Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)


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 → Y be a proper map of noetherian schemes which is smooth of relative dimension n. For any G ∈ D+ qc(Y ) (where D+ qc(−) denotes the derived category of bounded-below complexes with quasi-coherent homology), set f s (G):= f ∗G ⊗ωf [n]. Then for any F ∈ D+ qc(X) there is a natural bi-functorial isomorphismHomD+ qc(X) (F, f sG) ∼→ HomD+ qc(Y ) (Rf∗F, G) (1.1)where ωf is the top exterior power of the sheaf of relative differential forms for X over Y.In other words, fs : D+ qc(Y ) → D+ qc(X) is a right adjoint to Rf∗ : D+ qc(X) → 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 f∗ = Γ(X, −) 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 → Y of noetherian schemes, the functor Rf∗ has a right adjoint, i.e., there is a functor f × : D+ qc(Y ) → D+ qc(X) such that for any F ∈ D+ qc(X) and G ∈ D+ qc(Y ), there is a natural bi-functorial isomorphismHomD(X) (F, f ×G) ∼−→ HomD+ qc(Y ) (Rf∗F, G). (1.2)Thus if f is both proper and smooth, f s and f × agree, but not in general. The adjointness property of f ×, in effect, gives a duality statement and under properness assumption this is of considerable interest since Rf∗ then preserves coherence of homology. It is interesting that the restriction of (−) × to the category of proper maps appears to blend seamlessly with the restriction of (−) s to the category of smooth maps and so gives rise to a family of functors (−) ! which forms the central object of study in Grothendieck duality.Thus Grothendieck duality mainly concerns constructing a theory of (−) ! primarily determined by the following conditions. For now, let C denote the category of finite-type separated map of noetherian schemes.


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


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