We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T,−), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admits a calculus of left and right fractions. It follows that the Gabriel–Zisman localization of the quotient at the class of regular morphisms is abelian. We show that it is equiv...

We will give an introduction to Rota-Baxter algebras which started with a probability study of Spitzer in 1950s and found interesting applications in the work of Connes and Kreimer on renormalization of QFT. We will discuss their basic properties, the constructions of the free objects and applications to multiple zeta values by renormalization method. Some other applications will be given in Ku...

Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integer coefficients. In particular, there is a smooth complex projective variety X of dimension 7 and a torsion element of H(X,Z) which is not the class of a codimension-2 algebraic cycle [4]. In this paper, we provide a more systematic explanation for their examples: for every smooth complex algebraic variety X ...

In this paper we study modules with periodic free resolutions (that is, periodic modules) over an exterior algebra. We show that any module with bounded Betti numbers (that is, a module whose syzygy modules have a bounded number of generators) must have periodic free resolution of period 2, and that for graded modules the period is 1. We also show that any module with a linear Tate resolution i...

In this paper, we establish rigidity and vanishing theorems for Dirac operators twisted by E8 bundles. Introduction Let X be a closed smooth connected manifold which admits a nontrivial S1 action. Let P be an elliptic differential operator on X commuting with the S1 action. Then the kernel and cokernel of P are finite dimensional representation of S1. The equivariant index of P is the virtual c...

We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space wh...

We compute all the groups G⊗H, Tor(G, H), Hom(G, H), and Ext(G, H), where G and H can be any of the groups Z (the integers), Z/n = Z/nZ (the integers mod n), or Q (the rationals). All but one are reasonably accessible. Because all these functors are biadditive, these cases suffice to handle any finitely generated groups G and H. The emphasis here is on computation, not on the abstract definitio...