In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One main varieties satisfy rational Poincare ...

Let k be an arbitrary field, let G be a (smooth) linear algebraic group over k, and let U be a vector group over k on which G acts by automorphisms of algebraic groups. The action of G on U is said to be linear if there is a G-equivariant isomorphism of algebraic groups U ' Lie(U). Suppose that G is connected and that the unipotent radical of G is defined over k. If the G-module Lie(U) is simpl...

The bounded derived category of coherent sheaves D(X) is a natural triangulated category which can be associated with an algebraic variety X. It happens sometimes that two different varieties have equivalent derived categories of coherent sheaves D(X) ≃ D(Y ). There arises a natural question: can one say anything about motives of X and Y in that case? The first such example (see [4]) – abelian ...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.

To any triangulated category with tensor product (K,⊗), we associate a topological space Spc(K,⊗), by means of thick subcategories of K, à la Hopkins-Neeman-Thomason. Moreover, to each open subset U of this space Spc(K,⊗), we associate a triangulated category K(U), producing what could be thought of as a presheaf of triangulated categories. Applying this to the derived category (K,⊗) := (D(X),⊗...

Fix K a p-adic field and denote by GK its absolute Galois group. Let K∞ be the extension of K obtained by adding p-th roots of a fixed uniformizer, and G∞ ⊂ GK its absolute Galois group. In this article, we define a class of p-adic torsion representations of G∞, named quasi-semi-stable. We prove that these representations are “explicitly” described by a certain category of linear algebra object...

The structure of a k-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schwänzl and Vogt in [2] can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of (n-fold) operads in a k-fold monoidal category which generalizes the definition of operads in a braided category. Further...

A k-linear triangulated category A is called locally finite provided ∑ X∈indA dimk HomA(X,Y ) < ∞ for any indecomposable object Y in A. It has Auslander–Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander–Reiten triangles and contains loops, then its Auslander–Reiten quiver is of the form L̂n: · · · · ̂ n n− 1 2 1 By using this, we prove that the Aus...