Differentials and Cotangent Sheaves

Now we define the notion of differentials. Since analytical limits do not exist on algebraic varieties or schemes, defining them requires a purely algebraic approach, whose starting point is the Kähler differential module constructed in commutative algebra. In this post, we glue them into the cotangent sheaf of a scheme morphism, and through computations on tangent sheaves, Zariski tangent spaces, affine spaces, and projective spaces, we proceed to the definition of the canonical sheaf and the statement of Serre duality.

Kähler Differential Modules and Cotangent Sheaves

First, let us recall the algebraic tools at the affine level. For a ring \(A\) and an \(A\)-algebra \(B\), associated to \(B\) over \(A\), the Kähler differential module \(\Omega_{B/A}\) and the universal \(A\)-derivation \(d:B \rightarrow \Omega_{B/A}\) are defined. ([Commutative Algebra] §Differentials, ⁋Definition 3) Representing the functor of \(A\)-derivations \(\Der_A(B, -)\), this is the \(B\)-module characterized for any \(B\)-module \(M\) by the natural isomorphism

\[\Der_A(B, M)\cong \Hom_B(\Omega_{B/A}, M)\]

([Commutative Algebra] §Differentials, ⁋Lemma 2). That is, \(\Omega_{B/A}\) is generated by the elements \(\dd{b}\), with relations given by the Leibniz rule \(\dd{(xy)}=x\dd{y}+y\dd{x}\) and \(A\)-linearity, as a \(B\)-module; intuitively, \(\Omega_{B/A}\) can be thought of as regarding the elements of \(A\) as constants and each element of \(B\) as a function, and then imposing the Leibniz rule accordingly.

Since the above \(B\)-module comes from the structure of \(B\) as an \(A\)-algebra, or in other words from the ring homomorphism \(A\rightarrow B\), this can be viewed as a scheme morphism \(\Spec B\rightarrow \Spec A\). Then the \(B\)-module \(\Omega_{B/A}\) defined above translates, in the language of schemes, into a sheaf on \(\Spec B\), namely the quasi-coherent sheaf \(\widetilde{\Omega_{B/A}}\), and for a general scheme morphism \(\varphi:X\rightarrow S\), one glues these together to obtain \(\Omega_{X/S}\). (§Morphisms of Schemes, ⁋Proposition 1) Then, according to the intuition above, this can be understood as taking differentials while keeping the base \(S\) direction fixed and regarding only the fiber direction of \(\varphi:X\rightarrow S\) as functions.

Before beginning the main discussion, we first bring the algebraic tools into the form we will use. The Kähler differential module possesses the following two key exact sequences.

Proposition 1 (Cotangent exact sequence) Let an \(A\)-algebra \(B\) and a \(B\)-algebra \(C\) be given. Via the composition \(A \rightarrow B \rightarrow C\), viewing \(C\) as an \(A\)-algebra, the sequence of \(C\)-modules

\[\Omega_{B/A}\otimes_BC \rightarrow \Omega_{C/A} \rightarrow \Omega_{C/B} \rightarrow 0\]

is exact.

Proof

This exact sequence is precisely the cotangent sequence obtained in [Commutative Algebra] §Differentials, ⁋Proposition 8 (Cotangent sequence) by setting \(E=B\), \(E'=C\), and the base ring to \(A\).

The remaining exact sequence is as follows.

Proposition 2 (Conormal exact sequence) Given an \(A\)-algebra \(B\) with an ideal \(\mathfrak{a}\), let \(C=B/\mathfrak{a}\). Then the sequence of \(C\)-modules

\[\mathfrak{a}/\mathfrak{a}^2 \overset{\bar{d}}{\rightarrow} \Omega_{B/A}\otimes_BC \rightarrow \Omega_{C/A} \rightarrow 0\]

is exact, and the first morphism \(\bar{d}\) is given by \(f+\mathfrak{a}^2\mapsto \dd{f}\otimes 1\).

Proof

It suffices to apply [Commutative Algebra] §Differentials, ⁋Proposition 9 to the surjection \(\phi:B \rightarrow C=B/\mathfrak{a}\).

As mentioned above, we must glue these together to define Kähler differentials for a scheme morphism. Since this essentially requires reducing each of the base and the fiber to the affine case, we need gluing conditions for each of them.

Since the \(\Omega\) defined above is functorial with respect to commutative squares of ring homomorphisms ([Commutative Algebra] §Differentials, ⁋Proposition 6), an \(A\)-algebra homomorphism \(B \rightarrow B'\) induces a canonical \(B'\)-linear map \(\Omega_{B/A}\otimes_BB' \rightarrow \Omega_{B'/A}\), and a base change \(A \rightarrow A'\) induces a canonical \(B\)-module homomorphism \(\Omega_{B/A} \rightarrow \Omega_{B/A'}\). What is needed for gluing are the following cases where these canonical maps become isomorphisms.

Proposition 3 For an \(A\)-algebra \(B\), the following hold:

1. For any \(g\in B\), the canonical map gives a \(B_g\)-module isomorphism \((\Omega_{B/A})_g\cong\Omega_{B_g/A}\).
2. If the image of \(h\in A\) is invertible in \(B\) so that \(B\) can be viewed as an \(A_h\)-algebra, then the canonical map gives an isomorphism \(\Omega_{B/A}\cong\Omega_{B/A_h}\).

Proof

The first assertion is [Commutative Algebra] §Differentials, ⁋Proposition 7 for the multiplicative subset \(S=\{1, g, g^2,\ldots\}\), since the left-hand side \(\Omega_{B/A}\otimes_BB_g\) of that statement is equal to \((\Omega_{B/A})_g\). ([Commutative Algebra] §Properties of Localization, ⁋Lemma 1)

For the second assertion, applying Proposition 1 (Cotangent exact sequence) to the \(A\)-algebra \(A_h\) and the \(A_h\)-algebra \(B\) yields an exact sequence of \(B\)-modules

\[\Omega_{A_h/A}\otimes_{A_h}B \rightarrow \Omega_{B/A} \rightarrow \Omega_{B/A_h} \rightarrow 0\]

However, from the Leibniz rule, the universal derivation \(d:A \rightarrow \Omega_{A/A}\) satisfies

\[\dd{(1)}=\dd{(1\cdot 1)}=2\dd{(1)},\]

that is, \(\dd{(1)}=0\), and since it is \(A\)-linear, for any \(a\in A\) we have \(\dd{a}=a\cdot \dd{(1)}=0\), so \(\Omega_{A/A}=0\). Applying the first assertion to \(B=A\) and \(g=h\), we obtain \(\Omega_{A_h/A}\cong(\Omega_{A/A})_h=0\), so the first term of the sequence above vanishes, and the remaining map \(\Omega_{B/A} \rightarrow \Omega_{B/A_h}\) is an isomorphism.

Now, whenever \(\varphi(U)\subseteq V\) for affine open subsets \(U=\Spec B\subseteq X\) and \(V=\Spec A\subseteq S\), we need to glue together the local models \(\widetilde{\Omega_{B/A}}\) defined for each such pair. Gluing of this type was already encountered when constructing fiber products in §Fiber Products, ⁋Theorem 8, where we also verified restrictions on the factor side and on the base side before gluing the pieces together.

