Hereditary categories with serre duality which are. Serre duality for smooth maps, continued now for any o xcomplex f, and n 2z, ext n x f. Introduction to the strange duality conjecture for surfaces february 1, 2016 the goal of this document is to give a nontechnical and imprecise introduction to the strange duality conjecture for surfaces. The above discussion suggests that one may recast this result as a descenttheoretic statement along the etale cover spf e. One useful computational tool in the study of the knlocal picard group is the existence of a map pic n. Derived equivalences and serre duality for gorenstein. Find materials for this course in the pages linked along the left. To translate this statement into mathematical rigor, we just take the direct limit over all neighborhoods of x. Then on an arbitrary projective scheme x, we show that there is a coherent sheaf. Because of the adjoint property of f 1, we have a morphismf 1o y. Letgra denote the category of graded rightamodules. In mathematics, coherent duality is any of a number of generalisations of serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local theory. Serre duality was established for complex supermanifolds in, for projective supervarieties over a field in, and for projective supervarieties over a grassmann algebra in. Lecture 25 proof of serre duality mit opencourseware.
Throughout, swill be a smooth projective surface over c and k. On the way, we will spend time on chern classes and moduli of coherent sheaves. This duality induces the generalized serre functor on t, which is a linear triangle equivalence between two thick triangulated subcategories of t. Nov 01, 2009 derived equivalences and serre duality for gorenstein algebras derived equivalences and serre duality for gorenstein algebras abe, hiroki. Ringel, the diamond category of a locally discrete ordered set, proc. Serre duality manoj kummini we give a proof of the serre duality theorem using duality for. A natural notion arising here is that of a serre functor, effecting a certain duality in the triangulated category. In this formulation serre duality has been generalized to complex spaces.
Proving serre duality for finite flat covers of other spaces for which duality. This paper deals with the study of triangulated categories by the method of unwinding. On serre duality for compact homologically smooth dg algebras. This section provides the schedule of lecture topics and the lecture notes for each session. In this paper we classify noetherian hereditary abelian categories satisfying serre duality in the sense of bondal and kapranov. Serre duality, abels theorem, and jacobi inversion for. In particular, when f is locally free of finite rank on a projective nspace x, we. The quiver of projectives in hereditary categories with serre duality the quiver of projectives in hereditary categories with serre duality berg, carl fredrik. Serre duality for projective supermanifolds springerlink. Lecture notes in mathematics a collection of informal reports and seminars edited by a. Well deduce the serre duality of curves from a linear algebra observation.
We now sketch how the serre duality theorem iii, 7. Dg algebras, arising in this way, have to satisfy some compactness and smoothness conditions. It seems to be known in the more general setting of derived categories and morphisms of superschemes, though we are not aware of a reference. Note in particular that when f is a locally free, niterank o. When x is a smooth projective variety, we use the gorensteinness of a to prove a version of serre duality on proj a, the noncommutative p1 bundle defined by a. As a consequence we obtain a classification of saturated noetherian hereditary categories. So if the natural transformation is an isomorphism when both are locally free, it is an isomorphism when one is locally free and the other merely coherent since on a projective scheme, every coherent sheaf has a locally free presentation, and we can use the finite presentation trick.
When e has rank 2, we prove that a is gorenstein by computing the right derived functors of the internal hom functor. We will call this set of data the generalized serre duality on c. Csirik this paper was written in conjunction with r. Hereditary uniserial categories with serre duality. Derived equivalences and serre duality for gorenstein algebras derived equivalences and serre duality for gorenstein algebras abe, hiroki. In this note we prove the serre duality theorem for the cohomology of coherent. The quiver of projectives in hereditary categories with. Lecture 25 proof of serre duality free online course. In other words, 1 holds for i n and any coherent sheaf not just locally free coher. And best of all, it should arise from a cup product of some sort trace version of serre duality, which requires defining the cup product. Then we can obtain a natural map in one direction, f0. Reconstruction theorems bondalorlov, tannaka, gabriel.
