Let R be a commutative Noetherian ring, I and J ideals of R, M and N finitely generated R-modules, and c an integer. The linear growth property was first proved for primary decompositions of I^n in R as $n$ goes to infinity. Later, this property was proved for Tor_c(M, N/J^nN) and Ext^c(M,N/J^nN).

In this talk, we go over the definition of the linear growth property and some of the known cases where the property holds. Then we show that the linear growth property holds for Tor_c(M/I^mM, N/J^nN) as m and n vary.

Given a ring R of equal characteristic p > 0, one can associate an ideal called the "test ideal" which captures information about the singularities of Spec R. It has been known for about 2 decades that this ideal is closely related to the "multiplier ideal", an important tool for measuring singularities in complex algebraic geometry.

While the multiplier ideal is most often defined via a resolution of singularities, the test ideal has historically been defined by studying the action of Frobenius on the ring. In particular, while the connection between these ideals was known, the underlying reason for the connection was a mystery. In this talk I give a description of an ideal, which can be obtained via de Jong's alterations, which coincides with the test ideal in characteristic p and coincides with the multiplier ideal in characteristic 0. Some links with vanishing theorems, will also be discussed. Finally, I will discuss some connections between these ideas and open questions in mixed characteristic. This is joint work with Manuel Blickle and Kevin Tucker.