Hello to all,


I have been looking quite recently at the following theorem:
Let $X$ be a projective variety and $T$ a tilting object for $X$. If $A:=End(T)$ is the associated endomorphism algebra, then the functor
$RHom(T -): D^b(X) \rightarrow D^b(A)$ is in fact an equivalence. Now, this is proven (as in the claasical Bondal paper) by showing that the functor is fully faithful and essentially surjective. But in I have noticed another version where one defines a functor $-\otimes^L\_A T: D^b(A) \longrightarrow D^b(X)$. My question is could someone maybe give a definition of this functor (I of course know what all types of tensor-products are, but I'm not really sure how to "tensor" a module over a noncommutative ring $A$ together with a sheaf to obtain another sheaf.



Tensor product of sheaves and modules
What I've seen is a construction of a quasi-inverse for RHom(T,-), defined as $-\otimes\_A^L T$, as you wrote.
This last symbol should be interpreted as follows.
Given a left A-module M, we define a presheaf which with each U associates
$M\otimes^L\_A T(U)$.
Finally $M\otimes^L\_A T$ is defined as the sheafification of the former.


I should have a reference for this, let me check.


**Reference Added:**
A. A. Beilinson. Coherent sheaves on $\mathbb{P}^n$ and problems in linear
algebra. Funktsional.Anal. i Prilozhen., 12(3):68–69, 1978.
(it's this one, if I remember correctly, but I might not...)


