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Register a new userON cyclotomic elements and cyclotomic subgroups in K_{2} of a field
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The problem of expressing an element of K_2(F) in a more explicit form gives rise to many works. To avoid a restrictive condition in a work of Tate, Browkin considered cyclotomic elements as the candidate for the element with an explicit form. In this paper, we modify and change Browkins conjecture about cyclotomic elements into more precise forms, in particular we introduce the conception of cyclotomic subgroup. In the rational function field cases, we determine completely the exact numbers of cyclotomic elements and cyclotomic subgroups contained in a subgroup generated by finitely many different cyclotomic elements, while in the number field cases, using Faltings theorem on Mordell conjecture we prove that there exist subgroups generated by an infinite number of cyclotomic elements to the power of some prime, which contain no nontrivial cyclotomic elements.
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We prove that for a quasi-regular semiperfectoid $mathbb{Z}_p^{rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;mathbb{Z}_p)$ of $R$ to the topological cyclic homology $mathrm{TC}(R;mathbb{Z}_p)$ of $R$ identifies on $pi_2$ with a $q$-deformation of the logarithm.
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $infty$-categories) for additive $infty$-categories, (2) define the notion of nilpotent extensions for suitable $infty$-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas-Goodwillie-McCarthy theorem for stable $infty$-categories which are not monogenically generated (such as the stable $infty$-category of Voevodskys motives or the stable $infty$-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarkos notion of weight structures which provides a ring-with-many-objects analog of a connective $mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work of Hoyois-Krishna for homotopy $K$-theory, and establish new cases of Blancs lattice conjecture.
We observe that the necklace polynomials $M_d(x) = frac{1}{d}sum_{emid d}mu(e)x^{d/e}$ are highly reducible over $mathbb{Q}$ with many cyclotomic factors. Furthermore, the sequence $Phi_d(x) - 1$ of shifted cyclotomic polynomials exhibits a qualitatively similar phenomenon, and it is often the case that $M_d(x)$ and $Phi_d(x) - 1$ have many common cyclotomic factors. We explain these cyclotomic factors of $M_d(x)$ and $Phi_d(x) - 1$ in terms of what we call the emph{$d$th necklace operator}. Finally, we show how these cyclotomic factors correspond to certain hyperplane arrangements in finite abelian groups.
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
It is proved that with finitely many possible exceptions, each cyclotomic scheme over finite field is determined up to isomorphism by the tensor of 2-dimensional intersection numbers; for infinitely many schemes, this result cannot be improved. As a consequence, the Weisfeiler-Leman dimension of a Paley graph or tournament is at most 3 with possible exception of several small graphs.
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