The tensor product of a right and a left chain complex is totalized by direct sums with the Koszul differential

Depends on

• Chain complex in an abelian category
• The tensor product $M\otimes_R N$ from the additive group underlying the free $\mathbb Z$-module on $M\times N$, elementary tensors, and finite tensor sums

Used by

• Euler characteristic in a proper flat family is locally constant Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• A chain complex with coefficients obtained by tensoring Definition
• Bounded graded bimodule complexes and signed tensor totalization Definition
• Derived tensor product in the bounded above setting Definition
• Signed totalization of graded Aₘ-bimodule actions Definition
• The first-quadrant tensor double complex of two projective resolutions Definition
• The homology cross product for tensor complexes Definition
• Tor from a projective resolution of the left module Definition
• Tor from a projective resolution of the right module Definition
• Kunneth for two cyclic two-term complexes Example
• The simple module over dual numbers is not perfect Example
• Bounded above flat tensor complexes preserve quasi isomorphisms Lemma
• Finite projective complex for proper flat coherent cohomology Lemma
• Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
• Stalks, coproducts and right exactness of the abelian sheaf tensor product Lemma
• The cycle-boundary tensor sequence has the Kunneth kernel and cokernel Lemma
• The tensor-total differential is balanced, well defined, and squares to zero Lemma
• Universal finite projective cohomology complex over any base Lemma
• Kunneth Tor spectral sequence Theorem
• The Kunneth sequence splits nonnaturally Theorem
• The Kunneth theorem for free complexes over a PID Theorem
• The PID Kunneth sequence admits a section after choices Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Weibel, An Introduction to Homological Algebra (standard reference, not scraped)