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Concept
Grothendieck category
Summary
In mathematics, a Grothendieck category is a certain kind of , introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.
To every algebraic variety one can associate a Grothendieck category , consisting of the quasi-coherent sheaves on . This category encodes all the relevant geometric information about , and can be recovered from (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories.
By definition, a Grothendieck category is an with a . Spelled out, this means that
is an ;
every (possibly infinite) family of objects in has a coproduct (also known as direct sum) in ;
direct limits of short exact sequences are exact; this means that if a direct system of short exact sequences in is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always right-exact; the important point here is that we require them to be left-exact as well.)
possesses a generator, i.e. there is an object in such that is a faithful functor from to the . (In our situation, this is equivalent to saying that every object of admits an epimorphism , where denotes a direct sum of copies of , one for each element of the (possibly infinite) set .)
The name "Grothendieck category" neither appeared in Grothendieck's Tôhoku paper nor in Gabriel's thesis; it came into use in the second half of the 1960s in the work of several authors, including Jan-Erik Roos, Bo Stenström, Ulrich Oberst, and Bodo Pareigis. (Some authors use a different definition in that they don't require the existence of a generator.)
The prototypical example of a Grothendieck category is the ; the abelian group of integers can serve as a generator.
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