Nilpotence Theorem Research Articles

The Segal conjecture for topological Hochschild homology of Ravenel spectra

In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing the algebraic K-theory of X(n) using trace methods, which approximates the algebraic K-theory of the sphere spectrum in a precise sense. We solve the homotopy limit problem for topological Hochschild homology of T(n) under the assumption that the canonical map \(T(n)\rightarrow BP\) of homotopy commutative ring spectra can be rigidified to map of \(E_2\) ring spectra. We show that the obstruction to our assumption holding can be described in terms of an explicit class in an Atiyah-Hirzebruch spectral sequence.

Algebraic chromatic homotopy theory for BP∗BP-comodules

In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.

On the Rost nilpotence theorem for threefolds

We show that Rost nilpotence holds for a geometrically integral threefold X over a field k of characteristic 0 if and only if αk(X)*∘N( CH 0(Xk(X)))=0 for some integer N>0 for all correspondences α of degree 0 which vanish over some field extension of k. As a corollary we get the Rost nilpotence property for three-dimensional smooth projective geometrically integral schemes over a field of characteristic zero, which are birationally isomorphic to a toric model.

Thick tensor ideals of right bounded derived categories

Let $R$ be a commutative noetherian ring. Denote by $D^-(R)$ the derived category of cochain complexes $X$ of finitely generated $R$-modules with $H^i(X)=0$ for $i\gg0$. Then $D^-(R)$ has the structure of a tensor triangulated category with tensor product $-\otimes_R^L-$ and unit object $R$. In this paper, we study thick tensor ideals of $D^-(R)$, i.e., thick subcategories closed under the tensor action by each object in $D^-(R)$, and investigate the Balmer spectrum $Spc\,D^-(R)$ of $D^-(R)$, i.e., the set of prime thick tensor ideals of $D^-(R)$. First, we give a complete classification of the thick tensor ideals of $D^-(R)$ generated by bounded complexes, establishing a generalized version of the Hopkins-Neeman smash nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum $Spc\,D^-(R)$ and the Zariski spectrum $Spec\,R$, and study their topological properties. After that, we compare several classes of thick tensor ideals of $D^-(R)$, relating them to specialization-closed subsets of $Spec\,R$ and Thomason subsets of $Spc\,D^-(R)$, and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of $D^-(R)$ in the case where $R$ is a discrete valuation ring.

The nilpotence theorem for the algebraic K–theory of the sphere spectrum

We prove that in the graded commutative ring $K_{*}(\mathbb{S})$, all positive degree elements are multiplicatively nilpotent. The analogous statements also hold for $TC_{*}(\mathbb{S};\mathbb{Z}^{\wedge}_p)$ and $K_{*}(\mathbb{Z})$.

Residue fields for a class of rational E∞-rings and applications

Let A be an E∞-ring over the rational numbers. If A satisfies a noetherian condition on its homotopy groups π⁎(A), we construct a collection of E∞-A-algebras that realize on homotopy the residue fields of π⁎(A). We prove an analog of the nilpotence theorem for these residue fields. As a result, we are able to give a complete algebraic description of the Galois theory of A and of the thick subcategories of perfect A-modules. We also obtain partial information on the Picard group of A.

The local nilpotence theorem for 4-Engel groups revisited

‎The proof of the local nilpotence theorem for $4$-Engel groups was‎ ‎completed by G‎. ‎Havas and M‎. ‎Vaughan-Lee in 2005‎. ‎The complete proof on the other hand is spread over several articles and the aim of this paper is to give a‎ ‎complete coherent linear version‎. ‎In the process we are also‎ ‎able to make a few simplifications and in particular we are able to merge two of‎ ‎the key steps into one‎.

On a nilpotence conjecture of J. P. May

We prove a conjecture of J.P. May concerning the nilpotence of elements in ring spectra with power operations, i.e., $H_\infty$-ring spectra. Using an explicit nilpotence bound on the torsion elements in $K(n)$-local $H_\infty$-algebras over $E_n$, we reduce the conjecture to the nilpotence theorem of Devinatz, Hopkins, and Smith. As corollaries we obtain nilpotence results in various bordism rings including $M\mathit{Spin}_*$ and $M\mathit{String}_*$, results about the behavior of the Adams spectral sequence for $E_\infty$-ring spectra, and the non-existence of $E_\infty$-ring structures on certain complex oriented ring spectra.

