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Tor from a projective resolution of the right module
Definition
For a right -module with a specified projective resolution and a left -module , define the right-resolution construction The datum remains in the notation until the balance and change-of-resolution results prove that it may be suppressed.
Depends on
Used by
- Degree-zero Tor is the tensor product in either construction Proposition
- Each resolution-defined Tor construction is covariant in both variables Proposition
- Positive Tor vanishes when the resolved variable is projective Proposition
- A left module is flat exactly when Tor one against every right module vanishes Theorem
- The left and right projective constructions of Tor are naturally isomorphic Theorem
- Tor is symmetric over a commutative ring Theorem
Dependency tree · two levels
7 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)