Tensor product of sheaves and modules

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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.

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The functor should be $(\mathord-)\otimes^L_XT$, as $(\mathord-)$ does not have any possible action of $A$ on it... – Mariano Suárez-Alvarez Apr 19 2010 at 16:31
(Unless you swapped the domain and codomain of the functor!) – Mariano Suárez-Alvarez Apr 19 2010 at 16:33
I'm just curious, what is $D^a$? – Martin Brandenburg Apr 19 2010 at 17:55

1 Answer

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...)

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great, thanks, that was the only paper I couldn't get my hands on, I'll see if I can find it – louis de Thanhoffer de Völcsey Apr 19 2010 at 16:51
I had a hard time finding it as well! I had to ask my supervisor to ask the department for it. Unfortunately I don not have it with me at the moment, otherwise I could scan for you the interesting bit. – babubba Apr 19 2010 at 21:31
I've also found useful this paper by Alastair Craw while learning this stuff (and I'm still learning it) xxx.lanl.gov/abs/0807.2191 – babubba Apr 19 2010 at 21:34
@angeleirovero: I wonder how can you scan "the interesting bit" without essentially scanning the whole thing. It is one of those devilish short papers! – Mariano Suárez-Alvarez Apr 19 2010 at 22:14
@mariano: Well... I guess you can't. – babubba Apr 20 2010 at 7:46
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