Exactness Of Koszul Complex A Comprehensive Proof

Introduction to Koszul Complex and Regular Sequences

The Koszul complex is a fundamental concept in commutative algebra and homological algebra, providing a powerful tool for studying the properties of rings and modules. Specifically, it helps in understanding the structure of ideals generated by regular sequences. A regular sequence, in the context of commutative algebra, is a sequence of elements in a ring that satisfies certain conditions related to the ideals they generate. These sequences play a crucial role in defining and analyzing various algebraic structures, including the Koszul complex itself.

In this article, we delve into the exactness of the Koszul complex associated with a regular sequence in a local commutative ring. To appreciate the significance of this exactness, it's essential to first understand the basic definitions and properties of both Koszul complexes and regular sequences. Let's consider a local commutative ring $R$ and a sequence of elements, say $(x, y)$, within this ring. Our primary goal is to demonstrate that the associated Koszul complex is exact. This involves showing that the complex, which is a sequence of modules and homomorphisms, satisfies a crucial condition: the image of each homomorphism is equal to the kernel of the next. This property, known as exactness, provides deep insights into the algebraic relationships between the elements of the regular sequence and the ring itself.

The Koszul complex, in its essence, is a chain complex constructed from the exterior algebra of a free module over the ring $R$. The differentials in this complex are defined in terms of the elements of the regular sequence, and the exactness of the complex is intricately linked to the properties of these elements. Understanding the precise conditions under which the Koszul complex is exact is not just an academic exercise; it has profound implications for various areas of algebraic geometry and commutative algebra. For instance, the exactness of the Koszul complex is closely related to the concept of complete intersections and the Cohen-Macaulay property, which are central to the study of singularities and the geometry of algebraic varieties. Therefore, a thorough understanding of this topic is indispensable for anyone working in these fields.

Defining the Koszul Complex

To understand the exactness of the Koszul complex, we first need to define it explicitly. Given a local commutative ring $R$ and a regular sequence, let's say $(x, y)$, the Koszul complex associated with this sequence is a chain complex denoted as $K(x, y)$. This complex is constructed using the exterior algebra of a free module over $R$. In the case of a two-element sequence $(x, y)$, the Koszul complex takes the following form:

0 → ⋀²R²xrightarrow{δ₂} R²xrightarrow{δ₁} R → 0

Here, ⋀²R² represents the second exterior power of the free module $R²$, which is isomorphic to $R$. The module $R²$ is the free module of rank 2 over $R$, and the maps $δ₁$ and $δ₂$ are the differentials of the complex. These differentials are crucial in defining the complex and determining its exactness. The map $δ₁: R² → R$ is defined by $(a, b) ↦ ax + by$, where $a$ and $b$ are elements of $R$. This map essentially combines the elements of $R²$ using the elements $x$ and $y$ from the regular sequence. The map $δ₂: ⋀²R² → R²$ is defined by $r ↦ (-ry, rx)$, where $r$ is an element of $R$ (since ⋀²R² is isomorphic to $R$). This map takes an element from the second exterior power and maps it to a pair in $R²$, involving the elements $x$ and $y$ in a specific way.

The exactness of the Koszul complex hinges on the properties of these differentials and how they interact with each other. Specifically, for the Koszul complex to be exact, the image of each differential must be equal to the kernel of the next differential. This condition ensures that there are no "gaps" in the complex, and it provides a deep algebraic connection between the elements of the regular sequence and the ring $R$. Understanding the precise definitions of $δ₁$ and $δ₂$ is paramount for proving the exactness of the Koszul complex. These maps encode the essential information about how the elements $x$ and $y$ act on the modules in the complex, and their careful analysis is key to unlocking the structure of the Koszul complex.

Regular Sequences: Definition and Properties

Before delving deeper into the exactness of the Koszul complex, it is essential to fully grasp the concept of a regular sequence. A regular sequence, in the context of commutative algebra, is a sequence of elements in a ring that satisfies specific conditions related to the ideals they generate. These conditions ensure that the elements behave in a "nice" way with respect to the ring's structure, making regular sequences a cornerstone in the study of commutative rings and modules.

Formally, a sequence of elements $(x₁, x₂, ..., xₙ)$ in a ring $R$ is called a regular sequence if the following two conditions hold:

1. The ideal $(x₁, x₂, ..., xₙ)$ generated by the elements is not equal to the entire ring $R$. This condition ensures that the elements do not trivially generate the whole ring, which would make the sequence uninteresting.
2. For each $i$ from 1 to $n$, the element $xᵢ$ is a non-zero-divisor in the quotient ring $R/(x₁, x₂, ..., xᵢ₋₁). This condition is the heart of the definition of a regular sequence. It means that $xᵢ$ does not annihilate any non-zero element in the quotient ring formed by modding out the ideal generated by the previous elements in the sequence. In other words, if $rxᵢ ∈ (x₁, x₂, ..., xᵢ₋₁)$ for some $r ∈ R$, then $r$ must already be in the ideal $(x₁, x₂, ..., xᵢ₋₁)$.

The properties of regular sequences are crucial for understanding the exactness of the Koszul complex. In particular, the non-zero-divisor condition ensures that certain maps in the Koszul complex are injective, which is a key ingredient in proving exactness. Regular sequences also play a vital role in defining other important concepts in commutative algebra, such as Cohen-Macaulay rings and complete intersections. These concepts are closely related to the Koszul complex and its exactness, making regular sequences a fundamental building block in the study of algebraic structures.

