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On the exactness of products in the localization of (Ab.4) Grothendieck categories
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Let 931630271X&_mathId=si1.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C be an (Ab.4) Grothendieck category, that is, products are exact in 931630271X&_mathId=si1.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C. Given a hereditary torsion class 931630271X&_mathId=si2.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=43c93246293a6f1cef6eb88988180972" title="Click to view the MathML source">T⊆C, we study the exactness of products in the Gabriel localization 931630271X&_mathId=si3.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=2ee4060dd24246ece4e33ee5afc58d06" title="Click to view the MathML source">C/T of 931630271X&_mathId=si1.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C. We show that, under suitable assumptions on 931630271X&_mathId=si1.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C, the 931630271X&_mathId=si4.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=bf69ae9aa41901083f8f7ba6565be617" title="Click to view the MathML source">k+1-th derived functor of the product vanishes, provided the Gabriel dimension of 931630271X&_mathId=si3.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=2ee4060dd24246ece4e33ee5afc58d06" title="Click to view the MathML source">C/T is smaller than k  . As a consequence, we deduce that, under suitable hypotheses, the derived category 931630271X&_mathId=si18.gif&_user=111111111&_pii=S002186931630271X&_rdoc=1&_issn=00218693&md5=9e82d448e7f2ee5404d5941bd7d96907" title="Click to view the MathML source">D(C/T) is left-complete.

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