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Tor from a projective resolution of the left module
Definition
For a right -module and a left -module with a specified projective resolution , define the left-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
- The simple module over dual numbers is not perfect Example
- Tor does not take two left modules over an arbitrary ring without extra bimodule structure False statement
- Tor is not symmetric as a typed expression over every noncommutative ring False statement
- Vanishing Tor one does not require a projective factor False statement
- Dimension of the kernel modulo n-minus equals the next term (BGG 10.7) Lemma
- Finite projective complex for proper flat coherent cohomology Lemma
- The coinvariant algebra is a complete intersection with one-dimensional socle Lemma
- Tor with the trivial module is computed by the weak BGG resolution Lemma
- Verma-filtered objects are acyclic for n-minus coinvariants Lemma
- 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 right module is flat exactly when Tor one against every left module vanishes Theorem
- Hyper-Tor spectral sequence Theorem
- Kunneth Tor spectral sequence 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)