{"title":"The generalized Vaserstein symbol revisited","authors":"Tariq Syed","doi":"arxiv-2408.10164","DOIUrl":"https://doi.org/arxiv-2408.10164","url":null,"abstract":"We give a construction of a generalized Vaserstein symbol associated to any\u0000finitely generated projective module of rank $2$ over a commutative ring with\u0000unit.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dévissage Hermitian Theory","authors":"Satya Mandal","doi":"arxiv-2408.09633","DOIUrl":"https://doi.org/arxiv-2408.09633","url":null,"abstract":"We prove D'{e}vissage theorems for Hermitian $K$ Theory (or $GW$ theory),\u0000analogous to Quillen's D'{e}vissage theorem for $K$-theory. For abelian\u0000categories ${mathscr A}:=({mathscr A}, ^{vee}, varpi)$ with duality, and\u0000appropriate abelian subcategories ${mathscr B}subseteq {mathscr A}$, we\u0000prove D'{e}vissage theorems for ${bf GW}$ spaces, $G{mathcal W}$-spectra and\u0000${mathbb G}W$ bispectra. As a consequence, for regular local rings $(R, m,\u0000kappa)$ with $1/2in R$, we compute the ${BG}W$ groups ${mathbb\u0000G}W^{[n]}_k(spec{R})~forall k, nin {mathbb Z}$, where $n$ represent the\u0000translation.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Chow rings of quasi-split geometrically almost simple algebraic groups","authors":"Alexey Ananyevskiy, Nikita Geldhauser","doi":"arxiv-2408.09390","DOIUrl":"https://doi.org/arxiv-2408.09390","url":null,"abstract":"We compute the Chow ring of a quasi-split geometrically almost simple\u0000algebraic group assuming the coefficients to be a field. This extends the\u0000classical computation for split groups done by Kac to the non-split quasi-split\u0000case. For the proof we introduce and study equivariant conormed Chow rings,\u0000which are well adapted to the study of quasi-split groups and their homogeneous\u0000varieties.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Noe Barcenas, Luis Eduardo Garcia-Hernandez, Raphael Reinauer
{"title":"The Gromov-Lawson-Rosenberg Conjecture for Z/4xZ/4","authors":"Noe Barcenas, Luis Eduardo Garcia-Hernandez, Raphael Reinauer","doi":"arxiv-2408.07895","DOIUrl":"https://doi.org/arxiv-2408.07895","url":null,"abstract":"We prove the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4 by\u0000computing the connective real k-homology of the classifying space with the\u0000Adams spectral sequence and two types of detection theorems for the kernel of\u0000the alpha invariant: one based on eta-invariants, closely following work of\u0000Botvinnik-Gilkey-Stolz, and a second one based on homological methods. Along\u0000the way, we determine differentials of the Adams spectral sequence for\u0000classifying spaces involved in the computation, and we study the cap structure\u0000of the Adams spectral sequence for sub-hopf algebras of the Steenrod algebra\u0000relevant to the computation of connective real and complex k-homology.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alexander Kupers, Ezekiel Lemann, Cary Malkiewich, Jeremy Miller, Robin J. Sroka
{"title":"Scissors automorphism groups and their homology","authors":"Alexander Kupers, Ezekiel Lemann, Cary Malkiewich, Jeremy Miller, Robin J. Sroka","doi":"arxiv-2408.08081","DOIUrl":"https://doi.org/arxiv-2408.08081","url":null,"abstract":"In any category with a reasonable notion of cover, each object has a group of\u0000scissors automorphisms. We prove that under mild conditions, the homology of\u0000this group is independent of the object, and can be expressed in terms of the\u0000scissors congruence K-theory spectrum defined by Zakharevich. We therefore\u0000obtain both a group-theoretic interpretation of Zakharevich's higher scissors\u0000congruence K-theory, as well as a method to compute the homology of scissors\u0000automorphism groups. We apply this to various families of groups, such as\u0000interval exchange groups and Brin--Thompson groups, recovering results of\u0000Szymik--Wahl, Li, and Tanner, and obtaining new results as well.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Computational tools for Real topological Hochschild homology","authors":"Chloe Lewis","doi":"arxiv-2408.07188","DOIUrl":"https://doi.org/arxiv-2408.07188","url":null,"abstract":"In this paper, we construct a Real equivariant version of the B\"okstedt\u0000spectral sequence which takes inputs in the theory of Real Hochschild homology\u0000developed by Angelini-Knoll, Gerhardt, and Hill and converges to the\u0000equivariant homology of Real topological Hochschild homology, $text{THR}$. We\u0000also show that when $A$ is a commutative $C_2$-ring spectrum, $text{THR}(A)$\u0000has the structure of an $A$-Hopf algebroid in the $C_2$-equivariant stable\u0000homotopy category.