# Inverse of a tilting module

Let k be a field, A an associative unital k-algebra, ModA the category of left A-modules and Db(ModA) the bounded derived category. Let $A^◦$ be the opposite algebra and $A^e := A \bigotimes_k A^◦$ the enveloping algebra. Let T be a two-sided tilting complex: $T ∈ Db(ModA^e)$

How can I understand the structure of T^$:=RHom_A(T,A)$ and why is T^$\bigotimes_{A}^{L} T \simeq T \bigotimes_{A}^{L}$T^$\simeq A$?

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Could you supply a definition of tilting complex? –  Ben Webster Jan 26 '11 at 16:57
There are different definitions, for example: Let A be a ring. A tilting complex T over A is an object in Kb(P(A)) which satisfies the following conditions: (I) for all i ≠ 0, the set HomDb(A)(T, T[i]) of homomorphisms in Db(A) vanishes, (II) the category add(T) (that is, the full subcategory of all direct sums of direct summands of T inside Kb(P(A))) generates Kb(P(A)) as a triangulated category. By P(A) we denote the additive category of finitely generated projective left A-modules. –  Alex Jan 26 '11 at 17:44
And by Kb(P(A)) the homotopy category of complexes of finite length –  Alex Jan 26 '11 at 17:46
Are you sure the isomorphisms in your final question are correct? Left tensoring over A by A, oughtn't to do anything at all. –  Hugh Thomas Jan 26 '11 at 18:35
Yes, thank you! I've corrected this error. –  Alex Jan 26 '11 at 19:01
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