Shubhankar Sahai

Email: shubhankar (dot) sahai (dot) 72 (at) gmail (dot) com
Institutional email: ssahai (at) ucsd (dot) edu
Office: HSS 3073

I am a graduate student in the math department at UCSD.

I work in arithmetic and algebraic geometry. More specifically, my current work is in the intersection of derived algebraic geometry, p-adic Hodge theory and algebraic K-theory. I have some recent interests in (geometric) representation theory.

My advisor is Kiran Kedlaya.

Before graduate school, I worked as a technician in a physics lab and then as a math tutor.

Research

The PDF versions are more current than the arXiv versions.

Preprints

• Duality of differential operators and algebraic de Rham cohomology arXiv — Preprint
  with C. Ji, C. Kothari, O. Li, S. Makarova and S. Venkatesh.
  Notes

  This paper gives another proof of relative Poincaré duality in algebraic de Rham cohomology for smooth proper families X--->S, by generalising Serre duality to complexes over X where the differentials are finite order S-linear differential operators. The latter manoeuvre allows us to treat the relative de Rham complex as if it was a complex of O_X-modules. Then, via an examination of the Hodge-de Rham spectral sequence, one obtains a new proof of Poincaré duality. The lift of Serre duality to differential operators is obtained by combining recent ideas of Clausen on co-evaluation maps for de Rham cohomology along with more concrete constructions (due to de Jong). An obvious generalisation of this result is in progress with Caleb Ji, but is currently on hiatus.
• De Rham affineness of the Nygaard filtered prismatization in positive characteristic arXiv · PDF — Submitted
  Notes

  The PDF version has a more readable layout. Comments very welcome!

  The key idea in this paper is that the Nygaard filtered prismatization of an animated ring in positive characteristic is determined by its Nygaard filtered prismatic cohomology, functorially in the ring. The idea is not new and is already in the second paper of Bhatt-Lurie for unfiltered prismatization, but the observation that this is an intrinsic property of the functors in question (and not just the associated derived stacks) is. We name this property de Rham affineness. This concept will be useful in my future papers. Note that positive characteristic is not necessary and the same proof works mutatis mutandis relative to a perfectoid. It is also true relative to a bounded prism, provided one works with relative prismatic cohomology, but that's the subject of future work.
• Derived algebras on formal stacks and prismatic gauges arXiv · PDF — Preprint
  Notes

  A paper studying the theory of derived algebras (in the sense of Bhatt-Mathew and Raksit) in quasi-coherent sheaves on non-trivially formal derived stacks. The main goal is to develop a good working theory and prove classification theorems for derived algebras in prismatic gauges. Along the way we prove some basic theorems in (non-connective) formal derived geometry which seem to be missing from the literature.
• Tannakian reconstruction in derived algebraic geometry arXiv · PDF — Submitted
  Joint with J. Mundinger.
  Notes

  In the early 2010s, Lurie, Bhatt and Bhatt--Halpern-Leistner (building on a 'relative' perspective introduced by Lurie a decade earlier) proved strong reconstruction results for maps of spectral stacks from (certain) symmetric monoidal functors between their derived categories of quasi-coherent sheaves. Motivated by recent work in p-adic Hodge theory and Hochschild homology, one asks whether such reconstruction results exist in derived geometry, where the basic geometric objects are spectra of animated (i.e. simplicial commutative) rings rather than E_\infty-rings. In this paper, we answer this question by using some recent ideas of Nuiten-Toën.

In preparation

• Prismatic F-gauges and Frobenius liftable schemes Poster — In preparation
  Notes

  This paper gives a description of prismatic F-gauges on Frobenius liftable schemes in terms of big Fontaine-Laffaille modules. The main conceptual insight is that the Nygaard filtered prismatization of such a scheme is pulled back from the Hodge-filtered de Rham stack of its lift. Along the way we study a notion of p^n-de Rham stacks which deform prismatic cohomology to Z/p^n+1-schemes, with the case n=1 being relevant to our study.
• Cotangent-étale F_p-algebras and Cartier smoothness — In preparation
  Joint with R. Ishizuka.
  Notes

  Some counter-examples related to the notion of Cartier smoothness in algebraic K-theory are presented.