The actual gluing argument is as follows. First, shrinking \(U\) to the principal open subset defined by \(g\in B\), namely \(D(g)\), the first assertion of Proposition 3 gives

\[\Omega_{B_g/A}\cong (\Omega_{B/A})_g\]

so that \(\widetilde{\Omega_{B/A}}\vert_{D(g)}\cong \widetilde{\Omega_{B_g/A}}\) holds by §Quasi-Coherent Sheaves, ⁋Proposition 5. Shrinking the base side is similar: if we replace \(V\) with a principal open subset satisfying \(\varphi(U)\subseteq D(h)\), namely \(D(h)=\Spec A_h\) (\(h\in A\)), then the image of \(h\) does not belong to any prime ideal of \(B\) and is therefore invertible, so by the second assertion of Proposition 3, we obtain \(\Omega_{B/A}\cong\Omega_{B/A_h}\), and the local model does not change. Since these are all canonical isomorphisms arising from functoriality, we can cover the overlap of two charts by open sets that are principal on both sides to canonically identify the local models, and these identifications satisfy the cocycle condition on triple intersections. Thus the following definition uniquely determines a sheaf on \(X\). ([Topology] §Sheaves, ⁋Proposition 8)

Definition 4 For a morphism of schemes \(\varphi:X \rightarrow S\), the cotangent sheaf or sheaf of relative differentials on \(X\), \(\Omega_{X/S}\), is defined by setting, for all affine open subsets satisfying \(\varphi(U)\subseteq V\), namely \(U=\Spec B\subseteq X\) and \(V=\Spec A\subseteq S\),

\[\Omega_{X/S}\vert_U=\widetilde{\Omega_{B/A}}\]

to be the resulting \(\mathcal{O}_X\)-module. (§Quasi-Coherent Sheaves, ⁋Definition 4)

What this \(\Omega_{X/S}\) measures is revealed when we view \(\varphi\) as a family parametrized by \(S\) and change the base. The tool for this is that \(\Omega_{X/S}\) behaves well under base change.

Proposition 5 (Base change) For an \(A\)-algebra \(B\) and an \(A\)-algebra \(A'\), let \(B'=B\otimes_AA'\). Then the canonical \(B'\)-module isomorphism

\[\Omega_{B'/A'}\cong\Omega_{B/A}\otimes_BB'\]

holds.

Proof

For any \(B'\)-module \(M\), restricting an \(A'\)-derivation \(D:B' \rightarrow M\) to \(B\) gives an \(A\)-derivation \(B \rightarrow M\). Conversely, any \(A\)-derivation \(D_0:B \rightarrow M\) extends uniquely by the Leibniz rule to \(b\otimes a'\mapsto a'D_0(b)\), so this restriction gives \(\Der_{A'}(B', M)\cong\Der_A(B, M)\). By [Commutative Algebra] §Differentials, ⁋Lemma 2, the left-hand side of this isomorphism is \(\Hom_{B'}(\Omega_{B'/A'}, M)\), and since the right-hand side is

\[\Hom_B(\Omega_{B/A}, M)\cong\Hom_{B'}(\Omega_{B/A}\otimes_BB', M)\]

([Algebraic Structures] §Change of Scalars, ⁋Proposition 6), the two \(B'\)-modules represent the same functor and are therefore canonically isomorphic ([Category Theory] §Representable Functors, ⁋Proposition 8).

Since both sides are determined by their values on affine open sets and the identification is canonical, coming from the universal derivation, this isomorphism glues at the level of schemes. That is, for a morphism \(S' \rightarrow S\), letting \(X'=X\times_SS'\) and the projection \(\pi:X' \rightarrow X\), we have \(\Omega_{X'/S'}\cong \pi^\ast\Omega_{X/S}\). In particular, for a point \(s\in S\), setting \(S'=\Spec\kappa(s)\), \(X'\) is the fiber over \(s\), \(X_s\) (§Fiber Products, ⁋Definition 12); denoting the canonical morphism of this fiber into \(X\) by \(\iota:X_s \rightarrow X\), we obtain

\[\iota^\ast\Omega_{X/S}\cong\Omega_{X_s/\kappa(s)}\]

As the simplest example, for \(S=\Spec \mathbb{K}[\x]\) and \(X=\Spec \mathbb{K}[\x, \y]\), if we consider the projection \(X \rightarrow S\) onto the first coordinate, \(\Omega_{X/S}\) is a free module with basis \(\dd{\y}\) and rank \(1\) ([Commutative Algebra] §Differentials, ⁋Proposition 5).

When the family is non-trivial, \(\Omega_{X/S}\) records even the geometry of the fibers. For \(A=\mathbb{K}[t]\) and \(B=\mathbb{K}[t, \x, \y]/(\x\y-t)\), consider \(\varphi:X=\Spec B \rightarrow S=\Spec A\). Then the relation gives \(t=\x\y\), so \(B\cong \mathbb{K}[\x, \y]\) and \(X\) itself is an affine plane. However, to see this properly, it is best to examine how \(\varphi\) makes \(X\) into a family. Specifically, when \(a\in \mathbb{K}\) is not \(0\), over \(t=a\) the fiber \(X_a\) is the hyperbola \(\x\y=a\), but over \(t=0\) the fiber \(X_0\) degenerates into the two lines \(\x\y=0\).

Now let us see how these appear in \(\Omega_{X/S}\). Since \(B\) is the quotient of \(A[\x, \y]\) by the ideal \(\mathfrak{a}=(\x\y-t)\), applying Proposition 2 (Conormal exact sequence) to the \(A\)-algebra \(A[\x, \y]\) and its ideal \(\mathfrak{a}\) yields an exact sequence of \(B-modules\)

\[\mathfrak{a}/\mathfrak{a}^2 \overset{\bar{d}}{\rightarrow} \Omega_{A[\x, \y]/A}\otimes_{A[\x, \y]}B \rightarrow \Omega_{B/A} \rightarrow 0\]

Since the middle term \(\Omega_{A[\x, \y]/A}\) is a free module with basis \(\dd{\x}\) and \(\dd{\y}\) ([Commutative Algebra] §Differentials, ⁋Proposition 5), this term is \(B\dd{\x}\oplus B\dd{\y}\), and since \(\mathfrak{a}\) is generated by the single element \(\x\y-t\), its quotient \(\mathfrak{a}/\mathfrak{a}^2\) is as well. Now using that \(t\in A\) gives \(\dd{t}=0\), we have \(\bar{d}(\x\y-t)=\x \dd{\y}+\y \dd{\x}\), so this is the only relation, and

\[\Omega_{B/A}\cong\bigl(B\dd{\x}\oplus B\dd{\y}\bigr)/(\x \dd{\y}+\y \dd{\x})\]

Then we can see how this relation is resolved on each fiber. First, on \(t=a\), since \(\x\) is invertible, the relation becomes \(\dd{\y}=-(\y/\x)\dd{\x}\), and \(\Omega_{X_a/\mathbb{K}}\) is a free module with basis \(\dd{\x}\) and rank \(1\). On the other hand, on the fiber \(X_0\) at the origin, the situation is slightly different: in this case, for an \(x\)-axis point \(q\) (other than the origin), \(\mathfrak{m}_q\) does not contain \(\x\), and conversely, for a \(y\)-axis point \(q\), \(\mathfrak{m}_q\) does not contain \(\y\), so in both cases the relation is non-trivial over \(\kappa(q)\). That is, \(\Omega_{X/S}\otimes\kappa(q)\) is \(1\)-dimensional. At the origin \(p\) where the two lines meet, both \(\x\) and \(\y\) belong to \(\mathfrak{m}_p\), so the relation above vanishes and \(\Omega_{X/S}\otimes\kappa(p)\) becomes a vector space generated by \(\dd{\x}\) and \(\dd{\y}\), which is \(2\)-dimensional. In other words, the dimension of the fiber \(\Omega_{X/S}\otimes\kappa(q)\) jumps from \(1\) to \(2\) only at the point where the two lines meet, which is information not visible from the fact that on \(X\), taking differentials only relative to \(\mathbb{K}\), \(\Omega_{X/\mathbb{K}}\) is a free module of rank \(2\).

