I come across the following paragraph from the article *[Reminiscences of
Grothendieck and His
School][1]*, here is from the part of the interview by Luc Illusie,:

"
I was indeed looking for an Atiyah-Singer index
formula in a relative situation. A relative situation
is of course in Grothendieck’s style, so Cartan
immediately saw the point. I was doing something
with Hilbert bundles, complexes of Hilbert bundles
with finite cohomology, and he said, “It reminds me
of something done by Grothendieck, you should
discuss it with him.” I was introduced to him by
the Chinese mathematician Shih Weishu. He was in
Princeton at the time of the Cartan-Schwartz seminar
on the Atiyah-Singer formula; there had been a
parallel seminar, directed by Palais. We had worked
together a little bit on some characteristic classes.
And then he visited the IHÉS. He was friendly with
Grothendieck and proposed to introduce me.

So, one day at two o’clock I went to meet
Grothendieck at the IHÉS, at his office, which is
now, I think, one of the offices of the secretaries.
The meeting was in the sitting room which was
adjacent to it. I tried to explain what I was doing.
Then Grothendieck abruptly showed me some
naïve commutative diagram and said, “**It’s not leading
anywhere. Let me explain to you some ideas I
have.**” Then he made a long speech about finiteness
conditions in derived categories. I didn’t know anything
about derived categories! “**It’s not complexes
of Hilbert bundles you should consider. Instead,
you should work with ringed spaces and pseudocoherent
complexes of finite tor-dimension.**”
…(laughter)…It looked very complicated. But what
he explained to me then eventually proved useful
in defining what I wanted. I took notes but couldn’t
understand much.
"

Can someone explain why Atiyah-Singer index formula should be related to **ringed spaces and pseudocoherent
complexes of finite tor-dimension**? I know the definition of locally ringed spaces (from Hartshrone), but I do not know what is "pseudocoherent
complexes of finite tor-dimension" and how does it relate to the index theorem. This approach, as far as I know, is also absent from other discussions in literature (like the heat-kernel proof and the K-theory proof). 


  [1]: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CB8QFjAA&url=http%3A%2F%2Fpages.uoregon.edu%2Fvvologod%2Freminiscences1.pdf&ei=ONb2U6XMK8ifyASO7YE4&usg=AFQjCNH1Bs-F5xWtCIbF4bdZoNkCWQlEjw&sig2=TD-TSmaIZtz8osWD-o1BSQ&bvm=bv.73373277,d.aWw