arXiv subjects

Pramathanath Sastry

Publications and source records attributed to Pramathanath Sastry.

11 recordsLinked to original sources

arXiv2019

Grothendieck Duality and Transitivity I: Formal Schemes

For a proper map $f\colon X\to Y$ of noetherian ordinary schemes, one has a well-known natural transformation, ${\bf L}^*f^*(-)\overset{\bf L}{\otimes} f^!{\mathcal{O}}_Y\to f^!$, obtained via the projection formula, which extends, using Nagata's compactification, to the case where $f$ is separated and of finite type. In this paper we extend this transformation to the situation where $f$ is a pseudo-finite-type map of noetherian formal schemes which is a composite of compactifiable maps, and show it is compatible with the pseudofunctorial structures involved. This natural transformation has implications for the abstract theory of residues and traces, giving Fubini type results for iterated maps. These abstractions are rendered concrete in a sequel to this paper.

math.AG↗

arXiv2019

Grothendieck Duality and Transitivity II: Traces and Residues via Verdier's isomorphism

For a smooth map between noetherian schemes, Verdier relates the top relative differentials of the map with the twisted inverse image functor `upper shriek'. We show that the associated traces for smooth proper maps can be rendered concrete by showing that the resulting theory of residues satisfy the residue formulas (R1)--(R10) in Hartshorne's "Residues and Duality". We show that the resulting abstract transitivity map relating the twisted image functors for the composite of two smooth maps satisfies an explicit formula involving differential forms. We also give explicit formulas for traces of differential forms for finite flat maps (arising from Verdier's isomorphism) between schemes which are smooth over a common base, and use this to relate Verdier's isomorphism to Kunz and Waldi's regular differentials. These results also give concrete realisations of traces and residues for Lipman's fundamental class map via the results of Lipman and Neeman relating the fundamental class to Verdier's isomorphism.

math.AG↗

arXiv2015

Free resolutions of some Schubert singularities

In this paper we construct free resolutions of certain class of closed subvarieties of affine spaces (the so-called "opposite big cells" of Grassmannians). Our class covers the determinantal varieties, whose resolutions were first constructed by A. Lascoux (Adv. Math., 1978). Our approach uses the geometry of Schubert varieties. An interesting aspect of our work is its connection to the computation of the cohomology of homogeneous bundles (that are not necessarily completely reducible) on partial flag varieties.

math.AG↗

arXiv2010

Geometric Reductivity--A Quotient Space Approach

We give another proof that a reductive algebraic group is geometrically reductive. We show that a quotient of the semi-stable locus (by a linear action of a reductive algebraic group on a projective scheme) exists, and from this Haboush's Theorem (Mumford's Conjecture) follows.

math.AG↗

arXiv2007

Intermediate Jacobians and Hodge Structures of Moduli Spaces

Let SU_X(n,L) be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g>1 curve X. Let SU_X^s(n,L) denote the open subset parameterizing stable bundles. We show that for small i, the mixed Hodge structure on H^i(SU_X^s(n, L), Q) is independent of the degree of L, and hence pure of weight i. Moreover any simple factors is, up to Tate twisting, isomorphic to a summand of a tensor power of H^1(X,Q). A more precise statement for i = 3, yields a Torelli theorem complementing earlier work of several authors. This is a replacement of our preprint Intermediate Jacobians of Moduli spaces which contained a gap.

math.AG↗

arXiv2007

Applications of duality theory to cousin complexes

We use the anti-equivalence between Cohen-Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S_2 condition and the lowest cohomology N of its "dual" complex. We show that if a scheme has a Gorenstein complex satisfying certain coherence conditions, then in a finite étale neighborhood of each point, it has a dualizing complex. If the scheme already has a dualizing complex, then we show that the Gorenstein complex must be a tensor product of a dualizing complex and a vector bundle of finite rank. We relate the various results in [S] on Cousin complexes to dual results on coherent sheaves on formal schemes.

math.AG↗

arXiv2004

Pseudofunctorial behavior of Cousin complexes on formal schemes

On a suitable category of formal schemes equipped with codimension functions we construct a canonical pseudofunctor (-)^# taking values in the corresponding categories of Cousin complexes. Cousin complexes on such a formal scheme X functorially represent derived-category objects F by the local cohomologies H_x^{codim(x)}(F) (x \in X) together with "residue maps" from the cohomology at x to that at each immediate specialization of x; this representation is faithful when restricted to F which are Cohen-Macaulay (CM), i.e., H_x^i(F)=0 unless i=codim(x). Formal schemes provide a framework for treating local and global duality as aspects of a single theory. One motivation has been to gain a better understanding of the close relation between local properties of residues and global variance properties of dualizing complexes (which are CM). Our construction, depending heavily on local phenomena, is inspired by, but generalizes and makes more concrete, that of the classical pseudofunctor (-)^Δtaking values in residual complexes, on which the proof of Grothendieck's (global) Duality Theorem in Hartshorne's "Residues and Duality" is based. Indeed, it is shown in a subsequent paper by Sastry that (-)^# is a good concrete approximation to the fundamental duality pseudofunctor (-)^!. The pseudofunctor (-)^# takes residual complexes to residual complexes, so contains a canonical representative of (-)^Δ; and it generalizes as well several other functorial (but not pseudofunctorial) constructions of residual complexes which appeared in the 1990s.

