Grothendieck categories of enriched functors

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• 作者：Hassan Al Hwaeer^a ; ¹ ; ^{hassan1167@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; Grigory Garkusha^b ; ^{g.garkusha@swansea.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：18E15 ; 18E30 ; 18G55
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：450
• 期：Complete
• 页码：204-241
• 全文大小：581 K

文摘

It is shown that the category of enriched functors formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si1.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=322de076bbed39803d15e0c0ae2d1585" title="Click to view the MathML source">[C,V]er hidden">e"> is Grothendieck whenever formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si2.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=b53e8b52986a0de4063410f1ae674d14" title="Click to view the MathML source">Ver hidden">e"> is a closed symmetric monoidal Grothendieck category and formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si11.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=792bb60df382a1821996e12957e7e958" title="Click to view the MathML source">Cer hidden">e"> is a category enriched over formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si2.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=b53e8b52986a0de4063410f1ae674d14" title="Click to view the MathML source">Ver hidden">e">. Localizations in formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si1.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=322de076bbed39803d15e0c0ae2d1585" title="Click to view the MathML source">[C,V]er hidden">e"> associated to collections of objects of formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si11.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=792bb60df382a1821996e12957e7e958" title="Click to view the MathML source">Cer hidden">e"> are studied. Also, the category of chain complexes of generalized modules formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si26.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=43357713c86a19f8f7d20ac27203eff3" title="Click to view the MathML source">Ch(C_R)er hidden">e"> is shown to be identified with the Grothendieck category of enriched functors e="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si28.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=72659b9ba5b26f90d1cfffb5d24f00dd">eImage" height="16" width="141" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005529-si28.gif">er hidden">e">em">e>R,Chetchy="false">(Mode width="0.2em">e>Retchy="false">)etchy="false">] over a commutative ring <em>Rem>, where the category of finitely presented <em>Rem>-modules mod <em>R  em> is enriched over the closed symmetric monoidal Grothendieck category e="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si29.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=1098154ab7a9b692459688841f3789d5">eImage" height="16" width="80" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005529-si29.gif">er hidden">e">em">e>Retchy="false">) as complexes concentrated in zeroth degree. As an application, it is proved that formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si26.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=43357713c86a19f8f7d20ac27203eff3" title="Click to view the MathML source">Ch(C_R)er hidden">e"> is a closed symmetric monoidal Grothendieck model category with explicit formulas for tensor product and internal Hom-objects. Furthermore, the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of Hovey–Palmieri–Strickland 40">[14] is introduced. It is shown that the derived category of generalized modules formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005529&_mathId=si35.gif&_user=111111111&_pii=S0021869315005529&_rdoc=1&_issn=00218693&md5=478860c3747525b1c8313a1116018eed" title="Click to view the MathML source">D(C_R)er hidden">e"> over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category.