Résume | In modern algebraic geometry we encounter the notion of derived intersection of subschemes. This is a sophisticated way to encode what happens when two subschemes $Y_1$ and $Y_2$ of a given scheme X intersect non-transversely. The classical intersection multiplicity can be extracted from the derived intersection.
When the ambient scheme X is affine, it is not too hard to describe the derived intersection, by taking flat DG ring resolutions of the structure sheaves of the subschemes $Y_1$ or $Y_2$. This also works when the scheme X is quasi-projective. However, derived intersections in more general schemes X could only be treated using the very difficult homotopical methods of derived algebraic geometry.
Several months ago I discovered a ``cheap'' way to construct flat resolutions of sheaves of rings. The resolutions are by semi-pseudo-free sheaves of DG rings. The main advantage is that the geometry does not change: all the action takes place on the original topological space X.
Using semi-pseudo-free resolutions it is possible to produce derived intersections as above. It is also possible to get a direct presentation of the cotangent complex of a scheme (at least in characteristic 0). Presumably the derived adic completion of Shaul, so far studied only in the affine case, could be globalized using our our methods.
Lastly, the semi-pseudo-free resolutions give rise to a congruence on the category of sheaves of commutative DG rings on the space X, that we call relative quasi-homotopy. The functor from the homotopy category to the derived category turns out to be a faithful right Ore localization. This fact gives tight control on the derived category. It should be noted that in this situation there does not seem to exist a Quillen model structure, so the traditional approaches would fail.
In the talk I will explain the various ideas listed above. More details can be found in the eprint \href{http://front.math.ucdavis.edu/1608.04265}{arxiv:1608.04265}. |