On combinatorial and projective geometry springerlink. Therefore, by vector space, we shall always understand complex vector space unless otherwise specified. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. Jacobi operators on real hypersurfaces of a complex projective space cho, jong taek and ki, uhang, tsukuba journal of mathematics, 1998. Another example of a projective plane can be constructed as follows. I talked to hy bass, the author of the classic book algebraic ktheory, about what would be involved in writing such a book. From a thorough study of these cross ratios which, among other observations, includes a new matroid theoretic version and proof of the pappos. In algebraic topology, it is a cohomology theory known as topological ktheory. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory.
Pdf the bott inverted infinite projective space is. Hajac and tomasz maszczyk and albert jeuliang sheu and bartosz zielinski, year2020. Higher algebraic ktheory of schemes 249 theory of perfect complexes on schemes from sga 6. When people write algebraic ktheory today they mean the quillen version or other later developments of higher algebraic ktheory, and specify milnor ktheory when they mean that. The cohomology is zxx3 where x has degree 8, as you would expect. Lecture notes in algebraic topology pdf 392p this note covers the following topics. Surveys with accounts of the historical development include. It is also a fundamental tool in the field of operator algebras.
The real projective spaces in homotopy type theory arxiv. Cohomology of line bundles on projective space 463 18. Its probably wise to also include some simpler computations from ros94, x1. Because k0cclassifies the isomorphism classes in c and he wanted the name of the theory to reflect class, he used the first letter k in. It has been shown 2 that ktheory classifies the topological charge of the d brane gauge. Thus in the axiomatic development of the theory, the notions of distance and angle will play no part. Contents 1 introduction 2 2 motivic stable homotopy theory 4 3 the bott inverted in. All rings will be commutative with 1, and homomorphisms of. Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. This paper studies the algebraic ktheory of categories of gmodules on a scheme with an action of an algebraic group g. X ii a finite dimensional vector space structure on eac. This was before the general definition of algebraic ktheory by quillen.
As an application, we develop an intersection theory for quantum ruled surfaces. Also, does the answer depend on the choice of a definition of the infinite dimensional projective space how many possible definitions are there. Fibers, morphisms of sheaves back to work morphisms varieties. Foundations of projective geometry bernoulli institute. A geometric introduction to ktheory university of oregon. Talk projective modules over group rings for applications of algebraic ktheory in geometric topology, the most relevant rings are group rings of fundamental groups. By appointment, in 380383m third floor of the math building. In algebra and algebraic geometry, it is referred to as algebraic ktheory. Without subscripts, usually denotes complex ktheory whereas real ktheory is sometimes written as.
K theory of the infinite dimensional projective space. Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. Dominique arlettaz, algebraic ktheory of rings from a topological viewpoint. Its homotopy is unapproachable, because it is just two spheres stuck together, so you would pretty much have to know the homotopy groups of the spheres to. On the algebraic ktheory of some homogeneous varieties. Cross ratios constitute an important tool in classical projective geometry.
We define a category p n of quasicoherent sheaves on projective nspace and a notion of twisted structure sheaves o p n j. The bott inverted infinite projective space is homotopy algebraic ktheory article pdf available in bulletin of the london mathematical society 412 february 2009 with 40 reads. Vitaly tarasov, alexander varchenko submitted on 10 jan 2019. Pdf distinguished bases in the ktheory of multipullback.
In 1988 i wrote out a brief outline, following quillens paper higher algebraic ktheory i. From a build a topology on projective space, we define some properties of this space. They contain detailed information about the original object but are notoriously difficult to compute. As somebody who has been trying, for a long time, to understand the alexander polynomial, and who. However, one of the most important examples of the theory is the real pro. An introduction to berkovich analytic spaces and non. The packing problem in statistics, coding theory and. C cohx, the category of coherent sheaves on an algebraic variety x. In projective geometry, a hyper quadric is the set of points of a projective space where a certain quadratic. Potential theory on berkovich curves one obtains the berkovich projective line p1 berk by adjoining to a 1 berk in a suitable manner a point at in.
