ScalingStacks

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Corollary 12 For each k,m∈ℤk,m\in\mathbb{Z} there is an equality of isomorphism classes of abelian groups

ℋk,m​(L1⊔L2)=\displaystyle{\cal H}^{k,m}(L_{1}\sqcup L_{2})= ⊕i,j∈ℤ(ℋi,j​(L1)⊗ℋk−i,m−j​(L2))⊕\displaystyle{\mathop{\oplus}\limits_{i,j\in\mathbb{Z}}}({\cal H}^{i,j}(L_{1})\otimes{\cal H}^{k-i,m-j}(L_{2}))\oplus
⊕i,j∈ℤTor1ℤ​(ℋi,j​(L1),ℋk−i+1,m−j​(L2))\displaystyle{\mathop{\oplus}\limits_{i,j\in\mathbb{Z}}}{\mathrm{Tor}}_{1}^{\mathbb{Z}}({\cal H}^{i,j}(L_{1}),{\cal H}^{k-i+1,m-j}(L_{2}))

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Mikhail Khovanov

Original source: arXiv:math/9908171v2