Ext groups, denoted as $$\text{Ext}^n(A, B)$$, are mathematical constructions in homological algebra that measure the extent to which a module $A$ fails to be projective relative to another module $B$. They serve as a generalization of the concept of extensions and classify the equivalence classes of short exact sequences that can be formed with these modules. The presence of Ext groups provides insight into the structure and properties of modules, particularly in terms of their homomorphisms and extensions.
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Ext groups can be calculated using projective resolutions or injective resolutions of modules, linking them to other homological concepts.
The group $$\text{Ext}^1(A, B)$$ classifies extensions of module $B$ by module $A$, providing insight into how these modules can be combined.
If $$\text{Ext}^1(A, B) = 0$$, this implies that $A$ is a projective module in relation to $B$, meaning every short exact sequence involving them splits.
Ext groups satisfy certain functorial properties, allowing for their computation across different categories of modules and even between different rings.
The higher Ext groups $$\text{Ext}^n(A, B)$$ for $$n > 1$$ provide additional structure and information about more complex interactions between modules.
Review Questions
How do Ext groups relate to the concepts of projective and injective modules?
Ext groups provide important information about the relationship between modules, particularly regarding their projectivity and injectivity. For instance, if $$\text{Ext}^1(A, B) = 0$$, this indicates that the module $A$ is projective relative to $B$. Additionally, understanding how Ext groups behave can lead to insights into when extensions split or when a module can be represented as a direct sum, highlighting the importance of projective and injective properties in module theory.
Discuss the significance of $$\text{Ext}^1(A, B)$$ in classifying extensions of modules and how it impacts their structure.
$$\text{Ext}^1(A, B)$$ plays a crucial role in categorizing extensions because it describes the set of equivalence classes of extensions where module $B$ is extended by module $A$. This classification reveals whether certain modules can be combined without loss of properties. If there are non-trivial elements in $$\text{Ext}^1(A, B)$$, it indicates potential complexities in how these modules can be constructed together and may suggest deeper structural properties about the modules involved.
Evaluate the implications of non-zero higher Ext groups $$\text{Ext}^n(A, B)$$ for $$n > 1$$ on the relationships between modules A and B.
Non-zero higher Ext groups $$\text{Ext}^n(A, B)$$ for $$n > 1$$ indicate more complex relationships between modules $A$ and $B$, often reflecting higher-dimensional interactions beyond simple extensions. These higher groups can suggest intricate structures that may arise from multiple layers of extensions, highlighting how these modules interact over various levels. Understanding these implications can lead to insights into derived functors and more advanced homological techniques that explore the depths of module relationships.
An exact sequence is a sequence of module homomorphisms where the image of one map equals the kernel of the next, reflecting how modules relate to each other.