By Baeza R., et al. (eds.)

ISBN-10: 082183441X

ISBN-13: 9780821834411

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**Extra info for Algebraic and Arithmetic Theory of Quadratic Forms**

**Sample text**

2 the exact functors on Aop , that is the objects of D, may be identiﬁed with the exact functors on fun-A. 3) we see that the functors added are those which express non-exactness, hence on which exact functors are 0. e. for all D ∈ D, evD F = 0} in its kernel and hence which factors through the quotient map fun-A −→ fun(D) = fun-A/SD . Then one checks that this is an equivalence (note that already we had Aop −→ fun(D) via A → ((−, A), −)D ). 8) denote the locally coherent category corresponding, in the sense of 3, to A, so Ex((C fp )op , Ab), the above diagram may be reA C fp .

Then C is a deﬁnable subcategory of Mod-C fp . Therefore every deﬁnable subcategory of a ﬁnitely accessible category is a deﬁnable subcategory of a functor category. 12, every deﬁnable category is a deﬁnable subcategory of a locally coherent functor category. 4). 2. Let T be the category of torsion abelian groups. This is not a deﬁnable subcategory of Mod-Z (it is not closed under products) but it is ﬁnitely accessible with products (hence a deﬁnable category): the ﬁnitely presented objects are the ﬁnitely generated = ﬁnitely presented torsion groups and product is given by taking the torsion subgroup of the product in Mod-Z.

So let C be a Grothendieck category. A full subcategory T is a torsion subcategory if it is closed under extensions, quotient objects and direct sums. If it is also closed under subobjects then it is a hereditary torsion subcategory (or subclass). The objects of T are referred to as torsion and those in the corresponding torsionfree class/subcategory F = {D : (T , D) = 0} are the torsionfree objects. It is then the case that T = {C : (C, F) = 0} and the pair τ = (T , F) is referred to as a torsion theory, which is said to be hereditary if T is closed under subobjects, equivalently if F is closed under injective hulls.

### Algebraic and Arithmetic Theory of Quadratic Forms by Baeza R., et al. (eds.)

by Robert

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