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If E and F are reflexive sheaves of rank one, is their tensor product $E\otimes F$ reflexive? Thanks!

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I am pretty sure that if you try to compute examples in which $E$ and $F$ are not invertible you will see the answer. –  Angelo Apr 5 '12 at 13:23
    
For some generalizations to the commutative algebra setting, you could also try reading about symbolic powers of ideals. –  Karl Schwede Apr 5 '12 at 14:03
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1 Answer 1

Let $R=k[[x,y]]/(xy)$ be the node and $I=(x,y)$ the maximal ideal. Then $I$ is reflexive of rank $1$, but the tensor product $I \otimes_{R} I$ is not reflexive. Indeed, the self-tensor product admits the nonzero torsion section $x \otimes y$.

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