Two-sided tilting complexes for Gorenstein orders

by Alexander Zimmermann
Facult\'e de Math\'ematiques 
Universit\'e de Picardie Jules Verne
33, rue St Leu
F-80039 Amiens Cedex 1
France

This paper is submitted for publication. 

Work of J. Rickard proves that 
the derived module categories of two rings $A$ and $B$ are equivalent as 
triangulated categories if and only if there is a particular object $T$,
a so-called tilting complex, in
the derived category of $A$ such that $B$ is the endomorphism ring of $T$.
The functor inducing the equivalence 
however is not explicit by the knowledge of $T$. 
Suppose the derived categories of $A$ and $B$ are equivalent.
If $A$ and $B$ are $R$-algebras and projective of finite type over 
the commutative ring $R$, then 
Rickard proves the existence of a so-called two-sided tilting complex $X$,
which is an object in the derived category of bimodules. The left derived 
tensor product by $X$ is then an equivalence between the derived categories of 
$A$ and $B$.
There is no general explicit construction known to derive
$X$ from the knowledge of $T$. In an 
earlier paper S.~K\"onig and the author gave for a class of algebras 
a tilting complex $T$ by a general procedure with
prescribed endomorphism ring. Under some additional hypothesis
we construct in the present paper 
an explicit two-sided tilting complex whose restriction to one
side is any given one-sided tilting complex of the type described  in
the above cited paper. This provides two-sided tilting complexes for various 
cases of derived equivalences, making the functor inducing this equivalence 
explicit.