Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. On the other hand, k 1p o fq o f is a nontrivial theorem due to bass, milnor, and serre see. A universal characterization of higher algebraic ktheory. Let a be an associative ring and let a b be the localisation with respect to a set. Quillens higher algebraic ktheory i is probably the best source for understanding the basics and the original intuition. In this paper we prove a localization theorem for the k theory of commutative rings and of schemes, theorem 7. Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. 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. 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. We present an introduction with a few proofs to higher algebraic ktheory of. This fact implies that is acting freely on x kng, as we can see as follows. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Kowa is the free group or the free abelian groups on generators a.
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. This book is a general introduction to higher algebraic k groups of rings and algebraic varieties, which were first defined by quillen at the beginning of the 70s. References kst moritz kerz, florian strunk, georg tamme, algebraic ktheory and descent for blowups, arxiv. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic k theory. The connection between the ktheory localization theorem of thomason, trobaugh and yao and the smashing subcategories of bousfield and ravenel. Robert wayne thomason 5 november 1952 tulsa, oklahoma, u. 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. 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. These kgroups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory.
Higher algebraic ktheory of schemes and of derived categories r. Let a and b be two complicial biwaldhausen categories, each of which is closed under the formation of canonical pushouts and pullbacks, and let f. Thomason, thomas trobaugh, higher algebraic ktheory of schemes and of derived categories, the grothendieck festschrift, 1990, 247435. This implies a similar homotopy invariance property of the algebraic ktheory with coef. Algebraic transversality can be traced back to the work of higman, basshellerswan, stallings, casson and waldhausen on the algebraic ktheory of polynomial extensions and more general amalgamated free products. Our main result is the corepresentability of nonconnective ktheory by the base ring in the universal localizing motivator. Our emphasis is on the application of triangulated category methods in algebraic ktheory. Algebraic ktheory, algebraic cycles and arithmetic geometry. K theory of schemes based on the work of quillen, waldhausen, thomason and. Pdf higher algebraic ktheory after quillen, thomason. Call a sequence 1 in ba conation if it is a conation in a. Specz1m onto its big zariski site, and ktt is thomasontrobaugh ktheory, cf.
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. Thomason proved around 1980 that after inverting the bott element, algebraic ktheory with finite coefficients became isomorphic to etale ktheory. The fundamental structural theorems for higher ktheory are presented in chapter v. Higher algebraic ktheory after quillen, thomason and others. Our emphasis is on the application of triangulated category methods in algebraic k theory. In this paper we establish a universal characterization of higher algebraic k theory in the setting of small stable infinity categories. 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. 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. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic ktheory. We present an introduction with a few proofs to higher algebraic ktheory of schemes based on the work of quillen, waldhausen, thomason and others.
The proceedings of the london mathematical society is wholly owned and managed by the london mathematical society, a not. Trobaugh, higher algebraic ktheory of schemes and of derived categories, in the grothendieck festschrift, vol. Vectx of vector bundles that is, locally free sheaves of. An algebraic introduction to k theory this book is an introduction to k theory and a text in algebra. Pdf higher algebraic ktheory after quillen, thomason and. Ams transactions of the american mathematical society. See of ranicki 33, chapter 7 for a treatment of algebraic transversality in the case d1 when a. Kqwa is the free group or the free abelian groups on generators a. One checks that bequipped with this family of conations is an exact category making binto a fully exact. This implies a similar homotopy invariance property of the algebraic ktheory with coe.
Higher algebraic ktheory of admissible abelian categories and localization theorems dongyuan yao. Perfection in motivic homotopy theory elmanto 2020. As an application, we obtain for free higher chern characters, respectively. Dimensions of triangulated categories journal of ktheory. In this paper we establish a universal characterization of higher algebraic k theory in the setting of small stable.
They also have a better functoriality than algebraic cycles, and this has been. We show that the homotopy ktheory of such stacks is homotopy invariant. This book is a general introduction to higher algebraic kgroups of rings and algebraic varieties, which were first defined by quillen at the beginning of the 70s. So gauss, dirichlet, kummer, and dedekind were all actually studying algebraic ktheory of number. These k groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. Lectures on k theory pdf 95p this lecture note covers the following topics. However, it turns out that even working in a purely algebraic context, one requires techniques from homotopy theory to construct the higher kgroups and to perform computations.
