2025/02/01 by Safaeipour, Mahboubeh, Moghimi, Hosein Fazaeli, Rashedi, Fatemeh
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.00539
Let R be a commutative ring with identity, and let \R(R) denote the semiring of radical ideals of R. The radical functor \R, from the category of R-modules R-\boldsymbol\sfMod to the category of \R(R)-semimodules \R(R)-\boldsymbol\sfSemod, maps any complex \M=(Mn, fn)n≥ 0 of R-modules to a complex \R(\M)=(\R(Mn), \R(fn))n≥ 0 of \R(R)-semimodules, where \R(Mn) consists of radical submodules of Mn, and the \R(R)-semimodule homomorphisms \R(fn):\R(Mn)→ \R(Mn-1) are defined by \R(fn)(N)=\rad(fn(N)). The n-th radical homology of the complex (\R(Mn), \R(fn))n≥ 0, denoted Hn(\R(\M)), consists of radical submodules N of Mn such that fn(N) is contained in the radical of the zero submodule of Mn-1, and two such radical submodules are equivalent under the Bourne relation modulo the image of \R(fn+1). Hn(\R(-)) is regarded as a covariant functor from the category \boldsymbol\sfCh(R-\boldsymbol\sfMod) of chain complexes of R-modules to \R(R)-\boldsymbol\sfSemod, which acts identically on any pair of homotopic maps of complexes of R-modules. In particular, if \M and \M' are homotopically equivalent, then Hn(\R(\M)) and Hn(\R(\M')) are isomorphic \R(R)-semimodules. We provide conditions under which Hn(\R(-)) induces a long exact sequence of radical homology modules for any short exact sequence of complexes of R-modules, and satisfies the naturality condition for exact homology sequences. Finally, we introduce a projective resolution for an R-module M based on \R(R)-semimodules and give conditions under which such a projective resolution exists and is unique up to a homotopy.