The Stable Derived Category

Curve Counting via Stable Pairs in the Derived Category

For a nonsingular projective 3-fold X , we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X . The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten...

The Stable Derived Category of a Noetherian Scheme

For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal CohenMacaulay approximations, a construction of Tate cohomology, and an extension ...

The Derived Category of Quasi-coherent Sheaves and Axiomatic Stable Homotopy

We prove in this paper that for a quasi-compact and semiseparated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated forma...

Constructible Derived Category

B-branes and the Derived Category

In recent years, there has been much fruitful interplay between physics and geometry through string theory. Many questions in physics correspond to questions in pure geometry. The motivations in physics and geometry are often of a completely different nature, so that finding the precise dictionary between the two areas is key to making fundamental progress. The spectrum of the closed topologica...