I come across the following paragraph from the article Reminiscences of Grothendieck and His School, 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).
