Rankeya Datta

Department of Mathematics, Statistics and Computer Science
University of Illinois Chicago

Email: rankeya(at)uic(dot)edu
Office: SEO 411

I am a Research Assistant Professor in the Mathematics Department at University of Illinois Chicago. I recently completed my Ph.D. from the University of Michigan, where I was advised by Karen Smith.

I work at the interface of algebraic geometry and commutative algebra, specializing in the use of local prime characteristic methods (in commutative algebra) and their global variants. I often enjoy thinking about non-Noetherian objects such as valuation rings, perfectoid rings, and more recently, rings of differential operators.

Teaching

Papers

• On some permanence properties of (derived) splinters (joint with Kevin Tucker). In preparation.
• Hilbert-Kunz multiplicity of fibers and Bertini theorems (joint with Austyn Simpson).
• Annihilators of D-modules in mixed characteristic (joint with Nicholas Switala and Wenliang Zhang).
• Permanence properties of F-injectivity (joint with Takumi Murayama).
• Frobenius splitting of valuation rings and F-singularities of centers.
• Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic. To appear in Transactions of AMS.
• Excellence in Prime Characteristic (joint with Karen Smith). Contemporary Mathematics, Local and Global Methods in Algebraic Geometry, 712 (2018), 105--116.
• (Non)Vanishing results on local cohomology of valuation rings. Journal of Algebra 479-1(2017), 413--436. DOI 10.1016/j.jalgebra.2016.12.03.
• Frobenius and valuation rings (joint with Karen Smith). Algebra & Number Theory 10-5 (2016), 1057-1090. DOI 10.2140/ant.2016.10.1057; corrigendum: Correction to the article Frobenius and valuation rings. Algebra & Number Theory 11-4 (2017), 1003-1007. DOI 10.2140/ant.2017.11.1003.
• Free and very free morphisms on a Fermat hypersurface (joint with T. Bridges, J. Eddy, M. Newman, J. Yu). Involve 6 (2013), No. 4, 437-445.
• Polygons in quadratically closed rings and properties of n-adically closed rings (undergraduate thesis supervised by Aise Johan de Jong).

Thesis

Here is a copy of my dissertation. Corrections are welcome!

Notes

Notes are subject to change without notice.

• Very rough notes (prepared for a Grad seminar at UMich) on a proof of a theorem of Kunz, characterizing regularity of Noetherian rings in terms of flatness of Frobenius, using the surprising homological properties of perfect rings. Based loosely on a talk given by Bhatt at Gennady Lyubeznik's 60th birthday conference, the notes are a more verbose version of the original proof appearing in a paper by Bhatt-Scholze.
• Notes on Huber rings for a learning seminar at the University of Michigan (Winter 2017). Last updated Feb 18, 2017. A new section was added on uniform Huber rings (not discussed in the lectures), following Bhargav’s discussion of uniform K-Banach algebras in his course. In particular, we prove equivalence of categories results generalizing [Bha17, Thm 9.7 and Cor 9.9].
• On a vanishing result in sheaf cohomology. An example is given of a non-quasicompact scheme that violates a vanishing result in sheaf cohomology that holds for certain quasicompact spaces [Stacks Project, Tag02UX]. This example can be interpreted purely topologically (without mentioning schemes), and is incorporated in the latter form in Tag0BX0.
• Notes from a summer mini-course I taught at Michigan on notions of singularities in prime characteristic in 2016. The notes have not been proof-read and do not cover a lot of material.