In algebra and algebraic geometry, it is referred to as algebraic ktheory. His results include a proof that all infinite loop space machines are in some sense equivalent, and progress on the quillenlichtenbaum conjecture thomason did his undergraduate studies at michigan state university, graduating with. Oesterle linear simple lie algebras and ranks of operators. If x is a quasiseparated and quasi compact scheme, we consider the algebraic, thomasontrobaugh and bass k theory. The author shows that this definition of ktheory does coincide with that of k0 and k1 done. The connection between the k theory localization theorem of thomason, trobaugh and yao and the smashing. For schemes, there are two constructions which do not agree in full generality. Regulators and cycle maps in higherdimensional di erential. Regulators and cycle maps in higherdimensional di erential algebraic ktheory ulrich bunke and georg tammey september 25, 2015 abstract we develop di erential algebraic ktheory of regular arithmetic schemes. When i open a book on algebraic ktheory i am not an expert i see various complicated very ingenious constructions which become equivalent for mysterious to me. In this paper we establish a universal characterization of higher algebraic k theory in the setting of small stable. Specifically, we prove that connective algebraic k theory is the universal additive invariant, ie the universal functor with values in spectra which inverts morita equivalences, preserves filtered colimits and satisfies waldhausens additivity theorem. Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Ktheory of schemes based on the work of quillen, waldhausen, thomason and.
The reader will see a common theme throughout algebraic k theory, namely that of abelianization, which is very prevalent throughout modern mathematics. Specifically, we prove that connective algebraic ktheory is the universal additive invariant, i. As applications of this technology, we study the algebraic ktheory of associative rings in a wide range of homotopical contexts and of spectral delignemumford stacks. Pdf higher algebraic ktheory after quillen, thomason and. In algebraic topology, it is a cohomology theory known as topological k theory. This paper is concerned with higher algebraic ktheory and actions of algebraic groups g on such nice categories as the category of algebraic vector bundles on a scheme x. Quillens higher algebraic ktheory i is probably the best source for understanding the basics and the original intuition. An algebraic introduction to ktheory this book is an introduction to ktheory and a text in algebra. The same is done in a considerably, more general setting by a.
Quillens motivation of higher algebraic ktheory mathoverflow. In contrast, finding the correct definition of the higher kgroups of rings was a difficult. So gauss, dirichlet, kummer, and dedekind were all actually studying algebraic ktheory of number. This book gives a superb overview of algebraic k theory, and could be read by anyone who has taken a course in commutative algebra or a course in the theory of rings. Algebraic ktheory of brave new rings provides a very interesting. Algebraic ktheory plays an important role in many subjects, especially number theory, algebraic topology and algebraic geometry. For the topological equivariant ktheory, see topological ktheory in mathematics, the equivariant algebraic ktheory is an algebraic ktheory associated to the category. The purpose of this paper is to generalize the recent results of thomason and trobaugh on algebraic ktheory of schemes to certain noncommutative situa tions, that is, to establish a localization theorem and related results for algebraic ktheory of noncommutative rings and other noncommutative situations. We also explain in detail the comparison between the infinity categorical version of waldhausen ktheory and the classical definition. Trobaugh on algebraic ktheory of schemes to certain noncommutative situa. Idempotent completion of triangulated categories sciencedirect. Thomason trobaugh is also an excellent paper, but it is a fairly large paper and very fundamental so the first half of the paper is dedicated to construction of the basic objects. Additionally, thomason discovered that there is no analog of milnor ktheory for a general variety. Let a be an associative ring and let a b be the localisation with respect to a set.
Yevsey nisnevich, the completely decomposed topology on schemes and associated descent spectral sequences in algebraic ktheory, algebraic ktheory. As applications, we compute the ktheory and gtheory of. It is also a fundamental tool in the field of operator algebras. One checks that bequipped with this family of conations is an exact category making binto a fully exact. Almost the same question was already asked on mo motivation for algebraic k theory. As applications of this technology, we study the algebraic ktheory of associative rings in a wide range of homotopical contexts and of. An elementary introduction by max karoubi clay mathematics academy the purpose of these notes is to give a feeling of ktheory, a new interdisciplinary subject within mathematics.
It is also a fundamental tool in the field of operator. What we have in mind here is that the square diagram ka k ka p. Higher k theories, proceedings of the conference, battelle memorial institutes, seattle, wa, 1972, lecture notes in mathematics, 341, springer. Robert wayne thomason 5 november 1952 tulsa, oklahoma, u. Cartier p, illusie l, katz n m, laumon g, manin y, ribet k a, eds. Higher algebraic ktheory after quillen, thomason and. Using this, we give new, higher categorical proofs of the approximation, additivity, and fibration theorems of waldhausen in this context. The book takes the reader from the basics of the subject to the state of the art, including the calculation of the higher \ k\ theory of number fields and the. In mathematics, k theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. We present an introduction with a few proofs to higher algebraic ktheory of schemes based on the work of quillen, waldhausen, thomason and others. Higher algebraic ktheory of schemes and of derived categories r.
The fundamental structural theorems for higher ktheory are presented in chapter v. Higher algebraic ktheory of admissible abelian categories and localization theorems dongyuan yao. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic ktheory to topological hochschild homology thh and topological cyclic homology tc. The purpose of this paper is to generalize the recent results of thomason and trobaugh on algebraic ktheory of schemes to certain noncommutative situa tions, that is, to establish a localization theorem and related results for algebraic. A universal characterization of higher algebraic ktheory core. Higher algebraic ktheory of schemes and of derived categories. In this paper we establish a universal characterization of higher algebraic ktheory in the setting of small stable infinity categories. Download book pdf the grothendieck festschrift pp 247435 cite as. His results include a proof that all infinite loop space machines are in some sense equivalent, and progress on the quillenlichtenbaum conjecture. Pdf higher algebraic ktheory after quillen, thomason.
