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Large subdirect products of graded modules. (English) Zbl 0923.16035
Let \(\aleph\) be an infinite cardinal number. The \(\aleph\)-product of a family of modules \(\{M_i\}\) is \(\prod^\aleph M_i=\{x\in\prod M_i;\;|\text{supp }x|<\aleph\}\). Let \(R\) be a ring graded by a group \(G\) with neutral element \(e\). First an \(\aleph\)-product is developed for the category \(R\)-gr. A graded module \(M\) is called \(\aleph\)-finitely generated in \(R\)-gr if, for any subset \(S\) of \(M\) with \(| S|<\aleph\), there exists a finitely generated graded submodule \(N\) of \(M\) with \(S\subseteq N\); this concept is used to define finitely \(\aleph\)-presented modules in \(R\)-gr in the natural way. For an Abelian group \(G\), characterizations are given for the following properties: (1) \(M\) is \(\aleph\)-finitely generated in \(R\)-gr, (2) \(M\) is \(\aleph\)-finitely presented in \(R\)-gr, and (3) the graded \(\aleph\)-product of any flat right modules is flat (i.e., \(R\) is left \(\aleph\)-coherent in \(R\)-gr). Then the authors study when the graded \(\aleph\)-product of copies of a single gr-flat module remains flat. The \(\aleph\)-product is preserved by equivalences of locally finitely generated Grothendieck categories; this is used for the categories \(R\)-gr and \(R_e\)-mod when \(G\) is Abelian. When \(G\) is Abelian and \(R\) is strongly graded, \(R\) is \(\aleph\)-coherent in \(R\)-gr if and only if \(R_e\) is left \(\aleph\)-coherent. When \(R\) is strongly graded by a locally finite Abelian group \(G\), \(R\) is left \(\aleph\)-coherent if and only if \(R_e\) is left \(\aleph\)-coherent.

MSC:
16W50 Graded rings and modules (associative rings and algebras)
16S60 Associative rings of functions, subdirect products, sheaves of rings
16D40 Free, projective, and flat modules and ideals in associative algebras
16D90 Module categories in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
18E15 Grothendieck categories (MSC2010)
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
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