| Description |
Let R be a standard graded polynomial ring that is finitely generated over a field, let m be the homogeneous maximal ideal of R, and let I be a homogeneous prime ideal of R. A recent paper of Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of the thickenings R/It in characteristic 0 and provided a stabilization result for these cohomology modules. We explicitly construct isomorphisms between local cohomology modules of thickenings. Specifically, let k be a field, and consider an 2 ×3 matrix X = (xij) whose entries are in- dependent indeterminates over k. Define I2 to be the ideal of size two minors of X, and consider the residue class ring, R/I2, known as a determinantal ring. When the character- istic of k is 0, we relate the isomorphisms between local cohomology modules of thickenings of this determinantal ring with the Taylor series of natural log. When the characteristic of k is p > 0, we find the maps between local cohomology modules are remarkably different. Building on the work of Dao and Monta ̃no, we calculate the lengths of local cohomology modules of thickenings in the context that k is a field, X a 2 ×m matrix of indeterminates, and I2 the ideal of size two minors. Finally, we explicily construct isomorphisms between local cohomology modules of thick- enings when R/I is isomorphic to a particular family of Segre products. We relate these isomorphisms with Taylor series in generators of the ideal, I. iii |
