Related papers: Graded perfectoid rings
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via…
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We…
This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring $A=k[x_1, \dots ,x_n]$.…
In this paper we give a first study of perfect copositive $n \times n$ matrices. They can be used to find rational certificates for completely positive matrices. We describe similarities and differences to classical perfect, positive…
We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show…
A general notion of a quasi-finite algebra is introduced as an algebra graded by the set of all integers equipped with topologies on the homogeneous subspaces satisfying certain properties. An analogue of the regular bimodule is introduced…
We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a…
Non notherian Formal schemes of perfectoid type (for example $\mathbb{Z}_p[p^{1/p^\infty}]\langle X^{1/p^\infty} \rangle$ along with its multivariate version) with rational degree are constructed and are shown to be admissible. These formal…
We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to…
We generalise recent results about quasi-Cartan, Cartan and diagonal subalgebras by introducing graded versions. We show that there is a correspondence between graded algebraic quasi-Cartan/ Cartan/ diagonal pairs and certain graded twisted…
We consider more general framework than the corresponding one considered in our previous work on the Hodge-Iwasawa theory. In our current consideration we consider the corresponding more general base spaces, namely the analytic adic spaces…
We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several…
We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…
We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent homotopy categories of complexes. As an…
Perfectoid versions of Abel Jacobi and Reimann Roch Theorem are proved, and perfectoid Elliptic Curve is constructed. A Perfectoid Tate Curve is defined and its cohomology computed via a \v{C}ech complex. Furthermore, perfectoid Theta…
We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.
This survey is a final project of Twopole DRP in fall 20201. In this paper we try to understand a tiny part of the vast theory of perfecoid spaces, called perfectoid fields. We start by giving some motivation and historical background. Then…
For an (integral) perfectoid ring $R$ of characteristic $0$ with tilt $R^{\flat}$, we introduce and study a tilting map $(-)^{\flat}$ from the set of $p$-adically closed ideals of $R$ to the set of ideals of $R^{\flat}$ and an untilting map…
In this paper we develop a graded tilting theory for gauged Landau-Ginzburg models of regular sections in vector bundles over projective varieties. Our main theoretical result describes - under certain conditions - the bounded derived…
We identify an interesting special class of prime ideals in the finitary infinite symmetric group algebra. We show that the set of such ideals carries a semiring structure. Over the complex numbers, we establish a connection with spherical…