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See all on WikipediaK-theory - Wikipedia
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field … See more
The Grothendieck completion of an abelian monoid into an abelian group is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a suitable category and turning it into … See more
The subject can be said to begin with Alexander Grothendieck (1957), who used it to formulate his Grothendieck–Riemann–Roch theorem. It takes its name from … See more
K0 of a field
The easiest example of the Grothendieck group is the Grothendieck group of a point $${\displaystyle {\text{Spec}}(\mathbb {F} )}$$ for a field $${\displaystyle \mathbb {F} }$$. Since a vector bundle over this space is just a finite … See moreThere are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
Grothendieck group for compact Hausdorff spaces
Given a compact Hausdorff space $${\displaystyle X}$$ See moreThe other historical origin of algebraic K-theory was the work of J. H. C. Whitehead and others on what later became known as See more
Virtual bundles
One useful application of the Grothendieck-group is to define virtual vector bundles. For … See moreThe equivariant algebraic K-theory is an algebraic K-theory associated to the category $${\displaystyle \operatorname {Coh} ^{G}(X)}$$ of equivariant coherent sheaves on an algebraic scheme $${\displaystyle X}$$ with action of a linear algebraic group See more
Wikipedia text under CC-BY-SA license K-theory in nLab - ncatlab.org
Apr 7, 2023 · Most explicit constructions of K-theory spectra start with the data of an exact category, such as notably Quillen’s Q-construction and the Waldhausen S-construction. the …
With this approach, the K-groups of a ring Rare precisely the K-groups of the category of nitely generated projective R-modules. 4 At this point it may seem like we’re requiring a huge leap of …
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Of course, h is an isomorphism if there exists a homomorphism k: F ! E with k h = idE and. h k = idF . This clearly happens if and only if h is bijective and h 1 is continuous. The families E and …
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A rst idea of K-theory is to replace a monoid with a group by formally throwing in inverses. The K stands for \class," which is spelled with a K in German. 2. Maps of Monoids Definition 1.5. Let …
Additive K-Theory (Lecture 18) October 15, 2014 Let C be a pointed 1-category 1-category which admits nite colimits, let C 0 C be a full subcategory which is closed under nite colimits, and …
Here we outline Quillen’s construction of the K-theory of an exact category, as well as stating the main theorems we will be using. For the full treatment, see
The goal of this lecture is to give the basic definition of K-theory. The process of group comple- tion, which “completes” a commutative monoid M to an abelian group KpMq, loses information
Karnaugh map - Wikipedia
A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced it in 1953 [1] [2] as a refinement of Edward W. Veitch's 1952 Veitch chart, [3] [4] which itself was a …
May 5, 2000 · ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY To analyze K(Er~) we expect to use a localization sequence in algebraic K-theory to reduce to the algebraic K-theory …
In these lectures, we will sketch the foundations of a homotopy theory for algebraic varieties over a fixed field k. Morally, we want to replace the category Top of topological spaces by the …
SOME BASICS OF ALGEBRAIC K-THEORY AARON LANDESMAN 1. INTRODUCTION: ALGEBRAIC AND TOPOLOGICAL K THEORY In this paper, we explore the basics of …
K-theory (physics) - Wikipedia
K-theory classifies D-branes in noncompact spacetimes, intuitively in spacetimes in which we are not concerned about the flux sourced by the brane having nowhere to go.
TOPOLOGICAL K-THEORY ZACHARY KIRSCHE Abstract. The goal of this paper is to introduce some of the basic ideas sur-rounding the theory of vector bundles and topological K-theory. To …
In intersection theory one studies the geometry of the intersection Y \Z. This can be done using (Chow) cohomology by examining the product [Y ] [Z] 2 H (X), or with K-theory by studying the …
On the K-theory of pullbacks By Markus Land and Georg Tamme Abstract To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision …
This series of lectures gives an introduction to K-theory, including both topological K-theory and algebraic K-theory, and also discussing some as-pects of Hermitian K-theory and cyclic …
a quick and accessible introduction to K-theory, including how to cal- culate with it, and some of its additional features such as characteristic classes, the Thom isomorphism and Gysin maps.
1.1 What is K-theory? 1.1.1 Roughly speaking, K-theory is the study of functors (bridges) C nC n K Kn: (Nice categories) (category of Abelian groups → → ∈Z (See 2.4 (ii) for a formal definition …
Below is an outline of the paper. Sections 2.1–2.3 present notation, definitions and basic results from the liter-ature. Section 2.4 presents a particular model for computing the homotopy of …
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