As i recall, the cayley projective plane is painful to build, but it is a 2cell complex, with an 8cell and a 16cell. Topological ktheory of complex projective spaces uc davis. Explain the arithmetic roots of algebraic ktheory, following for example ros94, x1. Daniel grayson, quillens work in algebraic ktheory, j. This paper is concerned with an analogous result for the algebraic ktheory of spaces in the sense of waldhausen. It contains equivariant analogs of quillens basic results on the ktheory of schemes. A spectral sequence for ktheory, lecture notes in seminer of r. In the spherical model, a projective point correspondsto a pair of antipodalpoints on the sphere. An element of p is called a point and a set of linearly independent respectively dependent points of pn. We compute the ktheory of complex projective spaces.
Riemannroch, degrees of coherent sheaves, arithmetic genus, and. Homotopy classification of twisted complex projective spaces of dimension 4 mukai, juno and yamaguchi, kohhei, journal of the mathematical society of japan, 2005. Both methods have their importance, but thesecond is more natural. Adams operations 17 acknowledgments 19 references 19 1. In a comment on tom goodwillies question about relating the alexander polynomial and the iwasawa polynomial, minhyong kim makes the cryptic but tantalizing statement in brief, the current view is that the iwasawa polynomialpadic lfunction should be viewed as a path in ktheory space. This is called algebraic ktheory if we wish to generalize this. Ktheory of the infinite dimensional projective space. A family of vector spaces over x is a topological space e, together with. Higher algebraic ktheory of schemes and of derived. In mathematics, ktheory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. Tensor product of bundles gives ktheory a commutative ring structure.
Galois theory and projective geometry 5 projective space subject to pappus axiom is a projectivization of a vector space over a. An arrow from space a to space b implies that space a is also a kind of space b. The projective space pn thus contains more points than the a. Kvect gx is known as equivariant topological ktheory. Using k theory, adams and atiyah were able to give a simple proof that the only spheres which can be provided with hspace structures are s1, s3 and s7. Pdf perspectives on projective geometry download full. The basic building block of ktheory is the vector bundle. This problem is not only interesting from a geometrical point of view. Roughly speaking, projective maps are linear maps up toascalar. Like for example, just being able to write down a resolution by powers of twisting sheaves. Equivariant algebraic ktheory northeastern university. Hspaces, division algebras, and tangent bundles of spheres. Lecture notes in algebraic topology pdf 392p download book.
Gx, the category of gequivariant vector bundles on a topological space x. Chain complexes, homology, and cohomology, homological algebra, products, fiber bundles, homology with local coefficient, fibrations, cofibrations and homotopy groups, obstruction theory and eilenbergmaclane spaces, bordism, spectra, and generalized homology and spectral sequences. We will now investigate these additional points in detail. Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. Varvara karpova, complex topological ktheory, 2009. Using the theory of tutte groups as discussed in 6 it will be shown in this note that the concept of cross ratios extends naturally to combinatorial geometries or matroids. First of all, this is explained by the fact that this theory is the closest generalization of the theory of surfaces in a threedimensional projective space. A textbook account of topological ktheory with an eye towards operator k. Intuitively, we can think of a vector bundle as a way of assigning a vector space to each point of a topological space in a way that varies continuously. Vaserstein on k,theory in this note we prove the conjecture for n 3 which, of course, implies the cases n 1 and 2. An important new perspective on affine and projectivegeometry this innovative book treats math majors and math education studentsto a fresh look at affine and projective geometry from algebraic,synthetic, and lattice theoretic points of view. The corresponding variety x vg is clearly unirational. In this paper we study the \holomorphic ktheory of a projective variety.
Introduction a basic result in the ktheory of algebraic varieties is the computation of the kgroups for projective space bundles 2, 5, 11. Grotherdieck who first studied k0c then written kc where for a scheme x, c is the category px of locally free sheaves of oxmodules. Aderemi kuku, introduction to ktheory and some applications. Introduction to intersection theory in algebraic geometry lectures. Grothendieck group of finitely generated projective rmodules for a com mutative ring r if. Moreover, it is possible to derive a substantial part of stable homotopy theory from ktheory. This paper will presume a fair understanding of the basic ideas of algebraic topology, including homotopy theory and co. The expert might skip this, but should look at the characterization of perfect complexes as finitely presented objects 2. Then is defined to be the grothendieck group of the commutative monoid of isomorphism classes of finitedimensional kvector bundles over x under whitney sum. In other terms, every real matrix of determinant 1 depending continuously on 3. Equivariant quantum differential equation, stokes bases, and ktheory for a projective space authors.
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