蛇引理

叙述

考虑一阿贝尔范畴 A （例如阿贝尔群或模的范畴）中的交换图：

使得每一横列均为正合序列。此时存在一个联系 a,b,c 的核与上核的正合序列： （公式略）

此外，若 f 是单射，则 （公式略） 亦然；若 g' 是满射，则 coker b→ coker c 亦然。

引蛇出洞

并注意到：引理给出的正合序列可在此图中画成倒S状的蛇形。

构造连接同态

核间的同态与上核间的同态很容易构造，它们由该图的交换性自然导出，正合性也可以直接代定义验证。重点在于连接同态 d 及序列在该处的正合性。

对于模范畴的情形，同态 d 可如是构造：

选定 （公式略），并视之为 C 的元素；由于 g 是满射，存在 y∈ B 满足 g(y)=x。由图的交换性，我们有 g'(b(y))=c(g(y))=c(x)=0 （因为 （公式略））

于是 （公式略）。由于底部的横列正合，存在 z∈ A' 使得 f'(z)=b(y)。置 d(x):=z+im (a)。今须验证 d 是明确定义的，即 d(x) 不依赖 y,z 之选取；此外尚须验证它是个同态，及序列的正合性。

一旦完成以上几点验证，即证明了此引理在模范畴的情形。对一般情形，可利用核与上核的泛性；此外也能使用Mitchell嵌入定理，此定理断言任一阿贝尔范畴都能迁入某个环 R 的 R -模范畴。

函子性

在应用上，我们常常需要长正合列的“函子性”或曰“自然性”（就自然变换意义言之）；各种建构的函子性也是同调代数的基本哲学。此函子性可由蛇引理的函子性导出。

设交换图

The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms.

Statement

In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram:

where the rows are exact sequences and 0 is the zero object.

Then there is an exact sequence relating the kernels and cokernels of a, b, and c: （公式略） where d is a homomorphism, known as the connecting homomorphism.

Furthermore, if the morphism f is a monomorphism, then so is the morphism （公式略）, and if g' is an epimorphism, then so is （公式略）.

The cokernels here are: coker a=A'/im a, coker b=B'/im b, coker c=C'/im c.

Explanation of the name

To see where the snake lemma gets its name, expand the diagram above as follows:

and then the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering snake.

Construction of the maps

The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a connecting homomorphism d exists which completes the exact sequence.

In the case of abelian groups or modules over some ring, the map d can be constructed as follows:

Pick an element x ker c and view it as an element of C. Since g is surjective, there exists y B with g(y)=x. By commutativity of the diagram, we have g'(b(y))=c(g(y))=c(x)=0 (since x is in the kernel of c), and therefore b(y) is in the kernel of g'. Since the bottom row is exact, we find an element z A' with f'(z)=b(y). By injectivity of f', z is unique. We then define d(x)=z+im (a).

Naturality

In the applications, one often needs to show that long exact sequences are "natural" (in the sense of natural transformations). This follows from the naturality of the sequence produced by the snake lemma.

If

is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form

Example

Let k be field, V be a k -vector space. V k[t] -module by t:V→ V being a k -linear transformation, so we can tensor V and k over k[t]. V⊗ _(k[t])k=V⊗ _(k[t])(k[t]/(t))=V/tV=coker (t).

Given a short exact sequence of k -vector spaces 0→ M→ N→ P→ 0, we can induce an exact sequence M⊗ _(k[t])k→ N⊗ _(k[t])k→ P⊗ _(k[t])k→ 0 by right exactness of tensor product. But the sequence 0→ M⊗ _(k[t])k→ N⊗ _(k[t])k→ P⊗ _(k[t])k→ 0 is not exact in general. Hence, a natural question arises. Why is this sequence not exact?

According to the diagram above, we can induce an exact sequence （公式略） by applying the snake lemma. Thus, the snake lemma reflects the tensor product's failure to be exact.

In the category of groups

Whether the snake lemma holds in the category of groups depends on the definition of cokernel. If f:A→ B is a homomorphism of groups, the universal property of the cokernel is satisfied by the natural map B→ B/N(im f), where N(im f) is the normalization of the image of f. The snake lemma fails with this definition of cokernel: The connecting homomorphism can still be defined, and one can write down a sequence as in the statement of the snake lemma. This will always be a chain complex, but it may fail to be exact.

If one simply replaces the cokernels in the statement of the snake lemma with the (right) cosets A'/im a,B'/im b,C'/im c', the lemma is still valid. The quotients however are not groups, but pointed sets (a short sequence (X,x)→ (Y,y)→ (Z,z) of pointed sets with maps f:X→ Y and g:Y→ Z is called exact if f(X)=g⁻¹(z)).

Counterexample to snake lemma with categorical cokernel

Consider the alternating group A₅: this contains a subgroup isomorphic to the symmetric group S₃, which in turn can be written as a semidirect product of cyclic groups: （公式略）.

In popular culture

The proof of the snake lemma is taught by Jill Clayburgh's character at the very beginning of the 1980 film It's My Turn.