Tor groups, often denoted as $$\text{Tor}$$, are derived functors that measure the extent to which a functor fails to be exact. They arise in homological algebra when considering the tensor product of modules, specifically identifying obstructions to exactness in the sequence of modules. The significance of Tor groups extends to various existence theorems, providing insight into the structure of modules and their relationships.
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Tor groups are calculated using projective resolutions of modules, enabling the identification of higher Tor functors, such as $$\text{Tor}^n$$.
The first Tor group, $$\text{Tor}^1(M,N)$$, measures the failure of flatness when tensoring modules $$M$$ and $$N$$.
Tor groups exhibit properties like being functorial in both arguments and satisfying the universal coefficient theorem for homology.
In specific cases, such as over a Noetherian ring, if a module is flat, then all Tor groups involving that module vanish.
Tor groups are crucial for various existence results, particularly in deriving characteristics of projective and injective modules.
Review Questions
How do Tor groups relate to the concept of exact sequences in homological algebra?
Tor groups provide insight into the failure of a sequence to remain exact during tensoring. Specifically, when you take a short exact sequence of modules and tensor it with another module, any non-zero Tor groups indicate that exactness is lost at some point. This relationship highlights how Tor serves as a tool to measure deviations from exactness and helps in understanding how modules interact under tensor operations.
In what ways can Tor groups be utilized to determine properties of modules over a Noetherian ring?
In the context of Noetherian rings, Tor groups can reveal information about flatness and projective properties of modules. If all Tor groups involving a given module vanish, it indicates that the module is flat. Additionally, analyzing these groups can assist in understanding how various modules relate to each other in terms of projective resolutions and their respective homological dimensions.
Evaluate how the properties of Tor groups inform our understanding of projective and injective modules.
The properties of Tor groups significantly enhance our comprehension of projective and injective modules by linking them to exactness and flatness criteria. For instance, if a module is projective, its associated Tor groups with any other module will vanish. Conversely, injective modules can often be characterized through their interactions with these groups. This interplay allows mathematicians to establish deeper relationships between module structures and their homological properties, leading to broader insights within category theory.
A construction that takes two modules and produces another module, allowing for the study of bilinear maps and providing a way to combine module structures.
Ext Functor: A functor that measures the extent to which a module can be embedded into another module, often used alongside Tor to analyze module homomorphisms.