A short note on the weak Lefschetz property for Chow groups

Motivated by the Bloch-Beilinson conjectures, we formulate a certain covariant weak Lefschetz property for Chow groups. We prove this property in some special cases, using Kimura's nilpotence theorem.

The Rost nilpotence theorem for geometrically rational surfaces

We show that the Rost nilpotence theorem is true for geometrically rational surfaces over fields of characteristic zero.

Mini-Workshop: Thick Subcategories - Classifications and Applications

Thick subcategories of triangulated categories have been the main topic of this workshop. Triangulated categories arise in many areas of modern mathematics, for instance in algebraic geometry, in representation theory of groups and algebras, or in stable homotopy theory. We give three typical examples of such triangulated categories: In each case, there is a classification of thick subcategories under some appropriate conditions. Recall that a subcategory of a triangulated category is thick , if it is a triangulated subcategory and closed under taking direct factors. Historically, the first classification has been established by Hopkins and Smith for the stable homotopy category, using the nilpotence theorem. A similar idea was then applied by Hopkins and Neeman to categories of perfect complexes over commutative noetherian rings. Later, Thomason extended this classification to schemes. For stable categories of finite group representations, the classification of thick subcategories is due to Benson, Carlson, and Rickard. The format of the workshop has been a combination of introductory survey lectures and more specialized talks on recent progress and open problems. The mix of participants from different mathematical areas and the relatively small size of the workshop provided an ideal atmosphere for fruitful interaction and exchange of ideas. It is a pleasure to thank the administration and the staff of the Oberwolfach Institute for their efficient support and hospitality.

Localization of André–Quillen–Goodwillie towers, and the periodic homology of infinite loopspaces

Let K ( n ) be the n th Morava K -theory at a prime p, and let T ( n ) be the telescope of a v n -self map of a finite complex of type n. In this paper we study the K ( n ) * -homology of Ω ∞ X , the 0th space of a spectrum X, and many related matters. We give a sampling of our results. Let P X be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map s n ( X ) : L T ( n ) P ( X ) → L T ( n ) Σ ∞ ( Ω ∞ X ) + of commutative algebras over the localized sphere spectrum L T ( n ) S . The induced map of commutative, cocommutative K ( n ) * -Hopf algebras s n ( X ) * : K ( n ) * ( P X ) → K ( n ) * ( Ω ∞ X ) , satisfies the following properties. It is always monic. It is an isomorphism if X is n-connected, π n + 1 ( X ) is torsion, and T ( i ) * ( X ) = 0 for 1 ⩽ i ⩽ n - 1 . It is an isomorphism only if K ( i ) * ( X ) = 0 for 1 ⩽ i ⩽ n - 1 . It is universal. The domain of s n ( X ) * preserves K ( n ) * -isomorphisms, and if F is any functor preserving K ( n ) * -isomorphisms, then any natural transformation F ( X ) → K ( n ) * ( Ω ∞ X ) factors uniquely through s n ( X ) * . The construction of our natural transformation uses the telescopic functors constructed and studied previously by Bousfield and the author, and thus depends heavily on the Nilpotence Theorem of Devanitz, Hopkins, and Smith. Our proof that s n ( X ) * is always monic uses Topological André-Quillen Homology and Goodwillie Calculus in nonconnective settings.

Motivic decomposition of isotropic projective homogeneous varieties

We give a decomposition of the Chow motive of an isotropic projective homogeneous variety of a semisimple algebraic group in terms of twisted motives of simpler projective homogeneous varieties. As an application, we prove a generalization of Rost's nilpotence theorem.

A short proof of Rost nilpotence via refined correspondences

I generalize the notion of composition of algebraic correspondences using the refined Gysin homorphism of Fulton–MacPherson intersection theory. Using this notion, I give a short self-contained proof of Rost's “nilpotence theorem” and a generalization of one important proposition used by Rost in his proof of the theorem.

A nilpotence theorem for cycles algebraically equivalent to zero

A nilpotence theorem for cycles algebraically equivalent to zero

A nilpotence theorem for modules over the mod 2 steenrod algebra

A nilpotence theorem for modules over the mod 2 steenrod algebra

A nil implies nilpotent theorem for left ideals

A nil implies nilpotent theorem for left ideals

The Zelmanov nilpotence theorem for quadratic Jordan algebras

The Zelmanov nilpotence theorem for quadratic Jordan algebras