Homology product

An operation defined on the groups $ \mathop{\rm Tor} $ and $ \mathop{\rm Ext} $. One considers $ K $- algebras $ R, S $ and $ T = R \otimes _ {K} S $ over a commutative ring $ K $. The derived functors (cf. Derived functor) $ \mathop{\rm Tor} $ and $ \mathop{\rm Ext} $ over them may be combined with one another by means of four homomorphisms, known as homology products:

$$ \perp : \mathop{\rm Tor} _ {p} ^ {R} ( A, A ^ \prime ) \otimes \mathop{\rm Tor} _ {q} ^ {S} ( C, C ^ \prime ) \rightarrow \ \mathop{\rm Tor} _ {p + q } ^ {T} ( A \otimes C, A ^ \prime \otimes C ^ \prime ), $$

$$ \perp : \mathop{\rm Ext} _ {T} ^ {p + q } ( A \otimes C, \mathop{\rm Hom} ( A ^ \prime , C ^ \prime )) \rightarrow $$

$$ \rightarrow \ \mathop{\rm Hom} ( \mathop{\rm Tor} _ {p} ^ {R} ( A ^ \prime , A), \mathop{\rm Ext} _ {s} ^ {q} ( C, C ^ \prime )), $$

$$ \lor : \mathop{\rm Ext} _ {R} ^ {p} ( A, A ^ \prime ) \otimes \mathop{\rm Ext} _ {s} ^ {q} ( C, C ^ \prime ) \rightarrow \mathop{\rm Ext} _ {T} ^ {p + q } ( A \otimes C, A ^ \prime \otimes C ^ \prime ), $$

$$ \wedge : \mathop{\rm Tor} _ {p + q } ^ {T} ( \mathop{\rm Hom} ( A, C), A ^ \prime \otimes C ^ \prime ) \rightarrow $$

$$ \rightarrow \ \mathop{\rm Hom} ( \mathop{\rm Ext} _ {R} ^ {p} ( A, A ^ \prime ), \mathop{\rm Tor} _ {q} ^ {S} ( C, C ^ \prime )). $$

Here, $ A $ and $ A ^ \prime $ are right or left $ R $- modules, $ C $ and $ C ^ \prime $ are right or left $ S $- modules, while the symbol $ K $ is omitted in all functors. The last two homomorphisms are defined only if the algebras $ R $ and $ S $ are projective over $ K $ and if $ \mathop{\rm Tor} _ {n} ^ {K} ( A, C) = 0 $ for all $ n > 0 $. If certain supplementary restrictions are made, intrinsic products can be obtained which connect $ \mathop{\rm Tor} $ and $ \mathop{\rm Ext} $ over the same ring.

All four products can be obtained from formulas representing the functors $ \otimes $ and $ \mathop{\rm Hom} $ by replacing the arguments by the corresponding resolutions [1]. The multiplication $ \lor $ permits the following interpretation in terms of Yoneda products. Let

$$ 0 \rightarrow A ^ \prime \rightarrow X _ {1} \rightarrow \dots \rightarrow X _ {p} \rightarrow A \rightarrow 0, $$

$$ 0 \rightarrow C ^ \prime \rightarrow Y _ {1} \rightarrow \dots \rightarrow Y _ {q} \rightarrow C \rightarrow 0 $$

be exact sequences of $ R $- and $ S $- modules, respectively, that are representatives of the corresponding equivalence classes in $ \mathop{\rm Ext} _ {R} ^ {p} ( A, A ^ \prime ) $ and $ \mathop{\rm Ext} _ {S} ^ {q} ( C, C ^ \prime ) $. Multiplying the former tensorially from the right by $ C ^ \prime $ and the latter from the left by $ A $, one obtains exact sequences

$$ 0 \rightarrow A ^ \prime \otimes C ^ \prime \rightarrow X _ {1} \otimes C ^ \prime \rightarrow \dots \rightarrow \ X _ {p} \otimes C ^ \prime \rightarrow A \otimes C ^ \prime \rightarrow 0, $$

$$ 0 \rightarrow A \otimes C ^ \prime \rightarrow A \otimes Y _ {1} \rightarrow \dots \rightarrow A \otimes Y _ {q} \rightarrow A \otimes C \rightarrow 0, $$

which can be combined into the exact sequence

$$ 0 \rightarrow A ^ \prime \otimes C ^ \prime \rightarrow \ X _ {1} \otimes C ^ \prime \rightarrow \dots \rightarrow A \otimes Y _ {q} \rightarrow A \otimes C \rightarrow 0. $$

This sequence can be regarded as the representative of an equivalence class in the group

$$ \mathop{\rm Exp} _ {R \otimes S } ^ {p + q } ( A \otimes C, A ^ \prime \otimes C ^ \prime ). $$

The product $ \lor $ in the cohomology space $ H ( X, \mathbf Z ) $ of a topological space $ X $ with coefficients in the ring of integers $ \mathbf Z $ is known as the Alexander–Kolmogorov product or the $ \cup $- product.

References

[1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2]	S. MacLane, "Homology" , Springer (1963)