Although the definition above can be used immediately for calculations, it requires a choice of charts. A coordinate-free description was already obtained at the algebraic level, since [Commutative Algebra] §Differentials, ⁋Proposition 4 gives, for the multiplication \(m:B\otimes_AB \rightarrow B\) with kernel \(\mathfrak{I}\), the canonical isomorphism \(\mathfrak{I}/\mathfrak{I}^2\cong\Omega_{B/A}\). Translating this geometrically, \(\Spec(B\otimes_AB)\) is the fiber product of \(\Spec B\) over \(\Spec A\) (§Fiber Products, ⁋Lemma 2), and the morphism corresponding to \(m\) is the diagonal morphism \(\Delta\), so \(\mathfrak{I}\) is the ideal of \(\Delta\) and \(\mathfrak{I}/\mathfrak{I}^2\) is its conormal module. In other words, the differentials are read off from the first-order neighborhood of \(\Delta\), and transferring this to schemes yields a chart-independent description of \(\Omega_{X/S}\).

Proposition 6 For a separated morphism (§Valuation Rings, ⁋Definition 3) \(\varphi:X \rightarrow S\), the diagonal morphism \(\Delta:X \rightarrow X\times_SX\) is a closed embedding, and thus its image defines a closed subscheme of \(X\times_SX\). Letting \(\mathcal{I}\) be the ideal sheaf of this closed subscheme, the isomorphism with the conormal sheaf

\[\Omega_{X/S}\cong\Delta^\ast\bigl(\mathcal{I}/\mathcal{I}^2\bigr)\]

holds. Here, \(\Delta^\ast\) is the pullback (§Quasi-Coherent Sheaves, ⁋Definition 14).

Proof

Since both sides are determined by their values on affine charts, just as when defining \(\Omega_{X/S}\) above, it suffices to choose affine open charts \(U,V\) and show that the two sheaves are canonically identified on them. First, for an affine morphism \(\varphi:\Spec B \rightarrow \Spec A\), we compute \(\Delta^\ast(\mathcal{I}/\mathcal{I}^2)\). As seen above, in this case \(X\times_SX=\Spec(B\otimes_AB)\), and the diagonal morphism \(\Delta\) comes from the multiplication \(m:B\otimes_AB \rightarrow B\), \(b\otimes b'\mapsto bb'\). Now, letting \(\mathfrak{a}=\ker m\), the ideal sheaf of the image of \(\Delta\) is \(\widetilde{\mathfrak{a}}\), and from the exactness in §Quasi-Coherent Sheaves, ⁋Proposition 6, we have \(\mathcal{I}/\mathcal{I}^2\cong \widetilde{\mathfrak{a}/\mathfrak{a}^2}\).

Now let us show that \(\mathfrak{a}/\mathfrak{a}^2\cong \Omega_{B/A}\) as \(B\)-modules. For this, we first consider the \(A\)-linear map induced by the \(A\)-bilinear map \((b, c)\mapsto c\dd{b}\):

\[\theta:B\otimes_AB \rightarrow \Omega_{B/A};\qquad b\otimes c\mapsto c\dd{b}\]

Then for any \(b, b', c, c'\in B\),

\[\begin{aligned} \theta\bigl((b\otimes c)(b'\otimes c')\bigr)&=cc'\dd{(bb')}=bcc'\dd{b}'+b'cc'\dd{b}\\ &=m(b\otimes c)\theta(b'\otimes c')+m(b'\otimes c')\theta(b\otimes c) \end{aligned}\]

and since both sides are additive with respect to each argument, \(\theta(uv)=m(u)\theta(v)+m(v)\theta(u)\) holds for any \(u, v\in B\otimes_AB\). In particular, if \(u, v\in\mathfrak{a}=\ker m\), the right-hand side vanishes, so \(\theta(\mathfrak{a}^2)=0\). Therefore, \(\theta\) descends to \(\mathfrak{a}/\mathfrak{a}^2 \rightarrow \Omega_{B/A}\), and this map sends \(b\otimes 1-1\otimes b\) to \(\dd{b}\).

Conversely, defining \(\delta:B \rightarrow \mathfrak{a}/\mathfrak{a}^2\) by \(\delta(b)=(b\otimes 1-1\otimes b)+\mathfrak{a}^2\), for any \(b, b'\in B\), in \(B\otimes_AB\) we have

\[\begin{aligned} (bb'\otimes 1-1\otimes bb')&=(b\otimes 1-1\otimes b)(1\otimes b')+(b'\otimes 1-1\otimes b')(b\otimes 1)\\ &\equiv b'(b\otimes 1-1\otimes b)+b(b'\otimes 1-1\otimes b')\pmod{\mathfrak{a}^2} \end{aligned}\]

so \(\delta\) is an \(A\)-derivation. Thus by the universal property, a map \(\Omega_{B/A} \rightarrow \mathfrak{a}/\mathfrak{a}^2\) is induced; since \(\mathfrak{a}\) is generated by elements of the form \(b\otimes 1-1\otimes b\) and \(\Omega_{B/A}\) is generated by the \(\dd{b}\), checking on generators verifies that this is the inverse of \(\theta\).

Meanwhile, setting \(R=B\otimes_AB\), since \(\Delta\) is the closed embedding coming from the surjection \(m:R \rightarrow R/\mathfrak{a}\cong B\), what the pullback \(\Delta^\ast\) does on the affine scheme is to apply \(-\otimes_RB\) to an \(R\)-module. However, because \(\mathfrak{a}\) annihilates \(\mathfrak{a}/\mathfrak{a}^2\), its \(R\)-module structure is already given via \(R/\mathfrak{a}=B\), so \((\mathfrak{a}/\mathfrak{a}^2)\otimes_RB\cong \mathfrak{a}/\mathfrak{a}^2\). Consequently, we obtain the isomorphisms

\[\Delta^\ast\widetilde{\mathfrak{a}/\mathfrak{a}^2}\cong \widetilde{\mathfrak{a}/\mathfrak{a}^2}\cong \widetilde{\Omega_{B/A}}\]

where \(\widetilde{(-)}\) on the far left is the associated sheaf on \(\Spec R\), and the other two are on \(\Spec B\).

