#### Date of Submission

9-28-2019

#### Date of Award

9-28-2020

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

#### Supervisor

Nayak, Suresh (TSMU-Bangalore; ISI)

#### Abstract (Summary of the Work)

There are two parts to this thesis and both the parts involve working with derived categories over noetherian formal schemes. Beyond this there is no overlap between them and we discuss them separately.The first part concerns Grothendieck duality on noetherian formal schemes.Grothendieck duality is a vast generalisation of Serreduality in algebraic geometry. The main statements in this theory are expressed in the language of derived categories. We begin with an important special case.Let f : X â†’ Y be a proper map of noetherian schemes which is smooth of relative dimension n. For any G âˆˆ D+ qc(Y ) (where D+ qc(âˆ’) denotes the derived category of bounded-below complexes with quasi-coherent homology), set f s (G):= f âˆ—G âŠ—Ï‰f [n]. Then for any F âˆˆ D+ qc(X) there is a natural bi-functorial isomorphismHomD+ qc(X) (F, f sG) âˆ¼â†’ HomD+ qc(Y ) (Rfâˆ—F, G) (1.1)where Ï‰f is the top exterior power of the sheaf of relative differential forms for X over Y.In other words, fs : D+ qc(Y ) â†’ D+ qc(X) is a right adjoint to Rfâˆ— : D+ qc(X) â†’ D+ qc(Y ).In particular, if X is a smooth projective variety of dimension n over an algebraically closed field k and F is a locally free sheaf on X, we recover Serre duality by plugging in Y = Spec(k) and G = k so that fâˆ— = Î“(X, âˆ’) the global sections functor.The right-adjointness of f s however does not hold in general if the properness or the smoothness assumption on f is dropped. But it turns out we do have the following: For any separated finite-type map f : X â†’ Y of noetherian schemes, the functor Rfâˆ— has a right adjoint, i.e., there is a functor f Ã— : D+ qc(Y ) â†’ D+ qc(X) such that for any F âˆˆ D+ qc(X) and G âˆˆ D+ qc(Y ), there is a natural bi-functorial isomorphismHomD(X) (F, f Ã—G) âˆ¼âˆ’â†’ HomD+ qc(Y ) (Rfâˆ—F, G). (1.2)Thus if f is both proper and smooth, f s and f Ã— agree, but not in general. The adjointness property of f Ã—, in effect, gives a duality statement and under properness assumption this is of considerable interest since Rfâˆ— then preserves coherence of homology. It is interesting that the restriction of (âˆ’) Ã— to the category of proper maps appears to blend seamlessly with the restriction of (âˆ’) s to the category of smooth maps and so gives rise to a family of functors (âˆ’) ! which forms the central object of study in Grothendieck duality.Thus Grothendieck duality mainly concerns constructing a theory of (âˆ’) ! primarily determined by the following conditions. For now, let C denote the category of finite-type separated map of noetherian schemes.

#### Control Number

ISILib-TH488

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Singh, Saurabh Dr., "Some Topics Involving Derived Categories over Noetherian Formal Schemes." (2020). *Doctoral Theses*. 432.

https://digitalcommons.isical.ac.in/doctoral-theses/432

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843852