxiv.org/abs/math/0208172v2

Mathematics

Commutative Algebra

18 pages, to appear in Journal of Pure and Appl. Algebra. Following the comments of the referee, we removed the old section 6

Scientific paper

Let $(R,\fm,k)$ be a commutative noetherian local ring with dualizing complex
$\dua R$, normalized by $\Ext^{\depth(R)}_R(k,\dua R)\cong k$. Partly motivated
by a long standing conjecture of Tachikawa on (not necessarily commutative)
$k$-algebras of finite rank, we conjecture that if $\Ext^n_R(\dua R,R)=0$ for
all $n>0$, then $R$ is Gorenstein, and prove this in several significant cases.

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