Nakayama’s Lemma and Some Applications

Algebraic Geometry

Nakayama’s lemma is a powerful and useful tool in algebraic geometry. In this post, we will consider various versions of Nakayama’s lemma and some applications.

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December 2, 2020

Nakayama’s lemma is a powerful and useful tool in algebraic geometry. It sort of plays the role of the inverse function theorem in differential geometry.

In this post, we will consider various versions of Nakayama’s lemma and some applications.

1 Nakayama’s Lemma

We start with a standard technique due to Atiyah and Macdonald ().

Lemma 1 (Generalized Cayley–Hamilton Theorem) Let M be a finitely generated A-module, a an ideal of A and ϕ an element in EndA(M) such that ϕ(M)aM. Then there exist elements rkak such that ϕn+r1ϕn1++rn1ϕ+rn=0.

Proof. Let x1, x2,, xn be a basis of M. Then a endomorphism ϕ is represented by matrix (αij), i.e. ϕ(xi)=j=1naijxj. Since ϕ(M)aM, we may assume that αija. Denote by δij the Kronecker delta. Then the above equality is equivalent to j=1n(δijϕαij)xj=0.

By multiplying both sides the adjugate matrix, we find that det(δijϕαij)=0.

The desired equality follows by applying the Laplace expansion to the determinant.

Lemma 2 (Nakayama’s Lemma) Let a be an ideal of a commutative ring A and M a finitely generated A-module. If M=aM, then there exists aa such that m=am for all mM.

Proof. Take ϕ=id in the Lemma , and let a=ri, we see that m=am.

Corollary 1 Let A be a commutative local ring, m the maximal ideal and M a finitely generated A-module. If mM=M, then M=0.

Proof. By Lemma , there is an element rm such that m+rm=0 for all mM. Since 1+r is an unit in A, multiplying the inverse (1+r)1, we find that that m=0.

2 A Geometric Version of Nakayama’s Lemma

A good resource of various versions and application of Nakayama’s lemma is Mumford’s red book ().

Corollary 2 Let A be a commutative local ring, m the maximal ideal and M a finitely generated A-module. Let f1, , fn be elements in M such that f¯1, , f¯n generate M/mM. Then f1, , fn generate M. In particular, generators of m/m2 also generates m.

Proof. Let N=(f1,,fn) be the submodule in M. Then we have (N+mM)/mM=M/mM. It then follows that N+mM=M (by the third isomorphism theorem or the snake lemma). Therefore, M/N=(N+mM)/N=m(M/N).

Apply Nakayama’s Lemma to M/N, we get M/N=0 and hence M=N.

Let X be scheme and F be a OX-module. Then the map HomOX(OX,F)Γ(X,F),wwX(1) is an isomorphism. Indeed, if sΓ(X,F), then there is a unique homomorphism w:OXF such that wU(1)=s|U. This defines an inverse map.

Let (si), iI be a family of sections siΓ(X,F). We say that F is generated by the family, if the corresponding homomorphism OX(I)F is surjective.

Equivalently, F is generated by the family (si), iI if the family generates Γ(X,F) and for any closed point yX, (si) generate the stalk Fy as an OX,y-module.

The above corollary implies the following geometric version of Nakayama’s lemma.

Lemma 3 (Geometric Nakayama’s Lemma) Let X be a Noetherian scheme, x a closed point in X, U an open neighborhood of x, and F a coherent sheaf on X. Let a1, , ar be sections in F(U) such that the germs a¯1, , a¯r in Fx generate the sheaf F|x=Fk(x). Then there exist an open neighborhood Spec(A)p such that a1|Spec(A), , ar|Spec(A) generate F|Spec(A).

In particular, if F|x=0, then there exists an open subset V in X such that F|V=0.

Proof. Let V be an affine open neighborhood of x in U. Then F|V=M~, Fx=MOX,x, and F|x=Mk(x), where M=F(V) is a finitely generated O(V)-module. Since a¯1, , a¯n generate F|x, by Corollary @ref(cor:LiftingSection), we know that a1, , an generate M=F(V). Moreover, a1, , an generates Fx.

Those sections define a morphism O|VnF|V such that the morphism at the stalk level OxnFx is surjective.

By shrinking V, we may assume that O|VnF|V is surjective (see Lemma 17.9.4 in the Stack Project).

Therefore, a1, , an generate F|V.

In the lemma, the condition can be relax to any scheme X and F is quasicoherent sheaf of finite type (see for example Proposition 1).

This geometric version has some useful corollaries.

Corollary 3 Let X be a Noetherian scheme and F a coherent sheaf on X. Then dimk(x)F|x is an upper semi-continuous function, i.e. the set {x|dimk(x)F|xr}X is open for any r.

Proof. It follows directly from .

Corollary 4 Let X be a reduced Noetherian scheme and F a coherent sheaf on X. Then F is a free OX-module in some neighborhood of x if and only if e(x)=dimk(x)F|x is a constant near x.

Proof. If F is free in a neighborhood of x, then e(x) is constant over that neighborhood.