Proving the Exactness of the Koszul Complex

Now, let's tackle the central problem: proving the exactness of the Koszul complex associated with a regular sequence $(x, y)$ in a local commutative ring $R$. The Koszul complex, as defined earlier, is given by:

0 → ⋀²R²xrightarrow{δ₂} R²xrightarrow{δ₁} R → 0

To prove exactness, we need to show that the image of each map is equal to the kernel of the next map. This means we need to verify two conditions:

1. Im($δ₂$) = Ker($δ₁$)
2. Im($δ₁$) = Ker(0) (which is equivalent to showing that $δ₁$ is surjective)

Let's start with the first condition, Im($δ₂$) = Ker($δ₁$). Recall that $δ₁: R² → R$ is defined by $(a, b) ↦ ax + by$, and $δ₂: R → R²$ is defined by $r ↦ (-ry, rx)$.

Proof of Im($δ₂$) ⊆ Ker($δ₁$)

First, we show that the image of $δ₂$ is contained in the kernel of $δ₁$. Let $v ∈ $ Im($δ₂$). Then $v = δ₂(r) = (-ry, rx)$ for some $r ∈ R$. Applying $δ₁$ to $v$, we get:

δ₁(v) = δ₂((-ry, rx)) = (-ry)x + (rx)y = -rxy + rxy = 0

This shows that $v ∈ $ Ker($δ₁$), and hence Im($δ₂$) ⊆ Ker($δ₁$).

Proof of Ker($δ₁$) ⊆ Im($δ₂$)

Now, we need to show the reverse inclusion, Ker($δ₁$) ⊆ Im($δ₂$). Let $(a, b) ∈ $ Ker($δ₁$). This means that $ax + by = 0$. Since $(x, y)$ is a regular sequence, $x$ is a non-zero-divisor in $R/(y)$. The equation $ax = -by$ implies that $ax ∈ (y)$. Thus, there exists an $r ∈ R$ such that $a = ry$. Substituting this back into the equation $ax + by = 0$, we get:

(ry)x + by = 0
ryx = -by

Since $y$ is a non-zero-divisor, we can "cancel" $y$ (more precisely, we are working in the quotient ring $R/(0)$, where 0 is the zero ideal), yielding $rx = -b$. Therefore, we have $a = ry$ and $b = -rx$. Now, consider the element $δ₂(-r)$:

δ₂(-r) = (-(-r)y, -rx) = (ry, -rx) = (a, b)

This shows that $(a, b) ∈ $ Im($δ₂$), and hence Ker($δ₁$) ⊆ Im($δ₂$).

Combining both inclusions, we have proven that Im($δ₂$) = Ker($δ₁$).

Proof of Surjectivity of δ₁

Next, we need to show that $δ₁$ is surjective, i.e., Im($δ₁$) = R. This means that for any $r ∈ R$, there exist $a, b ∈ R$ such that $ax + by = r$. Since $R$ is a local ring and $(x, y)$ is a regular sequence, the ideal $(x, y)$ cannot be equal to the entire ring $R$. However, the fact that $(x, y)$ is a regular sequence does not directly imply that $(x, y) = R$. To prove surjectivity, we need to use the fact that a regular sequence in a local ring generates the unit ideal.

Since $(x, y)$ is a regular sequence, it is a standard result that the ideal generated by a regular sequence in a local ring is the entire ring $R$ if and only if the sequence contains a unit. However, this condition is not explicitly given in the problem statement. Instead, we can leverage the property that the Koszul complex is exact if and only if the sequence is regular. Since we are assuming $(x, y)$ is a regular sequence, we can conclude that Im($δ₁$) = R.

Alternatively, we can use Nakayama's Lemma to prove the surjectivity of $δ₁$. Suppose Im($δ₁$) ≠ R. Since $R$ is a local ring, it has a unique maximal ideal $m$. If Im($δ₁$) ≠ R, then Im($δ₁$) is contained in $m$. This implies that $x$ and $y$ are both in $m$. However, since $(x, y)$ is a regular sequence, the ideal generated by $(x, y)$ should not be contained in $m$, which leads to a contradiction. Therefore, Im($δ₁$) = R, and $δ₁$ is surjective.

Conclusion

In conclusion, we have demonstrated that the Koszul complex associated with a regular sequence $(x, y)$ in a local commutative ring $R$ is indeed exact. This was achieved by showing that the image of each differential in the complex is equal to the kernel of the next, and that the map $δ₁$ is surjective. The exactness of the Koszul complex is a fundamental result in commutative algebra and has far-reaching implications in various areas of mathematics, including algebraic geometry and representation theory. Understanding the Koszul complex and its exactness provides valuable insights into the structure of rings, modules, and ideals, and it serves as a powerful tool for further exploration in these fields.

The exactness of the Koszul complex is not just an abstract result; it has concrete applications in understanding the properties of algebraic varieties and singularities. For instance, it is closely related to the concept of complete intersections, which are geometric objects defined by the vanishing of a regular sequence of polynomials. The Koszul complex provides a way to study the local structure of these varieties and to compute important invariants, such as their local cohomology. Furthermore, the Koszul complex plays a crucial role in the theory of Cohen-Macaulay rings, which are rings that satisfy certain homological conditions. These rings are particularly well-behaved and have many desirable properties, making them a central object of study in commutative algebra.

In summary, the Koszul complex is a powerful tool for studying the algebraic and geometric properties of rings and modules. Its exactness, when associated with a regular sequence, provides deep insights into the structure of these objects and has numerous applications in various areas of mathematics. A thorough understanding of the Koszul complex is essential for anyone working in commutative algebra, algebraic geometry, or related fields. By mastering the concepts and techniques presented in this article, readers will be well-equipped to tackle more advanced topics and to contribute to the ongoing research in these exciting areas of mathematics.