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Regular version of the site
2017/2018

## Sheaves and Homological Algebra

Category 'Best Course for Career Development'
Category 'Best Course for New Knowledge and Skills'
Type: Optional course (faculty)
When: 3, 4 module
Language: English
ECTS credits: 5

### Course Syllabus

#### Abstract

This is an introduction to the theory of sheaves and supplying homological algebra - commonly used technique for handling locally defined objects on a manifold X. In algebraic and/or differential geometry and topology it allows to produce global geometric and topological invariants of X from those local data. In non-commutative geometry it gives various geometric style invariants of categories equipped with Grothendieck topologies

#### Learning Objectives

• Formation of a clear understanding of basic concepts and basic beam theory methods
• Familiarity with the categorical approach to geometry and topology, as well as elements noncommutative geometry
• The study of the theory of cohomology of beams and its application for the study of the links between global topological and geometric characteristics of varieties and local properties of natural beams on them. Comparison of singular and cellular cohomology with the de Rham and Dolbeau cohomology
• Forward and reverse images of beams. Leray spectral sequence
• Cohomology of coherent sheaves on projective algebraic varieties and their applications in algebraic geometry
• Understanding Grothendieck topologies and bundles on sites

#### Expected Learning Outcomes

• Get a general idea of the categorical approach to geometry and topology
• Study the basic geometric and topological methods of the theory of beams and the cohomology theory of bundles and the technique of computing the cohomology of bundles in algebraic and differential geometry
• Be ready to use the basic principles and methods of the theory of beams and related homological algebra in the subsequent professional activity as researcher or university teacher

#### Course Contents

• Categories, prebundle functors. Important examples: prebundles on open categories sets of topological space and on the simplicial category. Category functors, Ioneda's lemma, representable functors. Conjugate functors
• Bundles on topological spaces. Layers, ethereal space and gouging forearm. Direct and reverse image. Abelian bundles
• Complexes and (co) homology, long exact sequence (co) homology. Koszul complexes. Commutation homology with filtered limits. Spectral sequences of filtered complexes, bicomplexes and exact pairs
• Global sections, sluggish tufts and sluggish resolution of Godement. Cohomology and hypercohomology of bunches. Acyclic resolvents. Mayer sequence, Vietoris and Cech's resolvent. Acyclic Coverage and Cartan acyclicity criterion. Czech cohomology
• Soft and thin beams on paracompact Hausdorf spaces. Bundles differential forms, Poincaré’s lemma and De Rham theorem
• Higher direct images. Spectral Leray sequence
• Coherent bundles in algebraic geometries and their geometric applications. Acyclic Affinity Coatings algebraic varieties. Cohomology reversible pencils on projective spaces
• Grothendieck topologies and bundles on sites

• Exam

#### Interim Assessment

• Interim assessment (4 module)
The final grade for the discipline is put on a 10-point scale and is calculated by the formula: min (150, E + H) / 15 where E and H are the number of solved exam and home tasks, expressed in percent of the total number of exam and mandatory home tasks, respectively. With this task increased difficulty is not considered as a mandatory task in the denominator, but are counted as solved problems in the numerator, so E = 100 * (number of solved examination problems) / (total number of examination problems) H = 100 * (the number of all solved household tasks, including the increased difficulties) / (total number of compulsory household tasks, excluding tasks of increased difficulty). Thus, to obtain an estimate of 10 is sufficient, for example, to successfully solve 75%. Exam tasks and 75% of compulsory household tasks, or in another way to collect the amount E + H = 150%. When a smaller amount is recruited, the estimate decreases linearly

#### Recommended Core Bibliography

• Harder, G. (2011). Lectures on Algebraic Geometry I : Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces. [Place of publication not identified]: Springer Spektrum. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1245198

#### Recommended Additional Bibliography

• McLean, M. (2010). A spectral sequence for symplectic homology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1011.2478