Simple obstructions and cone reduction

Weak factorization and the Grothendieck group of Deligne–Mumford stacks

Motivic weight complexes for arithmetic varieties

The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Carathéodory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C 2, and on convex finite type domains in C n using the scaling method. Applications include an alternate proof of the Wong–Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C 2 and convex finite type domains in C n in terms of Euclidean parameters. Second, a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C 1-isometries of the Kobayashi and Carathéodory metrics on a smoothly bounded strongly pseudoconvex domain.

Every indecomposable module over a generalized uniserial ring is uniserial, hence local. This motivates one to study rings R satisfying the condition (*): R is a right artinian ring such that every finitely generated, indecomposable right R-module is local. The rings R satisfying (*) have been recently studied by Singh and Al-Bleahed (2004), they have proved some results giving the structure of local right R-modules. In this paper some more structure theorems for local right R-modules are proved. Examples given in this paper show that a rich class of rings satisfying condition (*) can be constructed. Using these results, it is proved that any ring R satisfying (*) is such that mod-R is of finite representation type. It follows from a theorem by Ringel and Tachikawa that any right R-module is a direct sum of local modules. If M is right module over a right artinian ring such that any finitely generated submodule of any homomorphic image of M is a direct sum of local modules, it is proved that it is a direct sum of local modules. This provides an alternative proof for that any right module over a right artinian ring R satisfying (*) is a direct sum of local modules.

In this paper, we prove that the bounded derived category D^b_{\mathrm {coh}}(Y) of coherent sheaves on a separated scheme Y of finite type over a field k of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: D^b_{\mathrm {coh}}(Y) is equivalent to a DG quotient D^b_{\mathrm {coh}}(\tilde{Y})/T, where \tilde{Y} is some smooth and proper variety, and the subcategory T is generated by a single object. The proof uses categorical resolution of singularities of Kuznetsov and Lunts [KL], and a theorem of Orlov [Or1] stating that the class of geometric smooth and proper DG categories is stable under gluing. We also prove the analogous result for \mathbb{Z}/2 -graded DG categories of coherent matrix factorizations on such schemes. In this case instead of D^b_{\mathrm {coh}}(\tilde{Y}) we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over \mathbb{A}_{\mathrm{k}}^1 .

Let k be a field of characteristic zero. Let V be a k-scheme of finite type, i.e., a k-variety, which is integral. We prove that if the associated arc scheme \({\mathcal{L}_{\infty}(V)}\) is reduced, then the \({\mathcal{O}_{V}}\)-Module \({\Omega_{V/k}^{1}}\) is torsion-free. Then if the k-variety V is assumed to be locally a complete intersection (lci), we deduce that the k-variety V is normal. We also obtain the following consequence: for every class \({\mathfrak{C}}\) of integral k-curves which satisfies the Berger conjecture, and for every \({\mathscr{C} \in \mathfrak{C}}\), the k-curve \({\mathscr{C}}\) is smooth if and only if \({\mathcal{L}(\mathscr{C})}\) is reduced.

Intersection theory in algebraic cobordism

Let $A$ be an integral $k$-algebra of finite type over a field $k$ of characteristic zero. Let ${\cal{F}}$ be a family of $k$-derivations on $A$ and $M_{\cal{F}}$ the $A$-module spanned by ${\cal{F}}$. In this paper, we generalize a result due to A. Nowicki and construct an element $\partial$ of $M_{\cal{F}}$ such that $\ker \partial=\cap_{d\in {\cal{F}}} \ker d$. Such a derivation is called ${\cal{F}}$-minimal. Then we establish a density theorem for ${\cal{F}}$-minimal derivations in $M_{\cal{F}}$.

Let A be a normal local ring which is essentially finite type over a field of characteristic zero. Let I⊂A be an ideal such that the Rees algebra RA(I) is Cohen–Macaulay and normal. In this paper we address the question: “When does RA(I) have rational singularities?” In particular, we study the connection between rational singularities of RA(I) and the adjoint ideals of the powers In (n∋ℕ).

In these notes we study properties of the multiplicity at points of a variety X over a perfect field. We focus on properties that can be studied using ramification method, such as discriminants and some generalized discriminants that we shall introduce. We also show how these methods lead to an alternative proof of resolution of singularities for varieties over fields of characteristic zero.

Duality and de Rham cohomology for graded [formula omitted]-modules

We introduce a class of C *-algebras which can be viewed as a generalization of the classical Cuntz-Krieger algebras. Our approach is based on a flexible “generators and relations”-concept. The main result is a canonical uniqueness theorem stating that the C *-algebras of this class are uniquely determined by their generators and relations. We can show that rank one Cuntz-Krieger algebras with infinitely large transition matrices fall in this class, and this provides an alternative proof of a result of Exel and Laca. Further we analyze a subclass of rank two Cuntz-Krieger algebras inspired by shifts of finite type in dimension two, with an infinite set of generators and relations.