文摘
In this note we consider a notion of relative Frobenius pairs of commutative rings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630357X&_mathId=si1.gif&_user=111111111&_pii=S002186931630357X&_rdoc=1&_issn=00218693&md5=2401681bab2aa2c2ab504151551b6860" title="Click to view the MathML source">S/Rclass="mathContainer hidden">class="mathCode">. To such a pair, we associate an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630357X&_mathId=si2.gif&_user=111111111&_pii=S002186931630357X&_rdoc=1&_issn=00218693&md5=c55740da3507d8834266bad420991608" title="Click to view the MathML source">Nclass="mathContainer hidden">class="mathCode">-graded R -algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630357X&_mathId=si29.gif&_user=111111111&_pii=S002186931630357X&_rdoc=1&_issn=00218693&md5=8820b053dd315680e0d277c704a35427" title="Click to view the MathML source">ΠR(S)class="mathContainer hidden">class="mathCode"> which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of S over R is 4 and R is Noetherian, we prove that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630357X&_mathId=si29.gif&_user=111111111&_pii=S002186931630357X&_rdoc=1&_issn=00218693&md5=8820b053dd315680e0d277c704a35427" title="Click to view the MathML source">ΠR(S)class="mathContainer hidden">class="mathCode"> is itself Noetherian and finite over its center and that each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630357X&_mathId=si4.gif&_user=111111111&_pii=S002186931630357X&_rdoc=1&_issn=00218693&md5=7ae366cd346a1e5921c473f9192aff37" title="Click to view the MathML source">ΠR(S)dclass="mathContainer hidden">class="mathCode"> is finitely generated projective. We also prove that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630357X&_mathId=si29.gif&_user=111111111&_pii=S002186931630357X&_rdoc=1&_issn=00218693&md5=8820b053dd315680e0d277c704a35427" title="Click to view the MathML source">ΠR(S)class="mathContainer hidden">class="mathCode"> is of finite global dimension if R and S are regular.