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Dissertations / Theses on the topic 'Riemann-Roch theorem'
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Shklyarov, Dmytro. "Hirzebruch-Riemann-Roch theorem for differential graded algebras." Diss., Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/1381.
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Schulze, Bert-Wolfgang, and Nikolai Tarkhanov. "The Riemann-Roch theorem for manifolds with conical singularities." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2505/.
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Xu, Quan. "On Deligne's functorial Riemann-Roch theorem in positive characteristic." Toulouse 3, 2014. http://thesesups.ups-tlse.fr/2365/.
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Dans cette thèse, on donne une nouvelle preuve d'une variante du théorème factoriel Deligne-Riemann-Roch dans le cas de caractéristique positive en utilisant des idées qui apparaissent dans la preuve du Pink et Rössler du théorème d'Adams-Riemann-Roch en caractéristique positive. La méthode du Pink et Rössler qui est valide en caractéristique positive et qui est complétement différente de preuve classique, nous permettra de montrer le théorème factoriel de Deligne-Riemann-Roch d'une façon plus facile et direct. Notre preuve est aussi partiellement compatible avec l'isomorphisme de Mumford<br>In this note, we give a proof for a variant of the functorial Deligne-Riemann-Roch theorem in positive characteristic based on ideas appearing in Pink and Rössler's proof of the Adams-Riemann-Roch theorem in positive characteristic (see [14]). The method of their proof appearing in [14], which is valid for any positive characteristic and which is completely different from the classical proof, will allow us to prove the functorial Deligne-Riemann-Roch theorem in a much easier and more direct way. Our proof is also partially compatible with Mumford's isomorphism
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Hahn, Tobias. "An arithmetic Riemann-Roch theorem for metrics with cusps." Aachen Shaker, 2009. http://d-nb.info/997223146/04.
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De, Gaetano Giovanni. "A regularized arithmetic Riemann-Roch theorem via metric degeneration." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19227.
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Das Hauptresultat dieser Arbeit ist ein regularisierter arithmetischer Satz von Riemann-Roch für ein hermitesches Geradenbündel, die isometrisch zum Geradenbündel den Spitzenformen vom geraden Gewicht ist, auf eine arithmetische Fläche, deren komplexe Faser isometrisch zu einer hyperbolischen Riemannschen Fläche ohne elliptische Punkte ist.
Der Beweis des Resultats erfolgt durch metrische Degeneration: Wir regularisieren die betreffenden Metriken in einer Umgebung der Singularitäten, wenden dann den arithmetischen Riemann-Roch-Satz von Gillet und Soulé an und lassen schließlich den Parameter gegen Null gehen. Durch die metrische Degeneration entsteht auf beiden Seiten der Formel ein divergenter Term. Die asymptotische Entwicklung der Divergenz berechnet sich auf der einen Seite direkt aus der Definition der glatten arithmetischen Selbstschnittzahlen.
Der divergente Term auf der anderen Seite ist die zeta-regularisierte Determinante des zu den regularisierten Metriken assoziierten Laplace-Operators, der auf den 1-Formen mit Werten in dem betrachteten hermitischen Geradenbündel operiert. Wir definieren und berechnen zuerst eine Regularisiereung des entsprechenden zu den singulären Metriken assoziierten Laplace-Operators; diese wird später im regularisierten Riemann-Roch-Satz auftauchen. Zu diesem Zweck passen wir Ideen von Jorgenson-Lundelius, D'Hoker-Phong und Sarnak auf die vorliegende Situation an und verallgemeinern diese.
Schließlich beweisen wir eine Formel für den zum betrachteten hermitischen Geradenbündel assoziierten Wärmeleitungskern auf der Diagonalen bei einer Modellspitze. Diese Darstellung steht im Zusammenhang mit einer Entwicklung nach zur Whittaker-Gleichung assoziierten Eigenfunktionen, die im Anhang bewiesen wird. Weitere Abschätzungen des zum betrachteten hermitischen Geradenbündel gehörigen Wärmeleitungskern auf der komplexe Faser der arithmetischen Fläche schließen den Beweis des Hauptresultats ab.<br>The main result of the dissertation is an arithmetic Riemann-Roch theorem for the hermitian line bundle of cusp form of given even integer weights on an arithmetic surface whose complex fiber is isometric to an hyperbolic Riemann surface without elliptic points.
