http://hdl.handle.net/10366/4541 Thu, 23 Apr 2026 08:59:47 GMT 2026-04-23T08:59:47Z http://hdl.handle.net/10366/167360 [EN]We consider projective, irreducible, non-singular curves over an algebraically closed field k. A cover Y → X of such curves corresponds to an extension / of their function fields and yields an isomorphism AY ≃ AX ⊗ of their geometric adele rings. The primitive element theorem shows that AY is a quotient of AX[T] by a polynomial. In general, we may look at quotient algebras AX {p} = AX [T ]/(p(T )) where p(T ) ∈ AX [T ] is monic and separable over AX , and try to characterize the field extensions / lying in AX {p} which arise from covers as above. We achieve this in two ways; the first, topologically, as those which embed discretely in AX {p}. The second is the characterization of such subfields as those which satisfy the additive analog of the product formula in classical adele rings. The technical machinery is based on the use of Tate topologies on the quotient algebras AX {p}. These are not locally compact, but we are able to define an additive content function as an index measuring the discrepancy of dimensions in commensurable subspaces. Wed, 01 Jan 2025 00:00:00 GMT http://hdl.handle.net/10366/167360 2025-01-01T00:00:00Z http://hdl.handle.net/10366/167359 [EN]Given the n-dimensional affine space over an arbitrary commutative ring k, G a group scheme flat and finitely presented over k and X ⊂ Ank a G-invariant affine and closed subscheme, we prove that the GIT quotient X/G is of finite type over k, even if k is not noetherian, provided G is linearly reductive. This is well-known if X → X/G is faithfully flat, which does not hold in general. We also explore the infinite- dimensional case. Concretely, we consider a G − k finitely presented projective module M and an arbitrary k-module N. We prove, under certain conditions on k and G, that the degrees of the generators of (S• (M ∨ ⊗ N ))G and the degrees of the generators of the ideal of relations are bounded. We encode this property into the notion of partially generated graded (pgg) algebra and we give their main properties. In particular, we prove the existence of canonical equivariant immersions of spectra of pgg algebras in certain projective spaces. Mon, 01 Jan 2024 00:00:00 GMT http://hdl.handle.net/10366/167359 2024-01-01T00:00:00Z http://hdl.handle.net/10366/164313 [EN] The aim of thiswork is to offer the definition and themain properties of a pseudo-characteristic polynomial of a finite potent endomorphism. From this polynomial we can characterize the spectrum of a bounded finite potent linear operator on a Hilbert space andwe study arithmetic properties of complete algebraic curves. In particular, we provide a new algebraic proof of the Residue Theorem. Wed, 01 Jan 2025 00:00:00 GMT http://hdl.handle.net/10366/164313 2025-01-01T00:00:00Z http://hdl.handle.net/10366/164140 [EN] The aim of this note is to offer a method for the explicit computation of all 1-inverses of an arbitrary square matrix. The main tool to obtain this algorithm is the characterization of the set of 1-inverses of a finite potent endomorphism. Sun, 01 Jan 2023 00:00:00 GMT http://hdl.handle.net/10366/164140 2023-01-01T00:00:00Z http://hdl.handle.net/10366/164137 [EN] The aim of this work is to show the consistency of all systems of nonhomogeneous linear difference equations of the form ϕ(x_n+1) = x_n + v_0, where ϕ ∈ End_k(V) is a finite potent endomorphism of an arbitrary vector space V and v_0 ∈ V. An algorithm to compute the set of solutions of these systems is given. In particular, the method offered is valid for computing the explicit solutions of the system of non-homogeneous difference equations A(x_n+1) = x_n + b, with A being a finite square matrix. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10366/164137 2022-01-01T00:00:00Z http://hdl.handle.net/10366/164136 [EN] The aim of this work is to offer necessary and sufficient conditions for the coincidence of the Drazin inverse and the DMP inverses of finite square complex matrices. This characterization is deduced from statements valid for finite potent endomorphisms and, in particular, several properties of matrices and generalized inverses are given. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10366/164136 2022-01-01T00:00:00Z http://hdl.handle.net/10366/164135 [EN] The aim of this work is to extend to finite potent endomorphisms the notion of G-Drazin inverse of a finite square matrix. Accordingly, we determine the structure and the properties of a G-Drazin inverse of a finite potent endomorphism and, as an application, we offer an algorithm to compute the explicit expression of all G-Drazin inverses of a finite square matrix. