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تسجيل مستخدم جديدRedshift and multiplication for truncated Brown-Peterson spectra
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We equip $mathrm{BP} langle n rangle$ with an $mathbb{E}_3$-$mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this $mathbb{E}_3$-ring is of chromatic height exactly $n+1$. Specifically, it is an fp-spectrum of fp-type $n+1$, which can be viewed as a higher height version of the Lichtenbaum-Quillen conjecture.
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We compute topological Hochschild homology of sufficiently structured forms of truncated Brown--Peterson spectra with coefficients. In particular, we compute $operatorname{THH}_*(operatorname{taf}^D;M)$ for $Min { Hmathbb{Z}_{(3)},k(1),k(2)}$ where $
operatorname{taf}^D$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Hill--Lawson. We compute $operatorname{THH}_*(operatorname{tmf}_1(3);M)$ when $Min { Hmathbb{Z}_{(2)},k(2)}$ where $operatorname{tmf}_1(3)$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Lawson--Naumann. We also compute $operatorname{THH}_*(Blangle nrangle;M)$ for $M=Hmathbb{Z}_{(p)}$ and certain $E_3$ forms $Blangle nrangle$ of $BPlangle nrangle$. For example at $p=2$, this result applies to the $E_3$ forms of $BPlangle nrangle$ constructed by Hahn--Wilson.
Fix the base field Q of rational numbers and let BP<n> denote the family of motivic truncated Brown-Peterson spectra over Q. We employ a local-to-global philosophy in order to compute the motivic Adams spectral sequence converging to the bi-graded ho
motopy groups of BP<n>. Along the way, we provide a new computation of the homotopy groups of BP<n> over the 2-adic rationals, prove a motivic Hasse principle for the spectra BP<n>, and deduce several classical and recent theorems about the K-theory of particular fields.
The dual Steenrod algebra has a canonical subalgebra isomorphic to the homology of the Brown-Peterson spectrum. We will construct a secondary operation in mod-2 homology and show that this canonical subalgebra is not closed under it. This allows us t
o conclude that the 2-primary Brown-Peterson spectrum does not admit the structure of an E_n-algebra for any n greater than or equal to 12, answering a question of May in the negative.
We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)). This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positiv
e characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on N^n. If the characteristic of k does not divide any of the a_i we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k=Z. To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write TC(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)) as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand. Updated: This is a substantial revision. We corrected several errors in the description of the Witt vectors on a truncation set on N^n and modified the key proofs accordingly. We also replaces several topological statement with purely algebraic ones. Most arguments have been reworked and streamlined.
Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this pape
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