The proof proceeds by metric degeneration: We regularize the metric under consideration in a neighborhood of the singularities, then we apply the arithmetic Riemann-Roch theorem of Gillet and Soulé, and finally we let the parameter go to zero. Both sides of the formula blow up through metric degeneration. On one side the exact asymptotic expansion is computed from the definition of the smooth arithmetic intersection numbers.
The divergent term on the other side is the zeta-regularized determinant of the Laplacian acting on 1-forms with values in the chosen hermitian line bundle associated to the regularized metrics. We first define and compute a regularization of the determinant of the corresponding Laplacian associated to the singular metrics, which will later occur int he regularized arithmetic Riemann-Roch theorem. To do so we adapt and generalize ideas od Jorgenson-Lundelius, D'Hoker-Phong, and Sarnak.
Then, we prove a formula for the on-diagonal heat kernel associated to the chosen hermitian line bundle on a model cusp, from which its behavior close to a cusp is transparent. This expression is related to an expansion in terms of eigenfunctions associated to the Whittaker equation, which we prove in an appendix. Further estimates on the heat kernel associated to the chosen hermitian line bundle on the complex fiber of the arithmetic surface prove the main theorem.
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Arruda, Rafael Lucas de [UNESP]. "Teorema de Riemann-Roch e aplicações." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/86493.
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arruda_rl_me_sjrp.pdf: 624072 bytes, checksum: 23ddd00e27d1ad781e2d1cec2cb65dee (MD5)<br>Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)<br>O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica<br>The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve
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Hahn, Tobias [Verfasser]. "An arithmetic Riemann-Roch theorem for metrics with cusps / Tobias Hahn." Aachen : Shaker, 2009. http://d-nb.info/1156518318/34.
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Arruda, Rafael Lucas de. "Teorema de Riemann-Roch e aplicações /." São José do Rio Preto : [s.n.], 2011. http://hdl.handle.net/11449/86493.
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Orientador: Parham Salehyan<br>Banca: Eduardo de Sequeira Esteves<br>Banca: Jéfferson Luiz Rocha Bastos<br>Resumo: O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica<br>Abstract: The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve<br>Mestre
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Porto, Anderson Corrêa. "Divisores sobre curvas e o Teorema de Riemann-Roch." Universidade Federal de Juiz de Fora (UFJF), 2018. https://repositorio.ufjf.br/jspui/handle/ufjf/6612.
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Previous issue date: 2018-02-08<br>O objetivo desse trabalho é o estudo de conceitos básicos da Geometria Algébrica sob o
ponto de vista clássico. O foco central do trabalho é o estudo do Teorema de Riemann-
Roch e algumas de suas aplicações. Esse teorema constitui uma importante ferramenta no
estudo da Geometria Algébrica clássica uma vez que possibilita, por exemplo, o cáculo do
gênero de uma curva projetiva não singular no espaço projetivo de dimensão dois. Para o
desenvolvimento do estudo do Teorema de Riemann-Roch e suas aplicações serão estudados
conceitos tais como: variedades, dimensão, diferenciais de Weil, divisores, divisores sobre
curvas e o anel topológico Adèle.<br>The goal of this work is the study of basic concepts of Algebraic Geometry from the
classical point of view. The central focus of the paper is the study of Riemann-Roch
Theorem and some of its applications. This theorem constitutes an important tool in the
study of classical Algebraic Geometry since it allows, for example, the calculation of the
genus of a non-singular projective curve in the projective space of dimension two. For
the development of the study of the Riemann-Roch Theorem and its applications we will
study concepts such as: varieties, dimension, Weil differentials, divisors, divisors on curves
and the Adèle topological ring.
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Kramer, Jürg [Gutachter], and Gerard [Gutachter] Freixas. "A regularized arithmetic Riemann-Roch theorem via metric degeneration / Gutachter: Jürg Kramer, Gerard Freixas." Berlin : Humboldt-Universität zu Berlin, 2018. http://d-nb.info/1182540503/34.
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Peixoto, Rafael 1983. "Algebras munidas de função peso e codigos de Goppa pontuais." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306309.