Birkhoffgrothendieck, riemannroch, serre duality mit. It can also be generalizred to not locally free sheaves and nonsmooth varieties best. Serre duality and applications 3 furthermore, if g is a sheaf of o ymodules, then f 1g is a f 1o ymodule. Serre duality is the statement, for a smooth projective integral variety and a locally free sheaf on. Noetherian hereditary categories satisfying serre duality.
Serre duality is the analog for coherent sheaf cohomology of poincare duality in topology, with the canonical line bundle replacing the orientation sheaf. First we will prove our earlier assertion that the optimal solution of a dual program gives a bound on the optimal value of the primal program. More precisely, the basic object of study is a triangulated category j. We now need to unwind what duality of compactly generated categories is. It is based on earlier works in several complex variables, however, we will observe it algebraically. Introduction to the strange duality conjecture for surfaces. An abelian krullschmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. We introduce a notion of generalized serre duality on a hom. Ringel, a ray quiver construction of hereditary abelian categories with serre duality, proc. Serre finiteness and serre vanishing for noncommutative p1bundles adam nyman abstract. Serre duality carl lian june 17, 2016 abstract notesforatalkonserredualityfollowingvakilsnotesvwithappeal tohandmtowardtheend.
A locally free coherent sheaf of rank n on p1 is isomorphic to. The category of strict polynomial functors was introduced by friedlander and suslin 15 in their work on the cohomology of nite group. An invertible sheaf is a locally free sheaf of rank one. The first kind of duality one might ask for because we used it in discussing curves and surfaces is serre duality for vector bundles. Lecture 25 proof of serre duality mit opencourseware free. As a side result we show that when our hereditary categories have no nonzero projectives or injectives, then the serre duality property is equivalent to the existence of almost split sequences. We need to show that s is a free rmodule, which is equivalent to s being a. Springer nature is making coronavirus research free. Serre duality is a special case of the duality called the coherent duality in a much general setting. In general there are plenty of nontrivial line bundles on a ne varieties, e. We show that every klinear abelian extfinite hereditary category with serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver. As in the commutative case, these facts, along with.
The secondnamed author was partially supported by nsf grantsc 2012 american mathematical. Indcoherent sheaves and serre duality ii introduction. Translated from funktsionalnyi analiz i ego prilozheniya, vol. The riemannroch theorem and serre duality 3 locally around x. This post goes through the statement and proof of serre duality for arbitrary projective schemes, as presented in chapter iii. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. In this paper, we describe a serre functor on the category of perfect modules over an arbitrary compact and smooth dg algebra and use it to prove the existence of a nondegenerate pairing on the hochschild homology of the dg algebra. Introduction the purpose of this paper is to develop some of the theory from r. Special cases of serreduality using l received by the editors june 15, 2010. Joseph lipman purdue university department of mathematics.
Duality for finite morphisms in this section, x and y are noetherian schemes and f. Serre duality for noncommutative p1bundles article pdf available in transactions of the american mathematical society 3574 november 2002 with reads how we measure reads. It can also be generalizred to not locally free sheaves and nonsmooth varieties best described using derived categories. Hereditary categories with serre duality which are generated. Indcoherent sheaves and serre duality ii 5 suppose g is proper. One useful computational tool in the study of the knlocal picard group is. We prove the serre duality theorem for the noncommutative projective scheme proj awhen ais a graded noetherian pi ring or a graded noetherian asgorenstein ring. For the second, we note that since x is geometric, p is an a. The serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties.
If f x is a vector bundle, f of the associated locally free sheaf, and. We carefully develop the theory of serre duality and dualizing sheaves. Hartshornes spring 1996 algebraic geometry course at uc berkeley. All projective cohenmacaulay kschemes of pure dimension n are. In algebraic geometry, a branch of mathematics, serre duality is a duality for the coherent sheaf. The answer will be yes, otherwise we wouldnt be discussing grothendieck duality.
Hereditary uniserial categories with serre duality springerlink. Observe that the category chas serre duality in the sense of 7 if and only if r c l. In this paper, we classify the hereditary uniserial categories with serre duality. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. Serre duality is a special case of the duality called the coherent duality in a much. Apr 06, 2014 serre duality is the statement, for a smooth projective integral variety and a locally free sheaf on. This includes the unconditional existence of a special object that we want to single out.
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