In this article, we revisit Weibel's conjecture for twisted $K$-theory. We also examine the vanishing of twisted negative $K$-groups for Pr"{u}fer domains. Furthermore, we observe that the homotopy invariance of twisted $K$-theory holds for (finite-dimensional) Pr"{u}fer domains.
{"title":"On the vanishing of Twisted negative K-theory and homotopy invariance","authors":"Vivek Sadhu","doi":"arxiv-2409.06228","DOIUrl":"https://doi.org/arxiv-2409.06228","url":null,"abstract":"In this article, we revisit Weibel's conjecture for twisted $K$-theory. We\u0000also examine the vanishing of twisted negative $K$-groups for Pr\"{u}fer\u0000domains. Furthermore, we observe that the homotopy invariance of twisted\u0000$K$-theory holds for (finite-dimensional) Pr\"{u}fer domains.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206308","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}
Anna Marie Bohmann, Teena Gerhardt, Cameron Krulewski, Sarah Petersen, Lucy Yang
The topological Hochschild homology of a ring (or ring spectrum) $R$ is an $S^1$-spectrum, and the fixed points of THH($R$) for subgroups $C_nsubset S^1$ have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work of Angeltveit, Blumberg, Gerhardt, Hill, Lawson and Mandell defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra.
{"title":"Equivariant Witt Complexes and Twisted Topological Hochschild Homology","authors":"Anna Marie Bohmann, Teena Gerhardt, Cameron Krulewski, Sarah Petersen, Lucy Yang","doi":"arxiv-2409.05965","DOIUrl":"https://doi.org/arxiv-2409.05965","url":null,"abstract":"The topological Hochschild homology of a ring (or ring spectrum) $R$ is an\u0000$S^1$-spectrum, and the fixed points of THH($R$) for subgroups $C_nsubset S^1$\u0000have been widely studied due to their use in algebraic K-theory computations.\u0000Hesselholt and Madsen proved that the fixed points of topological Hochschild\u0000homology are closely related to Witt vectors. Further, they defined the notion\u0000of a Witt complex, and showed that it captures the algebraic structure of the\u0000homotopy groups of the fixed points of THH. Recent work of Angeltveit,\u0000Blumberg, Gerhardt, Hill, Lawson and Mandell defines a theory of twisted\u0000topological Hochschild homology for equivariant rings (or ring spectra) that\u0000builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this\u0000paper, we study the algebraic structure of the equivariant homotopy groups of\u0000twisted THH. In particular, we define an equivariant Witt complex and prove\u0000that the equivariant homotopy of twisted THH has this structure. Our definition\u0000of equivariant Witt complexes contributes to a growing body of research in the\u0000subject of equivariant algebra.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206306","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}
In this article we describe the equivariant and ordinary topological $K$-ring of a toric bundle with fiber a $T$-{it cellular} toric variety. This generalizes the results in cite{su} on $K$-theory of smooth projective toric bundles. We apply our results to describe the equivariant topological $K$-ring of a toroidal horospherical embedding.
{"title":"Equivariant $K$-theory of cellular toric bundles and related spaces","authors":"V. Uma","doi":"arxiv-2409.05719","DOIUrl":"https://doi.org/arxiv-2409.05719","url":null,"abstract":"In this article we describe the equivariant and ordinary topological $K$-ring\u0000of a toric bundle with fiber a $T$-{it cellular} toric variety. This\u0000generalizes the results in cite{su} on $K$-theory of smooth projective toric\u0000bundles. We apply our results to describe the equivariant topological $K$-ring\u0000of a toroidal horospherical embedding.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206307","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}
In this brief note, we present an elementary construction of the first Chern class of Hodge--Tate crystals in line bundles using a refinement of the prismatic logarithm, which should be comparable to the one considered by Bhargav Bhatt. The key to constructing this refinement is Yuri Sulyma's norm on (animated) prisms. We explain the relation of this construction to prismatic Witt vectors, as a generalization of Kaledin's polynomial Witt vectors. We also propose the prismatic Hochschild homology as a noncommutative analogue of prismatic de Rham complex.
