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Dualizing complexes, Morita equivalence and the derived Picard group of a ring. (English) Zbl 0954.16006

The rings \(A\) and \(B\) are derived Morita equivalent if there is an equivalence of the derived categories \(F\colon D^b(\text{Mod-}A)\to D^b(\text{Mod-}B)\). A complex of bimodules \(T\) is called a tilting complex if the functor \(T\otimes^L_A-\colon D^b(\text{Mod-}A)\to D^b(\text{Mod-}B)\) is an equivalence. In the case \(A=B\) the isomorphism classes of tilting complexes form the derived Picard group \(\text{DPic}(A)\).
Some calculations of tilting complexes are shown for \(k\)-algebras. If \(A\) and \(B\) are \(k\)-algebras, \(A\) is local and \(T\in D(\text{Mod}(B\otimes A^o))\) is a tilting complex, then \(T\cong P[n]\) for some invertible bimodule \(P\) and integer \(n\). Also, if \(A\) is a commutative \(k\)-algebra, \(A\) and \(B\) are derived Morita equivalent, then they are Morita equivalent. These facts permit to describe the groups \(\text{DPic}_k(A)\) and \(\text{Pic}_k(A)\) in these cases. In particular, if \(A=A_1\times\cdots\times A_m\) and every \(A_i\) is a local \(k\)-algebra, then \(\text{DPic}_k(A)\cong\mathbb{Z}^m\times\text{Pic}_k(A)\) and \(\text{Pic}_k(A)\cong\operatorname{Aut}_k(A)\).
Let \(A\) be a Noetherian \(k\)-algebra. The dualizing complexes over \(A\) are introduced (as generalization of the commutative definition) and relations with tilting complexes are studied. It is proved that \((R,T)\mapsto R\otimes^L_AT\) is a right action of \(\text{DPic}(A)\) on the set of isomorphism classes of dualizing complexes, which gives a classification of such complexes.
In the last part finite \(k\)-algebras are considered. The complex \(T\in D^b(\text{Mod-}A^e)\) is tilting if and only if its dual \(T^*\) is a dualizing complex. For the algebra \(A=\left[\begin{smallmatrix} k &k\\ 0&k\end{smallmatrix}\right]\) some remarkable isomorphisms in \(D(\text{Mod-}A)\) and \(D(\text{Mod-}A^e)\) are shown, which in particular give \(t^3=s\), where \(t,s\in\text{DPic}(A)\) are the classes of \(A^*=\operatorname{Hom}_k(A,k)\) and \(A[1]\), respectively. This result is generalized in the appendix (by E. Kreines) to upper triangular \(n\times n\) matrices.

MSC:

16D90 Module categories in associative algebras
16E05 Syzygies, resolutions, complexes in associative algebras
18E30 Derived categories, triangulated categories (MSC2010)
16D20 Bimodules in associative algebras
16P10 Finite rings and finite-dimensional associative algebras
16S50 Endomorphism rings; matrix rings
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