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Flagged Perturbations and Anchored Resolutions","authors":"Keller VandeBogert","doi":"arxiv-2408.02749","DOIUrl":"https://doi.org/arxiv-2408.02749","url":null,"abstract":"In this paper, we take advantage of a reinterpretation of differential\u0000modules admitting a flag structure as a special class of perturbations of\u0000complexes. We are thus able to leverage the machinery of homological\u0000perturbation theory to prove strong statements on the homological theory of\u0000differential modules admitting additional auxiliary gradings and having\u0000infinite homological dimension. One of the main takeaways of our results is\u0000that the category of differential modules is much more similar than expected to\u0000the category of chain complexes, and from the K-theoretic perspective such\u0000objects are largely indistinguishable. This intuition is made precise through\u0000the construction of so-called anchored resolutions, which are a distinguished\u0000class of projective flag resolutions that possess remarkably well-behaved\u0000uniqueness properties in the (flag-preserving) homotopy category. We apply this\u0000theory to prove an analogue of the Total Rank Conjecture for differential\u0000modules admitting a ZZ/2-grading in a large number of cases.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Intersection of complete cotorsion pairs","authors":"Qikai Wang, Haiyan Zhu","doi":"arxiv-2408.01922","DOIUrl":"https://doi.org/arxiv-2408.01922","url":null,"abstract":"Given two (hereditary) complete cotorsion pairs\u0000$(mathcal{X}_1,mathcal{Y}_1)$ and $(mathcal{X}_2,mathcal{Y}_2)$ in an exact\u0000category with $mathcal{X}_1subseteq mathcal{Y}_2$, we prove that $left({rm\u0000Smd}langle mathcal{X}_1,mathcal{X}_2 rangle,mathcal{Y}_1cap\u0000mathcal{Y}_2right)$ is also a (hereditary) complete cotorsion pair, where\u0000${rm Smd}langle mathcal{X}_1,mathcal{X}_2 rangle$ is the class of direct\u0000summands of extension of $mathcal{X}_1$ and $mathcal{X}_2$. As an\u0000application, we construct complete cotorsion pairs, such as\u0000$(^perpmathcal{GI}^{leqslant n},mathcal{GI}^{leqslant n})$, where\u0000$mathcal{GI}^{leqslant n}$ is the class of modules of Gorenstein injective\u0000dimension at most $n$. And we also characterize the left orthogonal class of\u0000exact complexes of injective modules and the classes of modules with finite\u0000Gorenstein projective, Gorenstein flat, and PGF dimensions.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proof of a $K$-theoretic polynomial conjecture of Monical, Pechenik, and Searles","authors":"Laura Pierson","doi":"arxiv-2408.01390","DOIUrl":"https://doi.org/arxiv-2408.01390","url":null,"abstract":"As part of a program to develop $K$-theoretic analogues of combinatorially\u0000important polynomials, Monical, Pechenik, and Searles (2021) proved two\u0000expansion formulas $overline{mathfrak{A}}_a = sum_b\u0000Q_b^a(beta)overline{mathfrak{P}}_b$ and $overline{mathfrak{Q}}_a = sum_b\u0000M_b^a(beta)overline{mathfrak{F}}_b,$ where each of\u0000$overline{mathfrak{A}}_a$, $overline{mathfrak{P}}_a$,\u0000$overline{mathfrak{Q}}_a$ and $overline{mathfrak{F}}_a$ is a family of\u0000polynomials that forms a basis for $mathbb{Z}[x_1,dots,x_n][beta]$ indexed\u0000by weak compositions $a,$ and $Q_b^a(beta)$ and $M_b^a(beta)$ are monomials\u0000in $beta$ for each pair $(a,b)$ of weak compositions. The polynomials\u0000$overline{mathfrak{A}}_a$ are the Lascoux atoms, $overline{mathfrak{P}}_a$\u0000are the kaons, $overline{mathfrak{Q}}_a$ are the quasiLascoux polynomials,\u0000and $overline{mathfrak{F}}_a$ are the glide polynomials; these are\u0000respectively the $K$-analogues of the Demazure atoms $mathfrak{A}_a$, the\u0000fundamental particles $mathfrak{P}_a$, the quasikey polynomials\u0000$mathfrak{Q}_a$, and the fundamental slide polynomials $mathfrak{F}_a$.\u0000Monical, Pechenik, and Searles conjectured that for any fixed $a,$ $sum_b\u0000Q_b^a(-1), sum_b M_b^a(-1) in {0,1},$ where $b$ ranges over all weak\u0000compositions. We prove this conjecture using a sign-reversing involution.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"106 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A motivic Greenlees spectral sequence towards motivic Hochschild homology","authors":"Federico Ernesto Mocchetti","doi":"arxiv-2408.00338","DOIUrl":"https://doi.org/arxiv-2408.00338","url":null,"abstract":"We define a motivic Greenlees spectral sequence by characterising an\u0000associated $t$-structure. We then examine a motivic version of topological\u0000Hochschild homology for the motivic cohomology spectrum modulo a prime number\u0000$p$. Finally, we use the motivic Greenlees spectral sequence to determine the\u0000homotopy ring of a related spectrum, given that the base field is algebraically\u0000closed with a characteristic that is coprime to $p$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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