Vanishing of Tor over fiber products

Cohomology over Fiber products of local rings

Article history: Received 26 April 2007 Communicated by Michel Van Den Bergh

On the vanishing of Tor of the absolute integral closure

Let R be an excellent local domain of positive characteristic with residue field k and let R+ be its absolute integral closure. If Tor1 (R +, k) vanishes, then R is weakly F-regular. If R has at most an isolated singularity or has dimension at most two, then R is regular.

Non - vanishing of scalar products of Fourier -

A characterization of functions with vanishing averages over products of disjoint sets

Given α1, . . . , αm ∈ (0, 1), we characterize all integrable functions f : [0, 1] → C satisfying ∫ A1×···×Am f = 0 for any collection of disjoint measurable sets A1, . . . , Am ⊆ [0, 1] of respective measures α1, . . . , αm. We use this characterization to settle some of the conjectures in [S. Janson and V. Sós, More on quasi-random graphs, subgraph counts and graph limits, European J. Combin....

Vanishing ideals over rational parameterizations

Let R = K[y] = K[y1, . . . , yn] be a polynomial ring over an arbitrary field K and let F be a finite set {f1/g1, . . . , fs/gs} of rational functions in K(y), the quotient field of R, where fi (resp. gi) is in R (resp. R \ {0}) for all i. As usual we denote the affine and projective spaces over the field K by As and Ps−1, respectively. Points of the projective space Ps−1 are denoted by [α], wh...