In the case of a general \(\varphi\), if \(U=\Spec B\subseteq X\) and \(V=\Spec A\subseteq S\) are affine opens with \(\varphi(U)\subseteq V\), then \(\Delta(U)\subseteq U\times_VU\), which is an open subset of \(X\times_SX\). Restricting the ideal sheaf of the diagonal to this open subset again yields the kernel of the multiplication \(B\otimes_AB \rightarrow B\), so the above computation applies directly to give \(\Delta^\ast(\mathcal{I}/\mathcal{I}^2)\vert_U\cong \widetilde{\Omega_{B/A}}=\Omega_{X/S}\vert_U\). Since this identification is constructed canonically from the universal derivation, the identifications on different charts agree when shrinking or changing charts, and hence glue to a single global isomorphism.

As seen in the proof above, \(\mathcal{I}/\mathcal{I}^2\) is a sheaf on \(\Delta(X)\); since \(\Delta\) identifies \(X\) with its image, it is pulled back to a sheaf on \(X\) via \(\Delta^\ast\). Although in practical computations we compute it on affine opens as \(\widetilde{\Omega_{B/A}}\) according to Definition 4, the proposition above provides a coordinate-free description of this sheaf.

Just as algebraically \(\mathfrak{a}/\mathfrak{a}^2\) records first-order derivatives, the quotient \((B\otimes_AB)/\mathfrak{a}^{m+1}\) formed by higher powers records the Taylor expansion up to order \(m\), and the successive quotient \(\mathfrak{a}^m/\mathfrak{a}^{m+1}\) contains the \(m\)-th order terms. This is analogous to how on manifolds we read the cotangent space and higher-order terms via \(\mathfrak{m}_p\supset\mathfrak{m}_p^2\supset\cdots\) ([Manifolds] §Cotangent Space). Translating this geometrically leads to using the \(m\)-th infinitesimal neighborhood of the diagonal, \(\mathcal{O}_{X\times_SX}/\mathcal{I}^{m+1}\). Letting the two projections be \(p_1,p_2:X\times_SX\rightarrow X\) and setting

\[\mathcal{P}^m_{X/S}=p_{1\ast}\bigl(\mathcal{O}_{X\times_SX}/\mathcal{I}^{m+1}\bigr)\]

a function \(g\) on \(X\) gives a section \(j^m(g)\) of \(\mathcal{P}^m_{X/S}\) as the image of \(p_2^\ast g\), which we call the \(m\)-jet of \(g\). If we write \(g=a_0+a_1t+a_2t^2+\cdots\) near a point in terms of a local coordinate \(t\), then \(j^m(g)\) records \(a_0,\ldots,a_m\); thus \(j^{m-1}(g)\) vanishing at that point means that the order of vanishing of \(g\) is at least \(m\).

The two exact sequences from the previous section also carry over immediately to the sheaf level via the exactness of the associated sheaf functor (§Quasi-Coherent Sheaves, ⁋Proposition 6). For a composition of scheme morphisms \(X \rightarrow S' \rightarrow S\) and its first morphism \(\psi: X \rightarrow S'\), passing Proposition 1 (Cotangent exact sequence) to the associated sheaf on each affine open yields an exact sequence of \(\mathcal{O}_X\)-modules

\[\psi^\ast\Omega_{S'/S} \rightarrow \Omega_{X/S} \rightarrow \Omega_{X/S'} \rightarrow 0\]

and when a closed embedding \(\iota:Z\hookrightarrow Y\) is given by an ideal sheaf \(\mathcal{J}\), passing Proposition 2 (Conormal exact sequence) yields the conormal exact sequence

\[\mathcal{J}/\mathcal{J}^2 \rightarrow \iota^\ast\Omega_{Y/S} \rightarrow \Omega_{Z/S} \rightarrow 0\]

The middle term is the pullback of a sheaf on \(Y\) along \(\iota\) to \(Z\), and the first term \(\mathcal{J}/\mathcal{J}^2\) should also be understood as a sheaf on \(Z\). This is possible because \(\mathcal{O}_Y/\mathcal{J}\cong \iota_\ast \mathcal{O}_Z\) by §Closed Subschemes, ⁋Definition 5, and \(\mathcal{J}\) annihilates the quotient \(\mathcal{J}/\mathcal{J}^2\) so that its \(\mathcal{O}_Y\)-module structure is already given via \(\mathcal{O}_Y/\mathcal{J}\). Indeed, for an affine open subset \(\Spec B\subseteq Y\), \(\Spec A\subseteq S\) containing its image, and \(\mathfrak{a}=\mathcal{J}(\Spec B)\) and \(C=B/\mathfrak{a}\), the three terms are respectively the associated sheaves of \(\mathfrak{a}/\mathfrak{a}^2\), \(\Omega_{B/A}\otimes_BC\), and \(\Omega_{C/A}\), which recovers the sequence in Proposition 2 (Conormal exact sequence).

Since these two exact sequences are standard tools for computing sheaves of differentials, let us translate the algebraic intuition introduced earlier into geometric terms. We already noted that \(\Omega_{X/S}\) is differentiation carried out with the base directions fixed, treating only the fiber directions as functions. To examine the first exact sequence, suppose first that \(S'\rightarrow S\) maps \(s'\) to \(s\); then \(X_{s'}\) is always contained in \(X_s\). That is, in this situation, the directions measured over \(S'\) are narrower, and hence \(\Omega_{X/S'}\) is given as a quotient of \(\Omega_{X/S}\), where the part being quotiented out is the image of \(\psi^\ast \Omega_{S'/S}\).

In the second exact sequence, \(\mathcal{J}/\mathcal{J}^2\) is the conormal sheaf of \(Z\), and its dual corresponds to the normal bundle (more precisely, the normal sheaf) of \(Z\) in \(Y\). If \(S=\Spec A\), \(Y=\Spec A[\x_1,\ldots, \x_n]\), and \(\mathcal{J}\) is generated by \(f_1,\ldots, f_r\), then \(\iota^\ast\Omega_{Y/S}\) has \(\dd{\x_1},\ldots, \dd{\x_n}\) as a basis, and since \(\bar{d}\) is given by \(f_j\mapsto\sum_i(\partial f_j/\partial \x_i)\dd{\x_i}\), the matrix expressing this morphism with respect to these bases is precisely the Jacobian \((\partial f_j/\partial \x_i)\). That is, the differentials on \(Z\) are obtained by quotienting out the differentials of the ambient space by the part generated by the differentials of the equations, namely the directions normal to \(Z\), and dualizing this at a point recovers the description in [Algebraic Varieties] §Tangent Spaces and Smoothness, ⁋Proposition 2.

Tangent Sheaf and Zariski Tangent Space

Taking the dual of the cotangent sheaf gives the sheaf of tangent vectors. This corresponds to the tangent bundle on a variety.

Definition 7 For a scheme morphism \(\varphi:X \rightarrow S\), we define the tangent sheaf of \(X\) by

\[\mathcal{T}_{X/S}=\sHom_{\mathcal{O}_X}(\Omega_{X/S}, \mathcal{O}_X)\]

(§Quasi-Coherent Sheaves, ⁋Definition 2).

