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The work constructs natural lambda-operations satisfying special lambda-ring identities by reworking Quillen and Kratzer’s constructions for K-theory of diagrams of commutative rings. The second part applies these operations to relate motivic cohomology (Bloch’s higher Chow groups) with weight-graded pieces of algebraic K-theory via Adams eigenspace splittings and resulting isomorphisms after specific factorial localizations.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/lambda-operations-k-theory-and-motivic-cohomology-introduction/665536/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/lambda-operations-k-theory-and-motivic-cohomology-introduction/665536.png","ImageObject",300,407,{"name":92,"@type":93},"Marglet","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the paper’s main goal in the first part?","Question",{"text":112,"@type":113},"To construct natural lambda-operations for relative K-theory with supports and prove they satisfy special lambda-ring identities.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the construction relate to Quillen and Kratzer’s earlier work?",{"text":117,"@type":113},"It redoes Quillen and Kratzer’s constructions for K-theory of a commutative ring in the setting of K-theory of I-diagrams of commutative rings.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the second part apply to, and what is the key relationship?",{"text":121,"@type":113},"For a smooth quasi-projective variety over a field, it relates motivic cohomology defined as Bloch’s higher Chow groups to weight-graded pieces of K-theory after tensoring with Q and using Adams-operation eigenspace decompositions.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665536,1793304817,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},2336477982702,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Lambda-operations, K-theory and motivic cohomology  \nMarc Levine  \nIntroduction  \nThis paper is in two parts: in part one, our main object is to give a construction of natural 􀀕-operations for relative K-theory with supports, satisfying the special 􀀕-ring identities. In the second part, we give an application to the relation of motivic cohomology and algebraic K-theory of a smooth quasi-projective variety over a 􀀌eld.  \nThe main idea in the construction of the 􀀕-operations is to re-do the constructions of Quillen (Hiller gives an exposition of Quillen's construction in [H]) and Kratzer [Kr] for the K-theory of a commutative ring, in the setting of the K-theory of an I-diagram of commutative rings, i.e., a functor  \nA: I ! Rings  \nwith I a small category. This approach can contrasted with the construction of Soul􀀓e [So], where use is made of the closed model structure on the category of sheaves of simplicial sets. Our point of view is to look at the \\discrete\" setting, viewing for example a scheme as a 􀀌nite diagram of a􀀎ne schemes, and replace the use of topology and sheaf theory with the Mayer-Vietoris property of K-theory provided by the Thomason-Trobaugh theorem [T-T], together with the remark that the operations we construct are natural with respect to re􀀌nements.  \nWe rely on the general machinery of closed simplicial model categories, especially that of I-diagrams of simplicial sets: the category of functors  \nX: I ! S ;  \nwhere I is a 􀀌xed small category, and S is the category of simplicial sets. Much of the general theory, as well as the fundamental properties, of Quillen's +-construction carry over without change to the setting of I-diagram of commutative rings. However, at some point we need to work with an analog of 􀀌nite CW complexes, which restricts our theory to 􀀌nite categories, i.e., a category I whose nerve has only 􀀌nitely many non-degenerate simplices. Fortunately, the diagrams of a􀀎ne schemes needed to express the relative K-theory of anoetherian scheme, with supports in a closed subset, are all 􀀌nite.  \nBesides the works mentioned above, there is a recent construction of Lecomte [Le], building on previous work of Schechtman [Sc], giving Adams operations for the K-groups of schemes, which in turn gives a special 􀀕-ring structure for the rational K-groups. It seems that her construction would also give natural Adams operations for relative K-theory with supports, and for other groups expressable as 􀀌nite diagrams of schemes, or perhaps even for arbitrary small diagrams. There is also the unpublished work of Gillet and Soul􀀓e [G-S], which uses the sheaf-theoretic point of view. Together with the paper [J] of Jardine, it seems that the arguments of [G-S] would give a special 􀀕-ring structure for the K-groups of an arbitrary small diagram of schemes satisfying a certain cohomological 􀀌niteness (see  \npartially supported by the NSF, grant no. DMS-9401164, and the DFG Forschergruppe \\Arithmetik und Geometrie\"  \ne.g. [So2]) . Grayson ([Gr], [Gr2]) has constructed 􀀕-operations and Adams operations on the simplicial level, but it is not clear that these operations satisfy special 􀀕-ring identities. Our justi􀀌cation for producing this work is that there is currently nothing in the literature which constructs 􀀕-operations, satisfying the special 􀀕-ring identities, in the generality required for relative K-theory with supports.  \nThe argument given here consists of reworking the basic ingredients of the construction of Quillen in the setting of a diagram of rings. These ingredients are:  \ni) the special 􀀕-ring structure on the representation ring of a group  \nii) the universal property of the plus construction, and the structure of an H-space on BGL+  \niii) the construction of a \\universal\" natural transformation from the representation ring of 􀀙1 (X ) to the set of homotopy classes of maps [X; BGL+ ],  \nThe representation theory is quite straightforward and is discussed in x1 . The generalization of (ii) is done in x2","cbCaiaHTBS7vT5bC","https://ap.wps.com/l/cbCaiaHTBS7vT5bC","pdf",365103,67,"English","# Introduction\n## Construction of lambda-operations via diagrammatic K-theory\n## General machinery of closed simplicial model categories\n## Finiteness conditions and applications to motivic cohomology\n## Main theorem: motivic cohomology vs weight-graded K-theory","[{\"question\":\"What is the paper’s main goal in the first part?\",\"answer\":\"To construct natural lambda-operations for relative K-theory with supports and prove they satisfy special lambda-ring identities.\"},{\"question\":\"How does the construction relate to Quillen and Kratzer’s earlier work?\",\"answer\":\"It redoes Quillen and Kratzer’s constructions for K-theory of a commutative ring in the setting of K-theory of I-diagrams of commutative rings.\"},{\"question\":\"What does the second part apply to, and what is the key relationship?\",\"answer\":\"For a smooth quasi-projective variety over a field, it relates motivic cohomology defined as Bloch’s higher Chow groups to weight-graded pieces of K-theory after tensoring with Q and using Adams-operation eigenspace decompositions.\"}]","Lambda-operations, K-theory and motivic cohomology - Introduction | PDF",169]