For an acyelic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q, A) and prove that Mon(Q, A) has enough injective objects.
While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for cl...While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity,which has recently been investigated in the works of Enochs et al.(2016)and Estrada et al.(2017).More precisely,for a Grothendieck cosmos,i.e.,a bicomplete Grothendieck category V with a closed symmetric monoidal structure,we prove that the geometrically pure exact category(V,ε■)has enough relative injectives;in fact,every object has a geometrically pure injective envelope.We also show that for some regular cardinalλ,the tensor embedding yields an exact equivalence between(V,ε■)and the category ofλ-cocontinuous V-functors from Presλ(V)to V,where the former is the full V-subcategory ofλ-presentable objects in V.In many cases of interest,λcan be chosen to be■0 and the tensor embedding identifies the geometrically pure injective objects in V with the(categorically)injective objects in the abelian category of V-functors from fp(V)to V.As we explain,the developed theory applies,e.g.,to the category Ch(R)of chain complexes of modules over a commutative ring R and to the category Qcoh(X)of quasi-coherent sheaves over a(suitably nice)scheme X.展开更多
基金The first author was supported by the National Natural Science Foundation of China (Grant No. 11501465) and the Fundamental Research Funds for the Central Universities (XDJK2015C041), the second author was supported by the National Natural Science Foundation of China (Grant Nos. 11271257, 11201377) and the Natural Science Foundation of Shanghai (13ZR1422500).
文摘For an acyelic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q, A) and prove that Mon(Q, A) has enough injective objects.
基金supported by CONICYT/FONDECYT/INICIACIOóN(Grant No.11170394)。
文摘While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity,which has recently been investigated in the works of Enochs et al.(2016)and Estrada et al.(2017).More precisely,for a Grothendieck cosmos,i.e.,a bicomplete Grothendieck category V with a closed symmetric monoidal structure,we prove that the geometrically pure exact category(V,ε■)has enough relative injectives;in fact,every object has a geometrically pure injective envelope.We also show that for some regular cardinalλ,the tensor embedding yields an exact equivalence between(V,ε■)and the category ofλ-cocontinuous V-functors from Presλ(V)to V,where the former is the full V-subcategory ofλ-presentable objects in V.In many cases of interest,λcan be chosen to be■0 and the tensor embedding identifies the geometrically pure injective objects in V with the(categorically)injective objects in the abelian category of V-functors from fp(V)to V.As we explain,the developed theory applies,e.g.,to the category Ch(R)of chain complexes of modules over a commutative ring R and to the category Qcoh(X)of quasi-coherent sheaves over a(suitably nice)scheme X.