What we have in mind here is that the square diagram ka k ka p. Waldhausen to a certain type of rings up to homotopy, called brave new rings. Ktheory itself, rather than on these applications of algebraic ktheory. Higher algebraic ktheory after quillen, thomason and. Our theorem utilizes this sheaftheoretic description to reduce the. In this paper we prove a localization theorem for the ktheory of. Introduction the higher algebraic ktheories of quillen and thomason trobaugh are among the. Higher algebraic ktheory of schemes and of derived. Quillen q1 discovered the higher ktheory of exact categories in a form.
This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic k theory. Algebraic ktheory of brave new rings provides a very interesting. Ktheory of schemes based on the work of quillen, waldhausen, thomason and. Higher algebraic ktheory after quillen, thomason and others marco schlichting mathematics institute, zeeman building, university of warwick, coventry cv4 7al uk, m. W ith ou t a d ou b t, a p rim ary goal if n ot th e. Tabuada, higher ktheory via universal invariants, duke math. A more elementary proof of the theorem of merkujevsuslin is given in this edition. In this paper we establish a universal characterization of higher algebraic ktheory in the setting of stable. The equivariant k theory of toric varieties the equivariant k theory of toric varieties au, suanne. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers. A result of thomason tho93b asserts that after taking algebraic ktheory, the induced square of spectra kx kz kbl zx kpn zx i. However, it turns out that even working in a purely algebraic context, one requires techniques from homotopy theory to construct the higher k groups and to perform computations. Here kx denotes the bassthomasontrobaugh algebraic ktheory spectrum of perfect complexes on a scheme x. Dominique arlettaz, algebraic ktheory of rings from a topological viewpoint pdf.
Higher algebraic ktheory 291 we quote thomason and trobaugh s derived category theorem below for easy reference. 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. The higher algebraic kgroups of a ring r are defined to be the homotopy groups. Higher algebraic ktheory of admissible abelian categories. Adams, stable homotopy and generalized homology, univ. The equivariant k theory of toric varieties deepdyve. As applications, we compute the ktheory and gtheory of. Algebraic ktheory of brave new rings provides a very interesting link between algebraic ktheory of rings and geometry. Here kx denotes the bassthomason trobaugh algebraic ktheory spectrum of perfect complexes on a scheme x. Definitions of higher ktheory the higher algebraic k. Yevsey nisnevich, the completely decomposed topology on schemes and associated descent spectral sequences in algebraic ktheory, algebraic ktheory. Specifically, we prove that connective algebraic k theory is the universal additive invariant, i. The first is a formula for the equivariant k groups of an arbitrary affine toric variety, generalizing the known formula for smooth ones. Trobaugh, higher algebraic ktheory of schemes and of derived categorieshigher algebraic ktheory of schemes and of derived categories.
The higher algebraic ktheories of quillen and thomasontrobaugh. Algebraic ktheory, which is the main character of this book, deals mainly with studying the structure of rings. In this paper we establish a universal characterization of higher algebraic ktheory in the setting of small stable infinity categories. Higher algebraic ktheory of schemes and of derived categories. Algebraic k theory, which is the main character of this book, deals mainly with studying the structure of rings. On the negative ktheory of schemes in finite characteristic. 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. His results include a proof that all infinite loop space machines are in some sense equivalent, and progress on the quillenlichtenbaum conjecture. 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. Papers based on the sedano winter school on ktheory swisk, sedano, spain, january 2227, 2007 pp. 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.
Trobaugh and yao and the smashing subcategories of bousfield and ravenel. For the topological equivariant ktheory, see topological ktheory in mathematics, the equivariant algebraic ktheory is an algebraic ktheory associated to the category. Zx, which under the assumptions is locally free of rank n. Lecture notes on algebraic k theory pdf 252p download book.
They contain detailed information about the original object but are notoriously difficult to compute. I suggest looking at the introduction to waldhausens original paper on algebraic ktheory algebraic ktheory of generalized free products, part i, ann. Dimensions of triangulated categories journal of k. Waldhausen started out as a 3manifold theorist, and he realized that certain phenomena in the topology of 3manifolds would be explained if the whitehead groups of. Specifically, we prove that connective algebraic ktheory is the universal additive invariant, i.
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