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The small finitistic dimensions of commutative rings, III

Xiaolei Zhang

Abstract

The small finitistic dimension fPD$(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we show that a commutative ring $R$ has fPD$(R)\leq d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ext_R^i(R/I,R)=0$ for each $i=0,\dots,d$, then $Ext_R^i(R/I,R)=0$ for all $i\geq 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\leq \mbox{FP-}Id_RR$, the self-FP-injective dimension of $R$. We also give some applications of these results to (weak) $(n,d)$-rings, DW-rings and rings of Prufer type.

The small finitistic dimensions of commutative rings, III

Abstract

The small finitistic dimension fPD of a ring is defined to be the supremum of projective dimensions of -modules with finite projective resolutions. In this paper, we show that a commutative ring has fPD if and only if for any finitely generated ideal of , if for each , then for all As applications, we obtain that, for any commutative ring , fPD, the self-FP-injective dimension of . We also give some applications of these results to (weak) -rings, DW-rings and rings of Prufer type.
Paper Structure (7 sections, 15 theorems, 15 equations)

This paper contains 7 sections, 15 theorems, 15 equations.

Table of Contents

  1. introduction
  2. The main result
  3. Applications
  4. fPD(R) VS FP-id(R)
  5. fPD of weak (n,d)$\hbox{-}$rings
  6. New Characterizations of DW-rings
  7. Srong Prüfer rings VS Prüfer rings

Key Result

Lemma 2.2

S90$($Koszul duality$)$ For every module $M$, there is an isomorphism of complexes \xymatrix@C=1.2em@R=1.5em{ 0 \ar[r] & K_n(\textbf{x}, M) \ar[r] \ar[d]_{\phi_n} & \cdots \ar[r] & K_p(\textbf{x}, M) \ar[r]^{d_p^M} \ar[d]^{\phi_p} & K_{p-1}(\textbf{x}, M) \ar[r] \ar[d]^{\phi_{p-1}} & \cdots \ar[r] &

Theorems & Definitions (31)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • proof
  • Theorem 3.1
  • ...and 21 more