Notes on the Milnor conjectures
英文摘要:
These are some notes on the two Milnor conjectures and their proofs (due to Voevodsky, Orlov-Vishik-Voevodsky, and Morel).
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
Daniel Dugger
These are some notes on the two Milnor conjectures and their proofs (due to Voevodsky, Orlov-Vishik-Voevodsky, and Morel).
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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A note on the action of the primitive Milnor operations on the Dickson invariants
本文给出奇素数情形下本原 Milnor 运算作用于 $GL_n$ 在多项式代数 $P_n$ 中 Dickson 不变量代数生成元的公式。
We prove the Milnor conjecture for Lie groups and the Friedlander conjecture for complex algebraic Lie groups.
Let $\A$ be an arrangement of affine lines in $\C^2,$ with complement $\M(\A).$ The (co)homo-logy of $\M(\A)$ with twisted coefficients is strictly related to the cohomology of the Milnor fibre associated to the conified arrangement, endowed with the geometric monodromy. Although several partial results are known, even the first Betti number of the Milnor fiber is not understood. We give here a vanishing conjecture for the first homology, which is of a different nature with respect to the known results. Let $Γ$ be the graph of \emph{double points} of $\A:$ we conjecture that if $Γ$ is connecte…
WeintroducethenotionofderivedformalnoncommutativeZariski immersion for differential graded algebra with examples from topology. We il- lustrate the importance of such notion by reformulating the Friedlander-Milnor conjecture in terms of formal noncommutative Zariski immersions. This paper is based on the language developed by Dwyer, Greenless and Iyendar.
In his study of Halperin's toral-rank conjecture, M. R. Hilali conjectured that for any simply connected rationally elliptic space $X$, one must have $dimπ_*(X)\otimes \mathbb{Q} \leq dimH^*(X,\mathbb{Q})$. Let $(ΛV, d)$ denote a Sullivan minimal model of $X$ and $d_k$ the first non-zero homogeneous part of the differential $d$. In this paper, we use spectral sequence arguments to prove that if $(ΛV, d_k)$ is elliptic, then, there is no gaps in the $E_{\infty}$ term of the Milnor-Moore spectral sequence of $X$. Consequently, we confirm the Hilali conjecture when $V = V^{odd}$ or else when $k\g…
We prove that Friedlander's generalized isomorphism conjecture on the cohomology of algebraic groups, and hence the Isomorphism Conjecture for the cohomology of the complex algebraic Lie group G(C) made discrete, are equivalent to the existence of an isoperimetric inequality in the homological bar complex of G(F), where F is the algebraic closure of a finite field.
We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.
We note that Theorem 5.4 of the paper ``Characteristic cycles of perverse sheaves and Milnor fibers" (by P. Nang and K. Takeuchi, published online in Math. Zeitschrift, October 15, 2004) holds for "any" non-zero complex number $a$.
Notes on Milnor-Witt K-theory
本文为任意特征域建立 Milnor-Witt K-理论基础,利用 Grothendieck 迹映射处理不可分转移,完善其函子性公理。
An analogue of the Milnor conjecture for the de Rham-Witt complex in characteristic 2
本文提出了特征2域上de Rham-Witt复形与Milnor猜想的类比,用TRR的几何不动点的增广理想幂描述,并给出显式描述及同伦群计算。
We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve $C$. The relation to the zero set of Bernstein-Sato polynomial of the defining homogeneous polynomial for $C$ is also discussed. When $C$ has some non weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.
Hilton-Milnor's theorem in $\infty$-topoi
本文证明了经典Hilton-Milnor定理在任意$\infty$-topos中依然成立,其证明基于James分裂的一个版本,并仅使用$\infty$-范畴模型的基本构造。
We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster than for a hypersurface as above. Our conjecture is supported by the results due to L. Narv\' ez Macarro and M. Saito on the roots of Bernstein-Sato polynomials of such hypersurfaces, by all the examples computed so far, and by one p…
Milnor-Witt K-theory and Witt K-theory of a field
本文回顾了域的Milnor-Witt K-理论的基本计算,并利用Kato对Milnor二次型猜想的解答,给出了特征为2的域的Witt K-理论同构于对称双线性形式Witt环基本理想的Rees代数的初等证明。
We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main tools are the polar curves and the affine Lefschetz theory developped by H. Hamm, D.T. Lê and A. Némethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result). This is then…
This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results \begin{equation*} ΣΩΣX \simeq ΣX \vee (X\wedge ΣΩΣX) \quad \text{and} \quad Ω(X \vee Y) \simeq ΩX\times ΩY\times ΩΣ(ΩX \wedge ΩY) \end{equation*} in the maximal generality of an $\infty$-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these im…
We prove the Parshin's conjecture on the rational triviality of the higher algebraic $K$-theory of smooth projective varieties over finite fields. This is known to imply the Beilinson-Soulé conjecture for the fields of positive characteristic. Especially it implies that for a field $F$ of char $p$ and $n>0$, the only rationally non-trivial weight appearing in $K_n(F)\otimes \mathbb{Q}$ can be $n$, thus $K_n(F)\otimes \mathbb{Q}=K_n^M(F)\otimes \mathbb{Q}$ where $K_n^M(F)$ is the Milnor $K$-theory.
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