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
A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure...A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure-injective modules is studied.展开更多
In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can characteriz...In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prüfer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prüfer sheaves.展开更多
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.
文摘A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure-injective modules is studied.
基金Supported by National Nature Science Foundation of China(Grant Nos.11571286,11471269)the Natural Science Foundation of Fujian Province of China(Grant No.2016J01031)the Fundamental Research Funds for the Central Universities of China(Grant No.20720150006)
文摘In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prüfer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prüfer sheaves.