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10366/164135 2022-01-01T00:00:00Z http://hdl.handle.net/10366/164134 [EN] The aim of this work is to extend to finite potent endomorphisms the notion of the Drazin-Moore-Penrose inverse of a finite matrix. Several properties of this generalized inverse are proved that are also valid for some infinite matrices. Moreover, Drazin-Reflexive inverses associated with arbitrary reflexive generalized inverses of a finite potent endomorphism are introduced. Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/10366/164134 2021-01-01T00:00:00Z http://hdl.handle.net/10366/164132 [EN] The aim of this work is to prove the existence and uniqueness of a core-nilpotent decomposition of finite potent endomorphisms on arbitrary vector spaces. This decomposition generalized the well-known core-nilpotent decomposition of complex (n × n)-matrices. As an application we offer the definition and properties of the CMP inverse and of a new CR inverse for these linear maps. In particular, from the results of this work it is possible to compute the corenilpotent decomposition, the CMP and the CR inverses of some infinite matrices. Wed, 01 Jan 2020 00:00:00 GMT http://hdl.handle.net/10366/164132 2020-01-01T00:00:00Z http://hdl.handle.net/10366/164131 [EN] The aim of this work is to prove the existence of Drazin inverses for matrices associated with finite potent endomorphisms on arbitrary vector spaces. The result offered coincides with the classical Drazin inverse matrix for finite-dimensional vector spaces over C. Tue, 01 Jan 2019 00:00:00 GMT http://hdl.handle.net/10366/164131 2019-01-01T00:00:00Z http://hdl.handle.net/10366/164066 [EN]The aim of this work is to introduce the group inverse of a finite potent endomorphism on an infinite-dimensional vector space that generalizes the notion of group inverse of a square finite matrix. The existence and uniqueness of this inverse is proved, several properties are offered and the relations with Drazin inverse, CMP inverse and DMP inverses are studied. Tue, 01 Dec 2020 00:00:00 GMT http://hdl.handle.net/10366/164066 2020-12-01T00:00:00Z http://hdl.handle.net/10366/164064 [EN]The aim of this work is to develop a general theory of core-nilpotent endomorphisms of arbitrary vector spaces, such that endomorphisms of finite-dimensional vector spaces and finite potent endomorphisms of infinite-dimensional vector spaces are particular cases of the CN-endomorphisms studied in this theory. For these CN-endomorphisms, we introduce an index that generalizes the index of a finite square matrix and we prove the existence of the Drazin inverse and reflexive generalized inverses. In particular, we characterize all endomorphisms that have Drazin inverse on arbitrary vector spaces. Moreover, we offer a method to study infinite linear systems associated with CN-endomorphisms. Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/10366/164064 2021-01-01T00:00:00Z http://hdl.handle.net/10366/164061 [EN]The aim of this note is to offer an algorithm for studying solutions of infinite linear systems associated with group inverse endomorphisms. As particular results, we provide different properties of the group inverse and we characterize EP endomorphisms of arbitrary vector spaces from the coincidence of the group inverse and the Moore-Penrose inverse. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10366/164061 2022-01-01T00:00:00Z http://hdl.handle.net/10366/164058 [EN]The aim of this work is to offer a new solution to the problem of the classification of endomorphisms with an annihilating polynomial on infinite-dimensional vector spaces. For these endomorphisms we provide a family of invariants that allows us to classify them when the group of automorphisms acts by conjugation. Moreover, the description of a new method to construct Jordan bases is given. Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/10366/164058 2021-01-01T00:00:00Z http://hdl.handle.net/10366/164055 [EN]The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov — M. Kerr and the author — together with D. Hernández Serrano — can also be deduced from this general expression. Mon, 01 Jan 2018 00:00:00 GMT http://hdl.handle.net/10366/164055 2018-01-01T00:00:00Z http://hdl.handle.net/10366/164047 [EN]The aim of this work is to offer a method for computing reflexive generalized inverses of finite potent endomorphisms, that can be applied to obtain explicit solutions of infinite systems of linear equations. Sat, 15 Dec 2018 00:00:00 GMT http://hdl.handle.net/10366/164047 2018-12-15T00:00:00Z