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Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO"Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.

Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.

Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.

see http://mathoverflow.net/a/38256/: one should ask also for either affineness or finite type
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edit approved Mar 8, 2017 at 21:12
evgeny
edit approved Mar 8, 2017 at 21:12
evgeny
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Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.

Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.

Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.

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Marty
Marty
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Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.