Conversely, let U be a neighborhood of x such that e(y) is a constant for any yU. Then by shrinking U, we may assume that O|UeF|U is surjective. Let K be the kernel of this morphism. If K0, then there is a nonzero element s in K. Since KO|Ue and O|Ue is globally generated, we may view s as a global section of O|Ue. Since X is reduced, since X is reduced, the section s will be non-zero at some generic point yU. Localized at y, we get a short exact sequence 0KyOyeFy0, where Oy is a field because y is the generic point.

Because 0syKy. There is a contraction by comparing the ranks of the Oy-modules in the above short exact sequence.

Therefore, O|UeF|U is an isomorphism.

Corollary 5 Let X be a non-empty quasi-compact scheme and F is a coherent sheaf on X. If F is globally generated at all closed points, then F is globally generated at all points.

Proof. By Lemma , it suffices to show that there is a closed point y in the closure of every point x of X. Indeed, let U be an open neighborhood of y such that F|U is generated by it globally sections. Since yU, we see xU. Otherwise, if xUc, then y{x}Uc which implies a contradiction.

The fact that every point in X has a closed point in its closure follows from the quasi-compactness of X.

Since X is non-empty quasi-compact, it admits an irredundant finite open cover X=U1Ur. Restrict to {x}X, we get an irredundant finite open cover {x}=V1Vs. Because a prime ideal is always contained in a maximal ideal. In each affine open set, there is a closed point. Let x1 be a close point in V1. If it is a closed point of {x}, then we are done. Otherwise, let x2{x1} such that x2x1 and x2V1, where the closure is taken in the closed subset {x} but not in V1. With loss of generality, we may assume that x2V2. Repeating this procedures, as the open cover is finite, we know that there must be a closed point y in {x} which is also closed in X.

Note that without quasi-compactness, a closed subset of a scheme may not have a closed point (see Exercise 5.1.E).

3 The Terminology of Invertible Sheaf

We know that an invertible sheaf is a locally free rank 1 sheaf. The term invertible is from the following fact.

Proposition 1 Let X be a reduced Noetherian scheme. A coherent sheaf F is locally free of rank 1 if and only if there is another coherent sheaf G such that FOXG=OX.

Proof. If F is locally free of rank 1, then G=HomOX(F,OX) satisfies that FHomOX(F,OX)=HomOX(F,F)OX because F is locally free.

Conversely, for each point xX, we get FxOxGx=Ox. Tensor with k(x)=Ox/mx, we get F|xk(x)G|x=FxOxk(x)k(x)k(x)OxGx=k(x). Note that F|x=Fx/mxFx is a vector field over the residue field k(x). Therefore, F|x has rank 1 at every point xX. By Corollary @ref(cor:locally-free), F is locally free of rank 1.

A categorical definition of invertible sheaf can be found in 17.23 Invertible modules in the Stack Project.

4 The Theorem of the Square

Let f:XY be a morphism between schemes and F be a coherent sheaf on X. We denote Fy=F|f1(y).

Theorem 1 (Theorem of the Square) Let X be a complete varieties and Y a reduced Noetherian Scheme. Let L and M be two line bundles on X×Y. If for all closed points yY, we have LyMy there exists a line bundle N on Y such that LMpN, where p:X×YY is the projection onto Y.

Proof. From the statement, we are expect that N=p(LM1). Indeed, this is true. First, because X is complete and LyMy1 is trivial, then p(LM1)y=k(y) for any yY. By Grauert’s Theorem, we know that p(LM1) is locally free and moreover of rank 1.

We now show that the natural morphism α:pp(LM1)LM1 is an isomorphism. Let q be the restriction of p on the fiber Xy. Then by diagram chasing and the assumption, we have an isomorphism along the fiber Xy. (pp(LM1))|y=qq((LM1)|y)=OXy(LM1)|y=OXy. Consequently, for each closed point (x,y)X×Y, the morphism pp(LM1)|(x,y)OX×Y|(x,y) is an isomorphism.

Therefore, by Nakayama lemma, pp(LM1)OX×Y is surjective. Because pp(LM1) is of the rank 1, then the surjective morphism pp(LM1)OX×Y must be an isomorphism, i.e. pp(LM1)OX×Y.

Corollary 6 (See-Saw Principle) Suppose that, in addition to the hypotheses of Theorem of the Square, LxMx for some xX. Then LM.

Proof. Let N be the line bundle on Y such that L=MpN. Over {x}×Y, we have LxMx(pN)x. Therefore, (pN)xN is trivial, which implies that pN is trivial and LM.

The see-saw principle is very useful to prove the Theorem of cube which says that an invertible sheaf on the product X1×X2×X3 of three complete varieties is trivial if it is trivial along fibers to the three projections pi:X1×X2×X3Xi. Interested reader may find a proof, for example, in ().

References

Atiyah, Michael Francis, and Ian Grant Macdonald. 1969. Introduction to Commutative Algebra. Reading, Mass.: Addison-Wesley.
Milne, James S. 2008. “Abelian Varieties (V2.00).”
Mumford, David. 1999. The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobinians. 2nd expanded ed. Lecture Notes in Mathematics 1358. Berlin ; New York: Springer.
Serre, Jean-Pierre. 1955. “Faisceaux Algebriques Coherents.” Annals of Mathematics 61 (2): 197–278. https://doi.org/bsnrv2.
Vakil, Ravi. 2017. “Foundations of Algebraic Geometry.” http://math.stanford.edu/~vakil/216blog/index.html.