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Previous issue date: 2007<br>Resumo: O objetivo principal desta dissertação é apresentar o resultado central de R. Matsumoto sobre as álgebras munidas de função peso serem anéis de coordenadas a fim de curvas algébricas com exatamente um lugar de grau um no infinito. A partir disto, pode-se concluir que os códigos de avaliação, introduzidos por Ho/holdt, van Lint e Pellikaan, construídos sobre estas álgebras são um caso particular dos códigos geométricos de Goppa, isto e, códigos de Goppa pontuais. Para isto, utilizamos resultados sobre teoria de corpos de funções algébricas, de códigos geométricos de Goppa e de álgebra comutativa. Com a introdução dos conceitos de funções ordem e peso, nos é permitido descrever os códigos de avaliação e assim determinar cotas inferiores para a distancia mínima dos seus códigos duais, que em alguns casos são melhores que as cotas de Goppa<br>Abstract: The main objective of this text is to present the central result of R. Matsumoto concerning those algebra with a weight function being affine coordinate ring of an affine algebraic curve with exactly one place at infinity. From that statement one can conclude that the evaluation codes, introduced by Ho/holdt, van Lint e Pellikaan, constructed on this algebra are particular cases of geometric Goppa codes, that is, one point AG codes. For this, we use results of the algebraic function fields theory, geometric Goppa codes and commutative algebra. The introduction of the concepts of order and weight functions enable us to describe the evaluation codes and thus to determine lower bounds for the minimum distance of its duals codes, in same cases, are better than the Goppa¿s bounds<br>Mestrado<br>Mestre em Matemática
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Nazaikinskii, Vladimir, and Boris Sternin. "Relative elliptic theory." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2640/.
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This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms.
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Ho, Man-Ho. "On the differential Grothendieck-Riemann-Roch theorems." Thesis, Boston University, 2012. https://hdl.handle.net/2144/32024.
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Thesis (Ph.D.)--Boston University<br>PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.<br>We investigate aspects of differential K-theory. In particular, we give a direct proof that the Freed-Lott differential analytic index is well defined, and a short proof of the differential Grothendieck-Riemann-Roch theorem in the setting of Freed-Lott differential K-theory. We also construct explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory, define the Simons-Sullivan differential analytic index, and prove the differential Grothendieck-Riemann-Roch theorem in the setting of Simons-Sullivan differential K-theory.<br>2031-01-02
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Fischbacher-Weitz, Helena Beate. "Equivariant Riemann-Roch theorems for curves over perfect fields." Thesis, University of Southampton, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444966.
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Yokura, Shoji, and yokura@sci kagoshima-u. ac jp. "Verdier--Riemann--Roch for Chern Class and Milnor Class." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi933.ps.
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Hişil, Hüseyin. "Elliptic curves, group law, and efficient computation." Thesis, Queensland University of Technology, 2010. https://eprints.qut.edu.au/33233/1/H%C3%BCseyin_Hi%C5%9Fil_Thesis.pdf.
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This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.
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Usatine, Jeremy. "Arithmetical Graphs, Riemann-Roch Structure for Lattices, and the Frobenius Number Problem." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/hmc_theses/57.
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If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the construction for Lorenzini's Riemann-Roch structure and how it relates to the Riemann-Roch theorem for finite graphs shown by Matthew Baker and Serguei Norine.
We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of R. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of R.
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Manjunath, Madhusudan Verfasser], and Kurt [Akademischer Betreuer] [Mehlhorn. "A Riemann-Roch theory for sublattices of the root lattice An, graph automorphisms and counting cycles in graphs / Madhusudan Manjunath. Betreuer: Kurt Mehlhorn." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2012. http://d-nb.info/1052292607/34.
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Backman, Spencer Christopher Foster. "Combinatorial divisor theory for graphs." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/51908.
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Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as “combinatorial shadows” of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connections to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry.
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Auth, Matthew Leonard. "The quaternionic Riemann -Roch theorem." 2002. https://scholarworks.umass.edu/dissertations/AAI3039335.
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The aim is to define what it means to be a meromorphic section of a quaternionic holomorphic vector bundle over a compact Riemann surface and then prove a version of the Riemann-Rock theorem for divisors that generalizes the classical theorem. A meromorphic section of a quaternionic spin bundle provides Weierstrass data (modulo period conditions) for a conformal map into Euclidean three space with prescribed mean curvature half density.
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Rajeev, B. "Riemann Roch Theorem For Algebraic Curves." Thesis, 2005. https://etd.iisc.ac.in/handle/2005/1448.
