When ∏ is Isomorphic to ⊕

Abstract

For a category , we investigate the problem of when the coproduct ⊕ and the product functor ∏ from  ^I to  are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category  this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in  that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of  ^I which is isomorphic to  ^I but is not a full subcategory. If  is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.

Keywords:

• Coproduct
• Corings
• Frobenius functors
• Product

2000 Mathematics Subject Classification:

• Primary 18A30
• Secondary 16W30

ACKNOWLEDGMENTS

The author wishes to thank his Ph.D. advisor Professor Constantin Năstăsescu for very useful remarks on the subject as well as for his continuous support throughout the past years. He also wishes to thank the referee for useful remarks on the subject that helped improving the presentation.

He is partially supported by the bilateral project BWS04/04 “New Techniques in Hopf Algebra Theory and Graded Ring Theory” of the Flemish and Romanian governments and by the CNCSIS BD type grant BD86 2003-2005.

Notes

Communicated by R. Wisbauer.