A Cohen-Macaulay module is a type of module over a ring that satisfies specific depth conditions, making it an important object of study in commutative algebra and algebraic geometry. Such modules have the property that their depth equals their dimension, meaning they behave well with respect to regular sequences and exhibit desirable homological characteristics. This connection to depth highlights the module's structural integrity and the influence of regular sequences on its properties.
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Cohen-Macaulay modules arise naturally in the study of singularities and intersections in algebraic geometry.
If a module is Cohen-Macaulay, it implies that any finitely generated ideal over it is also Cohen-Macaulay, preserving this property through quotients.
One can determine whether a module is Cohen-Macaulay by analyzing its minimal free resolution and checking for conditions related to depth.
The concept of Cohen-Macaulay modules extends to graded modules, where graded pieces also maintain these depth properties.
Cohen-Macaulay modules often appear in contexts involving projective varieties, making them significant in understanding their geometric properties.
Review Questions
What properties characterize a Cohen-Macaulay module, and how do these properties relate to depth and regular sequences?
A Cohen-Macaulay module is characterized by having its depth equal to its dimension, which indicates a strong relationship with regular sequences. This means that there exists a regular sequence whose elements are non-zerodivisors on the module, allowing for coherent homological behavior. This property ensures that structures derived from these modules behave predictably under operations like localization and tensor products.
How does being a Cohen-Macaulay module influence the behavior of finitely generated ideals over that module?
When a module is Cohen-Macaulay, any finitely generated ideal over it will also share this property. This preservation means that one can utilize techniques from homological algebra to analyze the properties of these ideals. It allows for better control and understanding of their resolutions and syzygies since both ideals and modules retain similar structural characteristics, simplifying many proofs and applications in commutative algebra.
Evaluate the significance of Cohen-Macaulay modules in algebraic geometry, particularly in relation to projective varieties.
Cohen-Macaulay modules play a crucial role in algebraic geometry, especially when studying projective varieties. These varieties often exhibit Cohen-Macaulay properties due to their geometric configurations, leading to important insights into their singularities and intersection behavior. The rich interplay between algebraic properties provided by Cohen-Macaulay modules and geometric features allows mathematicians to formulate deeper results regarding dimension theory, intersection theory, and even deformation theory within algebraic geometry.
A regular sequence is a sequence of elements in a ring such that each element is a non-zerodivisor on the quotient by the ideal generated by the previous elements.