Centers of Noetherian Algebras and K-theory

I’ll start off a little vauge: Let E be a noncommutative ring which is finitely generated over its noetherian center Z. Denote by modE the category of finitely generated left E-modules and similarly for modZ. We have a functor F:modZ→modE which takes M to E⊗ZM, hence an induced map on (Quillen) K1-groups K1(F):K1(modZ)→K1(modE). I’m interested … Read more

Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let R be a commutative ring and Φ be a finite (also called spherical) reduced irreducible root system of rank ≥2. I will denote by St(Φ,R) the Steinberg group of type Φ over R, i. e. the quotient of the free product ∐α∈ΦXα of root subgroups Xα=⟨xα(a)∣a∈R⟩ modulo Chevalley commutator formulae: [xα(a);xβ(b)]=∏iα+jβ∈Φxiα+jβ(Ni,jα,βaibj), i,j∈N; (here Ni,jα,β … Read more

Waldhausen’s regular coherent groups: torsionfree non-examples and behaviour under taking products?

Waldhausen defined a group $G$ to be regular coherent, if for all regular noetherian rings $R$ the group algebra $RG$ is regular coherent. (see Waldhausen – Algebraic $K$-Theory of generalized free products III, IV, Section 19 or below for more details). Unsurprisingly this class of groups is mentioned regularly when the vanishing of his Nil-terms … Read more

Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions). The category of motives Mk over a field k, known as Grothendieck-Chow-Motives is based on choosing rational equivalence for ∼. But one can also choose other equivalence relations for ∼, and thus get a … Read more

Bounded self-adjoint perturbation of a p-summable spectral triple

I am new to the field of Noncommutative Geometry.I was reading the chapter on Spectral triple from the book ‘Elements of Noncommutative Geometry’ by Gracia-Bondía,Várilly and Figueroa.Now,after reading the basics of spectral triple,I have a felling that if I perturb the Dirac operator $\mathcal{D}$ of the Spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$ by a bounded self-adjoint operator $\mathcal{S}$,then … Read more

Defining structure maps of spectra by lifting from the homotopy category

Voevodsky’s original definition of the algebraic $K$-theory spectrum, $KGL$, was given as follows: The component spaces were fibrant replacements of the infinite Grassmannian $BGL$. The structure maps were then defined by using the projective bundle theorem for algebraic $K$-theory and the fact that you can lift maps between cofibrant-fibrant objects in the homotopy category to … Read more

Comparing real topological K-theory and algebraic K-theory

Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of finite CW complex $X$ such that $KO^i(X)\approx K^i_{alg}(R)$? Answer AttributionSource : Link , Question Author : rori , Answer Author : … Read more

On the Beilinson’s conjecture regarding the proper flat integral models

I had a question which seems to be straightforward but I wasn’t able to figure it out. In page 13 of this paper a conjecture of beilison is mentioned that if XZ is a proper flat model of XQ then the image of K′∗(X)⊗Q→K∗(X)⊗Q does not depend on the choice of X. Then it is … Read more

Question about an implication of Thomason’s étale descent spectral sequence

On page 5 of this paper by Dwyer and Mitchell, it is said that Thomason’s étale descent spectral sequence from his paper Algebraic K-theory and étale cohomology, which reads Heˊ where KX is the algebraic K-theory spectrum of X and \hat{L} denotes the \ell-completed Bousefield localization at topological K-theory, implies the natural isomorphism \pi_i\widehat{L}(KR) \cong … Read more