To see what this definition does, let us look at the affine case again. For \(X=\Spec B\) and \(S=\Spec A\), Definition 4 gives \(\Omega_{X/S}=\widetilde{\Omega_{B/A}}\). Since the global sections of \(\sHom\) constructed from this are \(\mathcal{O}_X\)-module homomorphisms, the global sections of \(\mathcal{T}_{X/S}\) are \(\Hom_{\mathcal{O}_X}(\widetilde{\Omega_{B/A}}, \widetilde{B})\cong\Hom_B(\Omega_{B/A}, B)\) (§Quasi-Coherent Sheaves, ⁋Theorem 7), and therefore from the definition we obtain

\[\mathcal{T}_{X/S}(X)\cong \Der_A(B, B)\]

([Commutative Algebra] §Differentials, ⁋Lemma 2). Repeating the same computation on the principal open set \(D(g)\), the first assertion of Proposition 3 yields \(\Der_A(B_g, B_g)\), so \(\mathcal{T}_{X/S}\) is the sheaf that locally collects the derivations of \(B\) over \(A\), and gluing them together also yields the general case.

\(\Omega_{B/A}\) corresponds by definition to differential \(1\)-forms. Then \(\Der_A(B,B)\) defined above corresponds to its dual, namely tangent vectors. Specifically, given any function \(b\in B\), an element of \(\Der_A(B,B)\), \(D\), gives its derivative \(D(b)\), and under the universal property, this correspondence is the same as specifying \(\dd{b}\mapsto D(b)\) to determine a \(B\)-linear map \(\Omega_{B/A} \rightarrow B\) ([Commutative Algebra] §Differentials, ⁋Lemma 2). That is, \(\Der_A(B, B)\) is by definition the dual module of \(\Omega_{B/A}\), namely \(\Hom_B(\Omega_{B/A}, B)\), and between the two modules the \(B\)-bilinear pairing

\[\langle -, -\rangle:\Omega_{B/A}\times \Der_A(B, B) \rightarrow B; \qquad \langle \dd{b}, D\rangle=D(b)\]

exists. Here, since \(\Omega_{B/A}\) is generated by elements of the form \(\dd{b}\), this pairing is determined by the equation above alone; if we fix \(D\) and vary \(b\), we obtain the operation of differentiating functions in the direction determined by \(D\), and conversely, if we fix \(b\) and vary \(D\), then \(\dd{b}\) becomes a function assigning the rate of change of \(b\) to each direction. This is the same structure as on a differentiable manifold, where a vector field assigns to a function its derivative, and a differential \(1\)-form takes a vector field as input to produce a function. How this pairing provides a dual basis is seen immediately in \(B=A[\x_1,\ldots, \x_n]\). In this case, \(\Omega_{B/A}\) is a free module with basis \(\dd{\x_1},\ldots, \dd{\x_n}\) ([Commutative Algebra] §Differentials, ⁋Proposition 5), and an \(A\)-derivation \(D\) is completely determined, by \(A\)-linearity and the Leibniz rule, by its values at \(\x_i\), namely \(D(\x_i)\), which, written using the dual basis, is expressed as \(D=\sum_iD(\x_i)\partial/\partial \x_i\).

On the other hand, while the two modules are dual to each other in this way when \(\Omega_{B/A}\) is a free module, in general information is lost upon taking the dual. In the previously seen case of \(A=\mathbb{K}[t]\) and \(B=\mathbb{K}[t, \x, \y]/(\x\y-t)\), an \(A\)-derivation must kill \(t\) and therefore must satisfy \(\x D(\y)+\y D(\x)=0\), and in \(B\cong \mathbb{K}[\x, \y]\), for \(w\in B\) the only solutions to this equation are \(D(\x)=\x w\) and \(D(\y)=-\y w\). That is, \(\Der_A(B, B)\) is a free module of rank \(1\) with basis \(\x\partial/\partial \x-\y\partial/\partial \y\), which, unlike the rank jumping to \(2\) at the origin for \(\Omega_{B/A}\), has rank \(1\) everywhere. Therefore, from \(\mathcal{T}_{X/S}\) one cannot recover \(\Omega_{X/S}\), and not only formal definitions such as universal properties, but also this difference in information makes the cotangent sheaf the more natural one.

For this reason, caution is required when computing the tangent space at a point as well. Given a \(\mathbb{K}\)-scheme \(X\) and a point \(x\), letting \(\kappa(x)\) be its residue field (§Schemes, ⁋Definition 5), we can consider the fiber \(\Omega_{X/\mathbb{K}}\otimes_{\mathcal{O}_X}\kappa(x)\) of the cotangent sheaf. Considering the fiber obtained by taking the dual first and then taking the fiber,

\[\mathcal{T}_{X/\mathbb{K}}\otimes_{\mathcal{O}_X}\kappa(x)\]

the canonical map

\[\mathcal{T}_{X/\mathbb{K}}\otimes_{\mathcal{O}_X}\kappa(x) \rightarrow \bigl(\Omega_{X/\mathbb{K}}\otimes_{\mathcal{O}_X}\kappa(x)\bigr)^\vee\]

always exists, and while this is an isomorphism if \(\Omega_{X/\mathbb{K}}\) is locally free in a neighborhood of \(x\), in general it is neither injective nor surjective. The previously seen family \(\x\y=t\) illustrates this difference: since \(\mathcal{T}_{X/S}\) is a free module of rank \(1\) generated by \(\x\partial/\partial \x-\y\partial/\partial \y\), its fiber at the origin \(p\) is \(1\)-dimensional, but this derivation itself vanishes at \(p\), so the canonical map above becomes the zero map, whereas the dual of \(\Omega_{X/S}\otimes\kappa(p)\) is the tangent space at the origin of the fiber \(X_0\), which is a \(2\)-dimensional vector space.

Therefore, between these two definitions, the correct one is the following definition.

Definition 8 Over a field \(\mathbb{K}\), for a scheme \(X\) and a point \(x\in X\), we define the Zariski tangent space at \(x\) by

\[T_xX=\bigl(\Omega_{X/\mathbb{K}}\otimes_{\mathcal{O}_X}\kappa(x)\bigr)^\vee=\Hom_{\kappa(x)}\bigl(\Omega_{X/\mathbb{K}}\otimes_{\mathcal{O}_X}\kappa(x), \kappa(x)\bigr)\]