D epartm en t of m athem atics, n orthw estern u n iversity, e van ston, u s a l ectu res given at the s chool on a lgebraic k theory an d its a pplication s t rieste, 14 25 m ay 2007 l n s 0823001. Algebraic ktheory and its applications graduate texts in. Higher algebraic ktheory after quillen, thomason and others. Lectures on k theory pdf 95p this lecture note covers the following topics. Some formulas in operator theory, involving determinants, are best understood in terms of algebraic ktheory.
Almost the same question was already asked on mo motivation for algebraic ktheory. On the vanishing of negative homotopy ktheory sciencedirect. Our emphasis is on the application of triangulated category methods in algebraic k theory. We introduce the gorenstein algebraic ktheory space and the gorenstein algebraic kgroup of a ring. Higher algebraic ktheory of admissible abelian categories. In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. Our emphasis is on the application of triangulated category methods in algebraic ktheory. Ktheory itself, rather than on these applications of algebraic ktheory. Lecture notes on algebraic k theory pdf 252p download book. Algebraic ktheory of brave new rings provides a very interesting link between algebraic ktheory of rings and geometry. The quillen ktheory of a scheme x x is defined as the algebraic ktheory of the exact category vect x vectx of vector bundles on x x using the quillen qconstruction thomason trobaugh ktheory.
Waldhausen to a certain type of rings up to homotopy, called brave new rings. Given a commutative ring r or more generally an algebraic variety, he introduced the nowadays called grothendieck group k 0r of r. In algebra and algebraic geometry, it is referred to as algebraic k theory. In this paper we establish a universal characterization of higher algebraic ktheory in the setting of stable. In this article we use the theories of relative algebraic geometry and of homotopical algebraic geometry cf. We present an introduction with a few proofs to higher algebraic ktheory of schemes based on the. Higher algebraic ktheory of schemes and of derived. Call a sequence 1 in ba conation if it is a conation in a. Thomason and thomas trobaugh to alexander grothendieck. We present an introduction with a few proofs to higher algebraic k theory of schemes based on the work of quillen, waldhausen, thomason and others.
On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic ktheory. When i open a book on algebraic k theory i am not an expert i see various complicated very ingenious constructions which become equivalent for mysterious to me. They contain detailed information about the original object but are notoriously difficult to compute. This theory was invented by alexander grothendieck1 bs in the 50s in order to solve some difficult problems in algebraic geometry the letter. In this paper we establish a universal characterization of higher algebraic k theory in the setting of small stable infinity categories.
Algebraic ktheory is a subject area in mathematics with connections to geometry, topology. Blumberg, david gepner, goncalo tabuada download pdf. In this paper we prove a localization theorem for the ktheory of commutative rings. In algebraic topology, it is a cohomology theory known as topological ktheory. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, called its derived category, or the derived category. This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic k theory. A more elementary proof of the theorem of merkujevsuslin is given in this edition. The question, i think, is whether basic theorems of algebraic ktheory is an appropriate new topic to draft, or whether it. A central part of the book is a detailed exposition of the ideas of quillen as contained in his classic papers higher algebraic ktheory, i, ii. Our approach is based on a new construction of a functorial, spectrum level beilinson regulator using di erential forms. The purpose of this paper is to generalize the recent results of thomason and. Higher algebraic ktheory algebraic ktheory goes back to grothendiecks work 20 on the riemannroch theorem. Thomason and thomas trobaugh to alexander grothendieck on his 60th birthday in this paper we prove a localization theorem for the atheory of com mutative rings and of schemes, theorem 7. Derived noncommutative algebraic geometry wikipedia.
Algebraic ktheory and its applications graduate texts in mathematics v. Thomason, thomas trobaugh, higher algebraic ktheory of schemes and of derived categories, the grothendieck festschrift, 1990, 247435. Algebraic k theory of gorenstein projective modules. In this paper we prove a localization theorem for the ktheory of commutative rings and of schemes, theorem 7.
On the other hand, k 1p o fq o f is a nontrivial theorem due to bass, milnor, and serre see. An algebraic introduction to k theory this book is an introduction to k theory and a text in algebra. Higher algebraic ktheory after quillen, thomason and others marco schlichting mathematics institute, zeeman building, university of warwick, coventry cv4 7al uk, m. The conjecture of serre on projective modules on polynomial rings, had initially nothing to do with k theory.
In mathematics, ktheory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. An overview lecture notes in mathematics on free shipping on qualified orders higher algebraic k theory. The algebraic ktheory of connective salgebras can be closely approximated by diagrams built from the algebraic ktheory of rings du, 5. A universal characterization of higher algebraic ktheory. Weibel conjectured the analogue of this theorem with the ktheory of bassthomason trobaugh in place of homotopy ktheory, originally formulated as a question in. Higher ktheories, proceedings of the conference, battelle memorial institutes, seattle, wa, 1972, lecture notes in mathematics, 341, springer. In fact theorem 1 is a special case of weibels original conjecture, as can be seen using the spectral sequence 2. Higher algebraic ktheory after quillen, thomason and others 3 b let abe an exact category, and let babe a full additive subcategory closed under extensions in a. Hagii to construct some categories of schemes defined under spec we define the categories of.