Remarks on modules approximated by G-projective modules

Let $R$ be a commutative Noetherian Henselian local ring. Denote by $\mathrm{mod} R$ the category of finitely generated $R$-modules, and by ${\mathcal G}$ the full subcategory of $\mathrm{mod} R$ consisting of all G-projective $R$-modules. In this paper, we consider when a given $R$-module has a right ${\mathcal G}$-approximation. For this, we study the full subcategory $\mathrm{rap}{\mathcal G}$ of $\mathrm{mod} R$ consisting of all $R$-modules that admit right ${\mathcal G}$-approximations. We investigate the structure of $\mathrm{rap}{\mathcal G}$ by observing ${\mathcal G}$, ${\mathcal G}^{\bot}$ and $\mathrm{lap}{\mathcal G}$, where $\mathrm{lap}{\mathcal G}$ denotes the full subcategory of $\mathrm{mod} R$ consisting of all $R$-modules that admit left ${\mathcal G}$-approximations. On the other hand, we also characterize $\mathrm{rap}{\mathcal G}$ in terms of Tate cohomologies. We give several sufficient conditions for ${\mathcal G}$ to be contravariantly finite in $\mathrm{mod} R$.Comment: 28 pages, to appear in J. Algebr

Syzygy modules with semidualizing or G-projective summands

Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.Comment: 13 pages, to appear in Journal of Algebr

Reconstruction from Koszul homology and applications to module and derived categories

Let R be a commutative noetherian ring. Let M be a finitely generated R-module. In this paper, we reconstruct M from its Koszul homology with respect to a suitable sequence of elements of R by taking direct summands, syzygies and extensions, and count the number of those operations. Using this result, we consider generation and classification of certain subcategories of the category of finitely generated R-modules, its bounded derived category and the singularity category of R.Comment: 14 pages, final version, to appear in Pacific J. Mat