文摘
Let md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C be an (Ab.4⁎) Grothendieck category, that is, products are exact in md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C. Given a hereditary torsion class md5=43c93246293a6f1cef6eb88988180972" title="Click to view the MathML source">T⊆C, we study the exactness of products in the Gabriel localization md5=2ee4060dd24246ece4e33ee5afc58d06" title="Click to view the MathML source">C/T of md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C. We show that, under suitable assumptions on md5=cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C, the md5=bf69ae9aa41901083f8f7ba6565be617" title="Click to view the MathML source">k+1-th derived functor of the product vanishes, provided the Gabriel dimension of 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 md5=9e82d448e7f2ee5404d5941bd7d96907" title="Click to view the MathML source">D(C/T) is left-complete.