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Rajeev, B. "Riemann Roch Theorem For Algebraic Curves." Thesis, 2005. http://etd.iisc.ernet.in/handle/2005/1448.
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Zheng, Weiqiong. "The Riemann Roch Theorem (for algebraic curves)." Thesis, 2017. http://hdl.handle.net/1885/173640.
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The Riemann-Roch theorem is a useful tool to calculate the dimension of the space of meromorphic functions with prescribed zeros and poles. There are severals versions of the theorem such as the Riemann-Roch theorem for line bundles, for (algebraic) curves, for surfaces and for higher dimensions. In this thesis, we will focus on the Riemann-Roch theorem for algebraic curves over an algebraically closed eld, which is a very important result in complex analysis and algebraic geometry. The study of the elds of rational functions on curves can be very useful in the proof. So we will recall some pre-knowledges in commutative algebra and some facts about a ne varieties. Then talk about function elds, discrete valuation rings and Weil di erentials to prove the theorem, using the methods of Andre Weil.
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Chang, Yu-Chan, and 張育展. "The Index Theorem from Gauss-Bonnet and Riemann-Roch Theorem." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/92526436832807455591.
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碩士<br>國立中央大學<br>數學研究所<br>98<br>In this thesis, we prove two important theorems in geometry. In chapter one, we state the Gauss-Bonnet theorem on even dimensional manifold and give the detail of the proof of two dimensional case. The proof is based on the paper "A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifold", published by S.S. Chern in 1943. A little history of this theorem is included. Chapter two and three mainly focus on Riemann-Roch theorem on one-dimensional complex manifold, Riemann surface. We establish some basics on Riemann surface in chapter two, such as divisors, holomorphic line bundles, sheaves and cohomology on sheaves, also Hodge theorem in the end of this chapter. The proof of Riemann-Roch is in the chapter three. In chapter four, we show a theorem by calculating two analytic indices of two operators, which give us Gauss-Bonnet and Riemann-Roch theorem. This theorem is the Atiyah-Singer index theorem, proved by Atiyah and Singer in 1963.
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Tsai, Yi-Chia, and 蔡宜家. "Some Applications of the Riemann-Roch Theorem on Compact Riemann Surfaces." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/45710602205964904433.
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碩士<br>國立成功大學<br>數學系應用數學碩博士班<br>95<br>This is an exposition of the Riemann-Roch theorem and its application to the study of compact Riemann surfaces.In particular, we show that all compact Riemann surfaces are algebraic.
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Liao, Sin-Ying, and 廖欣瑩. "Riemann-Roch-Hirzebruch Theorem For Circle Bundle Over A Riemann Surface." Thesis, 2017. http://ndltd.ncl.edu.tw/handle/6zvz75.
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碩士<br>國立中央大學<br>數學系<br>105<br>In a recent paper of Cheng, Hsiao and Tsai, they obtained a CR Riemann-Roch-Hirzebruch Theorem for CR manifolds with a circle action. In this thesis we first review the Riemann-Roch-Hirzebruch Theorem for Riemann surfaces. Then we discuss CR Riemann-Roch-Hirzebruch Theorem of Cheng, Hsiao and Tsai for a circle bundle over a Riemann surface in details.
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Huang, Hao-Wei, and 黃皓偉. "A Proof of Riemann-Roch Theorem by Algebraic Methods." Thesis, 2018. http://ndltd.ncl.edu.tw/handle/a24x7a.
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碩士<br>國立臺灣大學<br>數學研究所<br>106<br>We start from some basic notions, like sheaves and cohomology, and try to introduce and prove Riemann-Roch theorem in the 2-dimension case. The definition of cohomology of a sheaf is more difficult to compute in some situation. However, the Čech cohomology of a sheaf over a paracompact space is isomorphic to the usual definition of cohomology,and Čech cohomology gives us a more concrete way to think what the cohomology of a sheaf is. In chapter 3 we introduce the concept of twisted complexes. We will use it to compute Ext and the class in Čech cohomology which is in the statement of Riemann-Roch theorem, and identify this class with characteristic class Td in cochain level by direct computation.
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Kerber, Michael [Verfasser]. "The enumerative geometry of rational and elliptic tropical curves and a Riemann-Roch theorem in tropical geometry / Michael Kerber." 2009. http://d-nb.info/996406573/34.
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