math.AG↗

arXiv2004

Duality for Cousin Complexes

We relate the variance theory for Cousin complexes -^# developed by Lipman, Nayak and the author to Grothendieck duality for Cousin complexes. Specifically for a Cousin complex F on (Y, Δ)--with Δa codimension function on a formal scheme Y (noetherian, universally catenary)--and a pseudo-finite type map f:(X,Δ') --> (Y,Δ) of such pairs of schemes with codimension functions, we show there is a derived category map γ^!_f(F):f^#F --> f^!F, which is functorial as F varies over Cousin complexes on (Y,Δ), and induces an isomorphism f^#F = E(f^#F) --> E(f^!F). E here is the Cousin functor for the codimension function Δ. Further, we give conditions under which γ^!_f is an isomorphism. We also generalize the Residue Theorem of Grothendieck for residual complexes to Cousin complexes by defining trace as a sum of local residues when the map f is pseudo-proper.

math.AG↗

arXiv2003

Residues and duality for Cousin complexes

We construct a canonical pseudofunctor ^# on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is a concrete approximation to the restriction of the Grothendieck Duality pseudofunctor ^! to the full subcategory of the derived category having Cohen-Macaulay complexes as objects (a subcategory equivalent to the category of Cousin complexes, once a codimension function has been fixed). Specifically, for Cousin complexes M and any scheme map f:X -> Y as above, there is a functorial derived-category map γ: f^# M -> f^! M inducing a functorial isomorphism in the category of Cousin complexes f^# M \iso E(f^! M) (where E is the Cousin functor). γitself is an isomorphism if the complex f^! M is Cohen-Macaulay--which will be so whenever the map f is flat or whenever the complex M is injective. Also, f^# takes residual (resp. injective) complexes on Y to residual (resp. injective) complexes on X; and so the pseudofunctor ^# generalises--and makes canonical--the "variance theory" of residual complexes developed in Chapter VI of Hartshorne's "Residues and Duality." Moreover, we generalise the Residue Theorem of loc.cit., p.369 by defining a functorial Trace map of graded modules Tr_f(M): f_*f^# M -> M (a sum of local residues) such that whenever f is proper, Tr_f(M) is a map of complexes and the pair (f^# M, Tr_f(M)) represents the functor Hom(f_*C, M) of Cousin complexes C.

alg-geom↗

arXiv2000

Base change and Grothendieck duality for Cohen-Macaulay maps

Let $f:X\to Y$ be a Cohen-Macaulay map of finite type between Noetherian schemes, and $:Y'\to Y$ a base change map, with $Y'$ Noetherian. Let $f'$ be the base change of $f$ under $g$ and $g'$ the base change of $g$ under $f$. We show that there is a canonical isomorphism between ${g'}^*ω_f$ and $ω_{f'}$, where $ω_f$ and $ω_{f'}$ are the relative dualizing sheaves. The map underlying this isomorphism is easily described when $f$ is proper, and has subtler description when $f$ is not. If $f$ is smooth we show that this map between the dualizing sheaves corresponds to the canonical identification of differential forms. Our results generalize the results of B. Conrad in two directions - wedo not need the properness assumption, and we do not need to assume that theschemes involved carry dualizing complexes. Residual complexes do not appear in this paper.

math.AG↗

arXiv1996

Intermediate Jacobians of moduli spaces

Let $SU_X(n,L)$ be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g curve X. Let $SU_X^s(n,L)$ denote the open subset parametrizing stable bundles. We show that if g>3 and n > 1, then the mixed Hodge structure on $H^3(SU_X^s(n, L))$ is pure of type ${(1,2),(2,1)}$ and it carries a natural polarization such that the associated polarized intermediate Jacobian is isomorphic J(X). This is new when deg L and n are not coprime. As a corollary, we obtain a Torelli theorem that says roughly that $SU_X^s(n,L)$ (or $SU_X(n,L)$) determines X. This complements or refines earlier results of Balaji, Kouvidakis-Pantev, Mumford-Newstead, Narasimhan-Ramanan, and Tyurin.

alg-geom↗