This definition agrees with the description given in terms of the maximal ideal of the local ring on a variety. ([Algebraic Varieties] §Tangent Spaces and Smoothness, ⁋Definition 1) Suppose \(x\) is a point with residue field \(\kappa(x)=\mathbb{K}\), that is, a \(\mathbb{K}\)-rational point, and let \((\mathcal{O}_{X,x}, \mathfrak{m}_x)\) be its local ring. Analyzing the conormal exact sequence at the stalk shows that the canonical map \(\mathfrak{m}_x/\mathfrak{m}_x^2 \rightarrow \Omega_{X/\mathbb{K}}\otimes\kappa(x)\) is surjective; since this sequence has no reason to be exact on the left, injectivity must come from elsewhere. What provides it is the assumption that the point is \(\mathbb{K}\)-rational: in this case, \(\mathcal{O}_{X,x} \rightarrow \kappa(x)=\mathbb{K}\) splits as a \(\mathbb{K}\)-algebra homomorphism, so \(f\mapsto (f-\bar{f})+\mathfrak{m}_x^2\) becomes a \(\mathbb{K}\)-derivation and induces an inverse to the map above, whence \(\Omega_{X/\mathbb{K}}\otimes \kappa(x)\cong \mathfrak{m}_x/\mathfrak{m}_x^2\) holds. Then the Zariski tangent space is \((\mathfrak{m}_x/\mathfrak{m}_x^2)^\vee\), that is, the dual of the cotangent space \(\mathfrak{m}_x/\mathfrak{m}_x^2\). Whether the dimension at a point \(\dim_{\kappa(x)}T_xX\) is equal to the local dimension \(\dim \mathcal{O}_{X,x}\) at that point serves as the criterion for whether the point is nonsingular, and if \(\mathcal{O}_{X,x}\) is Noetherian, in general \(\dim_{\kappa(x)}T_xX\geq \dim \mathcal{O}_{X,x}\).

Sheaves of Differentials on Affine and Projective Spaces

To begin with, our simplest example is the following.

Proposition 9 For a ring \(A\), on the affine space \(\mathbb{A}^n_A=\Spec A[\x_1,\ldots, \x_n]\), the cotangent sheaf \(\Omega_{\mathbb{A}^n_A/A}\) is a free sheaf of rank \(n\)

\[\Omega_{\mathbb{A}^n_A/A}\cong \mathcal{O}_{\mathbb{A}^n_A}^{\oplus n}\]

having \(\dd{\x_1},\ldots, \dd{\x_n}\) as a basis.

Proof

Let \(B=A[\x_1,\ldots, \x_n]\). By Definition 4, we have \(\Omega_{\mathbb{A}^n_A/A}\cong \widetilde{\Omega_{B/A}}\), so it suffices to show that \(\Omega_{B/A}\) has \(\dd{\x_1},\ldots, \dd{\x_n}\) as a basis as a free \(B\)-module.

By definition, \(\Omega_{B/A}\) is generated by the elements \(\dd{f}\) (\(f\in B\)), and since \(d\) is an \(A\)-derivation, for any polynomial \(f\) the chain rule

\[\dd{f}=\sum_{i=1}^n\frac{\partial f}{\partial \x_i}\dd{\x_i}\]

holds. Therefore, \(\Omega_{B/A}\) is generated by \(\dd{\x_1},\ldots, \dd{\x_n}\). Meanwhile, to show that these are linearly independent over \(B\), we use the fact that for each \(j\), the \(j\)-th partial derivative \(\partial/\partial \x_j:B \rightarrow B\) is an \(A\)-derivation. By the universal property, this induces a \(B\)-linear map \(\partial_j:\Omega_{B/A} \rightarrow B\), and since \(\partial_j(\dd{\x_i})=\delta_{ij}\), if \(\sum_i b_i \dd{\x_i}=0\), applying \(\partial_j\) yields \(b_j=0\). Hence \(\dd{\x_1},\ldots, \dd{\x_n}\) is a free basis and \(\Omega_{B/A}\cong B^{\oplus n}\).

The same result holds even when the base is not affine. Since any scheme \(S\) is in a unique way a scheme over \(\Spec \mathbb{Z}\), we can define the relative affine space over \(S\) by the base change

\[\mathbb{A}^n_S=\Spec \mathbb{Z}[\x_1,\ldots, \x_n]\times_{\Spec \mathbb{Z}}S\]

and then over an affine open subset of \(S\), \(\Spec A\), this gives \(\mathbb{A}^n_A\). (§Fiber Products, ⁋Example 9) But by Definition 4, the cotangent sheaf is determined by the local model on each affine open subset of the base, so applying the above proposition to each chart yields \(\Omega_{\mathbb{A}^n_S/S}\cong \mathcal{O}_{\mathbb{A}^n_S}^{\oplus n}\).

That is, on affine space, the cotangent sheaf is a trivial bundle whose basis is formed by the differentials of the coordinate functions. When passing to projective space, the situation becomes more interesting: the cotangent sheaf of \(\mathbb{P}^n\) is no longer free, but it is expressed as a short exact sequence of twisting sheaves, namely the Euler exact sequence.

Theorem 10 (Euler exact sequence) Over a ring \(A\), for the projective space \(\mathbb{P}^n_A=\Proj A[\x_0,\ldots, \x_n]\) (§Projective Schemes, ⁋Definition 1), there exists a short exact sequence of \(\mathcal{O}_{\mathbb{P}^n_A}\)-modules:

\[0 \rightarrow \Omega_{\mathbb{P}^n_A/A} \rightarrow \mathcal{O}_{\mathbb{P}^n_A}(-1)^{\oplus(n+1)} \rightarrow \mathcal{O}_{\mathbb{P}^n_A} \rightarrow 0\]

Proof

For convenience of notation, we write \(\mathbb{P}^n=\mathbb{P}^n_A\) and work on the standard affine open subsets \(U_i=D_+(\x_i)\cong \Spec A[\x_0,\ldots, \x_n]_{(\x_i)}\). Since this coordinate ring, with variables \(\y^{(i)}_j=\x_j/\x_i\) (\(j\neq i\)), is a polynomial ring in \(n\) variables, \(U_i\) is an affine space over \(A\), namely \(\mathbb{A}^n_A\), and therefore \(\Omega_{\mathbb{P}^n/A}\vert_{U_i}\) is, by Proposition 9, a free sheaf having \(\dd{\y}^{(i)}_j\) (\(j\neq i\)) as a basis, of rank \(n\).

First, let us define a morphism \(\mathcal{O}(-1)^{\oplus(n+1)} \rightarrow \mathcal{O}\). Multiplying a section of \(\mathcal{O}(-1)\) by \(\x_i\) lands in \(\mathcal{O}\), and in this manner, considering the morphism

\[\mathcal{O}_{\mathbb{P}^n_A}(-1)^{\oplus(n+1)}\rightarrow \mathcal{O}_{\mathbb{P}^n_A};\qquad (s_0,\ldots, s_n)=\sum_{j=0}^n s_je_j\mapsto \sum_{j=0}^n s_j\x_j\tag{$\ast$}\]

it is surjective because on each \(U_i\), \(\x_i^{-1}e_i\) is sent to \(1\).

Now, to complete the exact sequence, we compute the kernel. On each \(U_i\), trivializing \(\mathcal{O}(-1)\) by \(\x_i^{-1}\), the above morphism is given by

\[(s_0,\ldots, s_n)\mapsto \sum_j s_j (\x_j/\x_i)\]

so the kernel is given by the condition \(\sum_j s_j(\x_j/\x_i)=0\) on the tuples \((s_0,\ldots, s_n)\). Here, since the coefficient of the \(i\)-th component is \(\x_i/\x_i=1\), \(s_i\) is uniquely determined by the other components, and the remaining \(s_j\) can vary freely. That is, with \(j\neq i\), the components \(s_j\) vary freely, so the kernel is a free module of rank \(n\).