{"title":"Prismatic logarithm and prismatic Hochschild homology via norm","authors":"Zhouhang Mao","doi":"arxiv-2409.04400","DOIUrl":"https://doi.org/arxiv-2409.04400","url":null,"abstract":"In this brief note, we present an elementary construction of the first Chern\u0000class of Hodge--Tate crystals in line bundles using a refinement of the\u0000prismatic logarithm, which should be comparable to the one considered by\u0000Bhargav Bhatt. The key to constructing this refinement is Yuri Sulyma's norm on\u0000(animated) prisms. We explain the relation of this construction to prismatic\u0000Witt vectors, as a generalization of Kaledin's polynomial Witt vectors. We also\u0000propose the prismatic Hochschild homology as a noncommutative analogue of\u0000prismatic de Rham complex.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206311","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}
We give a universal property of the construction of the ring of $p$-typical Witt vectors of a commutative ring, endowed with Witt vectors Frobenius and Verschiebung, and generalize this construction to the derived setting. We define an $infty$-category of $p$-typical derived $delta$-Cartier rings and show that the derived ring of $p$-typical Witt vectors of a derived ring is naturally an object in this $infty$-category. Moreover, we show that for any prime $p$, the formation of the derived ring of $p$-typical Witt vectors gives an equivalence between the $infty$-category of all derived rings and the full subcategory of all derived $p$-typical $delta$-Cartier rings consisting of $V$-complete objects.
{"title":"Witt vectors and $δ$-Cartier rings","authors":"Kirill Magidson","doi":"arxiv-2409.03877","DOIUrl":"https://doi.org/arxiv-2409.03877","url":null,"abstract":"We give a universal property of the construction of the ring of $p$-typical\u0000Witt vectors of a commutative ring, endowed with Witt vectors Frobenius and\u0000Verschiebung, and generalize this construction to the derived setting. We\u0000define an $infty$-category of $p$-typical derived $delta$-Cartier rings and\u0000show that the derived ring of $p$-typical Witt vectors of a derived ring is\u0000naturally an object in this $infty$-category. Moreover, we show that for any\u0000prime $p$, the formation of the derived ring of $p$-typical Witt vectors gives\u0000an equivalence between the $infty$-category of all derived rings and the full\u0000subcategory of all derived $p$-typical $delta$-Cartier rings consisting of\u0000$V$-complete objects.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"183 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206309","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}
We introduce a version of Koszul duality for categories, which extends the Koszul duality of operads and right modules. We demonstrate that the derivatives which appear in Weiss calculus (with values in spectra) form a right module over the Koszul dual of the category of vector spaces and orthogonal surjections, resolving conjectures of Arone--Ching and Espic. Using categorical Fourier transforms, we then classify Weiss towers. In particular, we describe the $n$-th polynomial approximation as a pullback of the $(n-1)$-st polynomial approximation along a ``generalized norm map''.
{"title":"Koszul duality and a classification of stable Weiss towers","authors":"Connor Malin, Niall Taggart","doi":"arxiv-2409.01335","DOIUrl":"https://doi.org/arxiv-2409.01335","url":null,"abstract":"We introduce a version of Koszul duality for categories, which extends the\u0000Koszul duality of operads and right modules. We demonstrate that the\u0000derivatives which appear in Weiss calculus (with values in spectra) form a\u0000right module over the Koszul dual of the category of vector spaces and\u0000orthogonal surjections, resolving conjectures of Arone--Ching and Espic. Using\u0000categorical Fourier transforms, we then classify Weiss towers. In particular,\u0000we describe the $n$-th polynomial approximation as a pullback of the $(n-1)$-st\u0000polynomial approximation along a ``generalized norm map''.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206310","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}
In this article, we use the Bruhat and Schubert cells to calculate the cellular homology of the maximal compact subgroup $K$ of a connected semisimple Lie group $G$ whose Lie algebra is a split real form. We lift to the maximal compact subgroup the previously known attaching maps for the maximal flag manifold and use it to characterize algebraically the incidence order between Schubert cells. We also present algebraic formulas to compute the boundary maps which extend to the maximal compact subgroups similar formulas obtained in the case of the maximal flag manifolds. Finally, we apply our results to calculate the cellular homology of $mbox{SO}(3)$ as the maximal compact subgroup of $mbox{SL}(3, mathbb{R})$ and the cellular homology of $mbox{SO}(4)$ as the maximal compact subgroup of the split real form $G_2$.
