Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Operator algebras and conjugacy problem for the pseudo-Anosov automorphisms of a surface

Abstract: The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied To each pseudo-Anosov automorphism f, we assign an AF-algebra A(f) (an operator algebra) It is proved that the assignment is functorial, ie every f', conjugate to f, maps to an AF-algebra A(f'), which is stably isomorphic to A(f) The new invariants of the conjugacy of the pseudo-Anosov automorphisms are obtained from the known invariants of the stable isomorphisms of the AF-algebras Namely, the main invariant is a triple (L, [I], K), where L is an order in the ring of integers in a real algebraic number field K and [I] an equivalence class of the ideals in L The numerical invariants include the determinant D and the signature S, which we compute for the case of the Anosov automorphisms A question concerning the p-adic invariants of the pseudo-Anosov automorphism is formulated

Regular formal modules in one-dimensional local fields

Abstract: This paper considers the problem of the description of unramified extensions of a local field which, together with the main field, do not contain nontrivial roots of isogeny of the corresponding formal group defined over a ring of integers of this field. This problem originated from investigation of extensions without higher ramification for multiplicative formal groups in the paper by Z.I. Borevich (1962).

Macaulayfication of Noetherian schemes

Abstract: To reduce to resolving Cohen-Macaulay singularities, Faltings initiated the program of "Macaulayfying" a given Noetherian scheme $X$. For a wide class of $X$, Kawasaki built the sought Cohen-Macaulay modifications, with a crucial drawback that his blowups did not preserve the locus $\mathrm{CM}(X) \subset X$ where $X$ is already Cohen-Macaulay. We extend Kawasaki's methods to show that every quasi-excellent, Noetherian scheme $X$ has a Cohen-Macaulay $\widetilde{X}$ with a proper map $\widetilde{X} \rightarrow X$ that is an isomorphism over $\mathrm{CM}(X)$. This completes Faltings' program, reduces the conjectural resolution of singularities to the Cohen-Macaulay case, and implies that every proper, smooth scheme over a number field has a proper, flat, Cohen-Macaulay model over the ring of integers.

Generalized orbifold Euler characteristics on the Grothendieck ring of varieties with actions of finite groups

Abstract: The notion of the orbifold Euler characteristic came from physics at the end of 80's. There were defined higher order versions of the orbifold Euler characteristic and generalized ("motivic") versions of them. In a previous paper the authors defined a notion of the Grothendieck ring $K_0^{\rm fGr}(Var)$ of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from $K_0^{\rm fGr}(Var)$ to the Grothendieck ring $K_0(Var)$ of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.