Now, on \(U_i\), if we define the morphism \(\Omega_{\mathbb{P}^n/A}\vert_{U_i} \rightarrow \mathcal{O}(-1)^{\oplus(n+1)}\vert_{U_i}\) by

\[d\Bigl(\frac{\x_j}{\x_i}\Bigr)\longmapsto \frac{1}{\x_i}\Bigl(e_j-\frac{\x_j}{\x_i}e_i\Bigr)\]

the image of the right-hand side under \((\ast)\) is

\[\frac{1}{\x_i}\left(\x_j-\frac{\x_j}{\x_i}\x_i\right)=0\]

so its image is contained in the kernel computed above. Moreover, since both sides are indexed by \(j\neq i\) as free modules of rank \(n\) and a basis is mapped to a basis, the image of this morphism coincides precisely with the kernel of (\(\ast\)). These locally defined morphisms agree with each other on \(U_i\cap U_k\): indeed, using the two relations

\[\frac{\x_l}{\x_i}=\frac{\x_l}{\x_k}\cdot\frac{\x_k}{\x_i},\qquad d\Bigl(\frac{\x_k}{\x_i}\Bigr)=-\Bigl(\frac{\x_k}{\x_i}\Bigr)^2d\Bigl(\frac{\x_i}{\x_k}\Bigr)\]

to expand \(\dd{(\x_l/\x_i)}\) in terms of the basis on \(U_k\) and then applying the correspondence above, the \(e_k\) terms cancel to yield \(\x_i^{-2}(\x_ie_l-\x_le_i)\), which equals the value on \(U_i\). Therefore, these glue together to a global morphism \(\Omega_{\mathbb{P}^n/A} \rightarrow \mathcal{O}(-1)^{\oplus(n+1)}\) to form an exact sequence, and since exactness is a local property, they form a short exact sequence of sheaves.

As in Proposition 9, we can also extend the base to an arbitrary scheme. For \(\mathbb{P}^n_S=\mathbb{P}^n_\mathbb{Z}\times_{\Spec \mathbb{Z}}S\) and the projection \(\pi:\mathbb{P}^n_S \rightarrow \mathbb{P}^n_\mathbb{Z}\), if we define \(\mathcal{O}_{\mathbb{P}^n_S}(d)=\pi^\ast\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}(d)\), then the generators \(\x_i^d\) and transition functions \((\x_i/\x_j)^d\) on each chart transfer directly. In particular, when the base is affine, this agrees with the twisting sheaf in §Sheaf Cohomology of Schemes, ⁋Definition 6, and we also have \(\Omega_{\mathbb{P}^n_S/S}\cong\pi^\ast\Omega_{\mathbb{P}^n_\mathbb{Z}/\mathbb{Z}}\) by Proposition 5 (Base change). Meanwhile, if we consider the sequence obtained by applying Theorem 10 (Euler exact sequence) to \(A=\mathbb{Z}\):

\[0 \rightarrow \Omega_{\mathbb{P}^n_\mathbb{Z}/\mathbb{Z}} \rightarrow \mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}(-1)^{\oplus(n+1)} \rightarrow \mathcal{O}_{\mathbb{P}^n_\mathbb{Z}} \rightarrow 0\]

we have already verified in the proof above that \(\Omega_{\mathbb{P}^n_\mathbb{Z}/\mathbb{Z}}\) is, on each \(U_i\), a free sheaf of rank \(n\), and the other two sheaves are also locally free by definition; in particular, the last sheaf \(\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}\) is free and hence projective ([Multilinear Algebra] §Projective, Injective, and Flat Modules, ⁋Proposition 4), so this sequence splits on each \(U_i\) ([Multilinear Algebra] §Exact Sequences, ⁋Proposition 10). Since split exact sequences are preserved by additive functors and exactness is a local property, applying \(\pi^\ast\) yields, for an arbitrary scheme \(S\), the Euler exact sequence:

\[0 \rightarrow \Omega_{\mathbb{P}^n_S/S} \rightarrow \mathcal{O}_{\mathbb{P}^n_S}(-1)^{\oplus(n+1)} \rightarrow \mathcal{O}_{\mathbb{P}^n_S} \rightarrow 0\]

Canonical sheaf

When the cotangent sheaf is locally free, its top exterior power is a sheaf of rank \(1\), that is, an invertible sheaf. This unique invertible sheaf obtained in this way controls a significant part of the geometry of \(X\), and in the world of varieties, this corresponds to the canonical line bundle defined as the top exterior power of the cotangent bundle ([Algebraic Varieties] §Canonical Line Bundle, ⁋Definition 5). Even for schemes, given an \(\mathcal{O}_X\)-module, we have already defined its exterior power \(\bigwedge^r\mathcal{F}\) by sheafifying the exterior power on each open set (§Quasi-Coherent Sheaves, ⁋Definition 2), and showed that this preserves quasi-coherence and that for a rank \(n\) locally free sheaf \(\mathcal{E}\), \(\bigwedge^r\mathcal{E}\) is a locally free sheaf of rank \(\binom{n}{r}\). In particular, in the case \(r=n\), the determinant \(\det\mathcal{E}=\bigwedge^n\mathcal{E}\) is an invertible sheaf (§Quasi-Coherent Sheaves, §§Locally Free Sheaves and Invertible Sheaves).

Definition 11 Over a field \(\mathbb{K}\), let \(X\) be a scheme such that the cotangent sheaf \(\Omega_{X/\mathbb{K}}\) is a locally free sheaf of rank \(n\). Then on \(X\), we define the canonical sheaf \(\omega_X\) to be the top exterior power

\[\omega_X=\bigwedge\nolimits^n\Omega_{X/\mathbb{K}}=\det\Omega_{X/\mathbb{K}}\]

By the preceding observation, \(\omega_X\) is an invertible sheaf. More generally, for a morphism of schemes \(\varphi:X \rightarrow S\), when \(\Omega_{X/S}\) is a locally free sheaf of rank \(n\), we define the relative canonical sheaf \(\omega_{X/S}=\det\Omega_{X/S}\) by the same formula, and the case \(S=\Spec \mathbb{K}\) is the absolute case as above.

The tool used in practice to compute canonical sheaves is the fact that the determinant decomposes into a tensor product along a short exact sequence.

Proposition 12 Given a short exact sequence of locally free sheaves on a scheme \(X\)

\[0 \rightarrow \mathcal{E}' \rightarrow \mathcal{E} \rightarrow \mathcal{E}'' \rightarrow 0\]

suppose that \(\mathcal{E}'\) and \(\mathcal{E}''\) have ranks \(r\) and \(s\), respectively. Then \(\mathcal{E}\) is a locally free sheaf of rank \(r+s\), and there exists an isomorphism

\[\det\mathcal{E}\cong \det\mathcal{E}'\otimes_{\mathcal{O}_X}\det\mathcal{E}''\]

Proof

We first verify the rank. Since \(\mathcal{E}''\) is locally free, each point admits isomorphisms \(\mathcal{E}''\vert_U\cong\mathcal{O}_U^{\oplus s}\) and \(\mathcal{E}'\vert_U\cong\mathcal{O}_U^{\oplus r}\) on an open neighborhood \(U\). Because surjectivity of sheaves guarantees surjectivity only at the level of stalks, in order to lift each basis section of \(\mathcal{E}''\vert_U\) to a section of \(\mathcal{E}\), we may need to shrink \(U\) further. Since the basis is finite, intersecting the open neighborhoods to which each section lifts yields such a \(U\) again. On this shrunken \(U\), lifting the basis sections of \(\mathcal{E}''\vert_U\) to \(\mathcal{E}\vert_U\) yields a splitting of the surjection \(\mathcal{E}\vert_U \rightarrow \mathcal{E}''\vert_U\), so that \(\mathcal{E}\vert_U\cong\mathcal{E}'\vert_U\oplus\mathcal{O}_U^{\oplus s}\cong\mathcal{O}_U^{\oplus(r+s)}\), and hence \(\mathcal{E}\) is a locally free sheaf of rank \(r+s\).