{"title":"Cellular homology of compact groups: Split real forms","authors":"Mauro Patrão, Ricardo Sandoval","doi":"arxiv-2408.16795","DOIUrl":"https://doi.org/arxiv-2408.16795","url":null,"abstract":"In this article, we use the Bruhat and Schubert cells to calculate the\u0000cellular homology of the maximal compact subgroup $K$ of a connected semisimple\u0000Lie group $G$ whose Lie algebra is a split real form. We lift to the maximal\u0000compact subgroup the previously known attaching maps for the maximal flag\u0000manifold and use it to characterize algebraically the incidence order between\u0000Schubert cells. We also present algebraic formulas to compute the boundary maps\u0000which extend to the maximal compact subgroups similar formulas obtained in the\u0000case of the maximal flag manifolds. Finally, we apply our results to calculate\u0000the cellular homology of $mbox{SO}(3)$ as the maximal compact subgroup of\u0000$mbox{SL}(3, mathbb{R})$ and the cellular homology of $mbox{SO}(4)$ as the\u0000maximal compact subgroup of the split real form $G_2$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206315","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}
Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park
We apply our previous results on ``saturated descent'' to express a wide range of logarithmic cohomology theories in terms of the infinite root stack. Examples include the log cotangent complex, Rognes' log topological cyclic homology, and Nygaard-complete log prismatic cohomology. As applications, we show that the Nygaard-completion of the site-theoretic log prismatic cohomology coincides with the definition arising from log ${rm TC}$, and we establish a log version of the ${rm TC}$-variant of the Beilinson fiber square of Antieau--Mathew--Morrow--Nikolaus.
{"title":"Logarithmic TC via the Infinite Root Stack and the Beilinson Fiber Square","authors":"Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park","doi":"arxiv-2408.15627","DOIUrl":"https://doi.org/arxiv-2408.15627","url":null,"abstract":"We apply our previous results on ``saturated descent'' to express a wide\u0000range of logarithmic cohomology theories in terms of the infinite root stack.\u0000Examples include the log cotangent complex, Rognes' log topological cyclic\u0000homology, and Nygaard-complete log prismatic cohomology. As applications, we\u0000show that the Nygaard-completion of the site-theoretic log prismatic cohomology\u0000coincides with the definition arising from log ${rm TC}$, and we establish a\u0000log version of the ${rm TC}$-variant of the Beilinson fiber square of\u0000Antieau--Mathew--Morrow--Nikolaus.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206313","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}
We present a procedure that constructs, in a combinatorial manner, a chain complex of free modules over a polynomial ring in finitely many variables, modulo an ideal generated by quadratic monomials. Applying this procedure to two specific rings and one family of rings, we demonstrate that the resulting chain complex is indeed an exact chain complex and thus a free resolution. Utilizing this free resolution, we show that, for these rings, the injective dimension is infinite, as modules over itself. Finally, we propose the conjecture that this procedure always yields a free resolution.
{"title":"Combinatorial free chain complexes over quotient polynomial rings","authors":"Daniel Bravo","doi":"arxiv-2408.14695","DOIUrl":"https://doi.org/arxiv-2408.14695","url":null,"abstract":"We present a procedure that constructs, in a combinatorial manner, a chain\u0000complex of free modules over a polynomial ring in finitely many variables,\u0000modulo an ideal generated by quadratic monomials. Applying this procedure to\u0000two specific rings and one family of rings, we demonstrate that the resulting\u0000chain complex is indeed an exact chain complex and thus a free resolution.\u0000Utilizing this free resolution, we show that, for these rings, the injective\u0000dimension is infinite, as modules over itself. Finally, we propose the\u0000conjecture that this procedure always yields a free resolution.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206312","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}
We compute the Hochschild homology of the differential graded category of perfect curved modules over suitable curved rings, giving what might be termed "de Rham models" for such. This represents a generalization of previous results by Dyckerhoff, Efimov, Polishchuk, and Positselski concerning the Hochschild homology of matrix factorizations. A key ingredient in the proof is a theorem due to B. Briggs, which represents a "curved version" of a celebrated theorem of Hopkins and Neeman. The proof of Briggs' Theorem is included in an appendix to this paper.
{"title":"On the Hochschild Homology of Curved Algebras","authors":"Benjamin Briggs, Mark E. Walker","doi":"arxiv-2408.13334","DOIUrl":"https://doi.org/arxiv-2408.13334","url":null,"abstract":"We compute the Hochschild homology of the differential graded category of\u0000perfect curved modules over suitable curved rings, giving what might be termed\u0000\"de Rham models\" for such. This represents a generalization of previous results\u0000by Dyckerhoff, Efimov, Polishchuk, and Positselski concerning the Hochschild\u0000homology of matrix factorizations. A key ingredient in the proof is a theorem\u0000due to B. Briggs, which represents a \"curved version\" of a celebrated theorem\u0000of Hopkins and Neeman. The proof of Briggs' Theorem is included in an appendix\u0000to this paper.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206314","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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