The Bass-Quillen conjecture in dimension three but characteristic ? 2,3 via a question of A. Suslin.
The Construction of a Module of Finite Projective Dimension from a Finitely Generated Module of Finite Injective Dimension.
The F-method and a branching problem for generalized Verma modules associated to
The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras , and generalized conformal -Verma modules of scalar type. As a result, we classify the -singular vectors for this class of -modules.
The group of commutativity preserving maps on strictly upper triangular matrices
Let be the algebra of all strictly upper triangular matrices over a unital commutative ring . A map on is called preserving commutativity in both directions if . In this paper, we prove that each invertible linear map on preserving commutativity in both directions is exactly a quasi-automorphism of , and a quasi-automorphism of can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.
The Serre Problem for Discrete Hodge Algebras.
The structure of indecomposable injective modules.
Torsion äußerer Potenzen von Moduln der homologischen Dimension l.
Torsion Part of ℤ-module
In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].
Torsion Z-module and Torsion-free Z-module
In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].
Torsion-free abelian groups and completely decomposable -adic modules
Two applications of dualizing complexes over local rings
Two Notes on Flatness.
Two properties of the power series ring.