We now construct a morphism

\[\lambda:\det\mathcal{E}'\otimes_{\mathcal{O}_X}\det\mathcal{E}'' \rightarrow \det\mathcal{E}\]

Given sections over an open set \(V\), namely \(\alpha\in(\det\mathcal{E}')(V)\) and \(\bar{t}_1\wedge\cdots\wedge\bar{t}_s\in(\det\mathcal{E}'')(V)\), we shrink \(V\) sufficiently to lift each \(\bar{t}_i\) to \(t_i\in\mathcal{E}(V)\), and then define

\[\lambda\bigl(\alpha\otimes(\bar{t}_1\wedge\cdots\wedge\bar{t}_s)\bigr)=\alpha\wedge t_1\wedge\cdots\wedge t_s\]

Here, \(\alpha\) on the right-hand side denotes the section transferred via the morphism induced by the inclusion \(\mathcal{E}'\hookrightarrow\mathcal{E}\), namely \(\det\mathcal{E}' \rightarrow \bigwedge^r\mathcal{E}\). This value is independent of the choice of lifts. Since the difference between two lifts is a section of \(\mathcal{E}'\), changing \(t_i\) to \(t_i+a_i\) (\(a_i\in\mathcal{E}'(V)\)) results in a difference that is a sum of terms having at least one \(a_i\) as a factor. However, on a local splitting, \(\alpha\) can be expressed on \(\mathcal{E}'\vert_V\) in terms of a basis \(f_1,\ldots, f_r\) as a multiple of \(f_1\wedge\cdots\wedge f_r\), and since each \(a_i\) is a linear combination of the \(f_j\) over \(\mathcal{O}_V\), such a term contains some \(f_j\) twice and vanishes. Therefore, the locally defined \(\lambda\) agree on overlaps and glue to a global morphism.

It suffices to check locally that \(\lambda\) is an isomorphism. On \(U\) as above, let \(\mathcal{E}'\vert_U\) have basis \(f_1,\ldots, f_r\), and let \(\mathcal{E}''\vert_U\) have basis \(\bar{g}_1,\ldots, \bar{g}_s\) with lifts \(g_1,\ldots, g_s\). Then by the splitting, \(f_1,\ldots, f_r, g_1,\ldots, g_s\) is a basis of \(\mathcal{E}\vert_U\). By [Multilinear Algebra] §Tensor Algebra, ⁋Proposition 13, \(\det\mathcal{E}\vert_U\) is, with basis \(f_1\wedge\cdots\wedge f_r\wedge g_1\wedge\cdots\wedge g_s\), a free sheaf of rank \(1\), and similarly \((\det\mathcal{E}'\otimes\det\mathcal{E}'')\vert_U\) has basis \((f_1\wedge\cdots\wedge f_r)\otimes(\bar{g}_1\wedge\cdots\wedge\bar{g}_s)\). Since \(\lambda\) sends the latter basis to the former basis, it is an isomorphism on \(U\), and thus also an isomorphism globally.

Applying this to the Euler exact sequence immediately computes the canonical sheaf of projective space.

Example 13 Over a field \(\mathbb{K}\), consider the projective space \(\mathbb{P}^n\). Applying Theorem 10 (Euler exact sequence) to \(A=\mathbb{K}\), we obtain the Euler exact sequence; as seen in its proof, \(\Omega_{\mathbb{P}^n/\mathbb{K}}\) is a free sheaf on each \(U_i=D_+(\x_i)\) with basis \(\dd{\y}^{(i)}_j\) (\(j\neq i\)) of rank \(n\), so it is locally free, and therefore \(\omega_{\mathbb{P}^n}\) is defined. Since the ranks of the three terms \(\Omega_{\mathbb{P}^n/\mathbb{K}}\), \(\mathcal{O}(-1)^{\oplus(n+1)}\), and \(\mathcal{O}_{\mathbb{P}^n}\) in the Euler exact sequence are \(n\), \(n+1\), and \(1\) respectively, by Proposition 12

\[\det\bigl(\mathcal{O}(-1)^{\oplus(n+1)}\bigr)\cong \omega_{\mathbb{P}^n}\otimes_{\mathcal{O}_{\mathbb{P}^n}}\det\mathcal{O}_{\mathbb{P}^n}\cong\omega_{\mathbb{P}^n}\]

holds. Repeatedly applying the same proposition to the direct sum, the left-hand side is \(\mathcal{O}(-1)^{\otimes(n+1)}\), and since the transition functions of \(\mathcal{O}(d)\) are \((\x_i/\x_j)^d\) (the description following §Sheaf Cohomology of Schemes, ⁋Definition 6), this is \(\mathcal{O}_{\mathbb{P}^n}(-n-1)\). Therefore, we obtain

\[\omega_{\mathbb{P}^n}\cong\mathcal{O}_{\mathbb{P}^n}(-n-1)\]

This agrees with the calculation performed using the transition functions of \(n\)-forms on varieties. ([Algebraic Varieties] §Canonical Line Bundle, ⁋Example 8)

The reason the canonical sheaf occupies a special place among invertible sheaves is that it mediates duality between cohomology groups. Just as the fundamental class gives Poincaré duality in topology, \(\omega_X\) plays that role on projective schemes.

Theorem 14 (Serre duality) Over an algebraically closed field \(\mathbb{K}\), for an \(n\)-dimensional integral projective scheme \(X\) (§Closed Subschemes of Projective Space, ⁋Definition 7, §Algebraic Structure of Schemes, §§Reduced Schemes and Integral Schemes), suppose that \(\Omega_{X/\mathbb{K}}\) is a locally free sheaf of rank \(n\). Then on \(X\), for any locally free sheaf \(\mathcal{E}\) and any \(0\leq i\leq n\), there exists an isomorphism

\[H^i(X, \mathcal{E})\cong H^{n-i}\bigl(X, \omega_X\otimes_{\mathcal{O}_X}\mathcal{E}^\vee\bigr)^\ast\]

Here, \(\mathcal{E}^\vee=\sHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{O}_X)\), and \((-)^\ast\) denotes the dual of a finite-dimensional \(\mathbb{K}\)-vector space. (§Sheaf Cohomology of Schemes, ⁋Theorem 9)

The proof of this theorem proceeds by showing that the pairing formed by the trace map and cup product on \(\mathbb{P}^n\) is a perfect pairing, and then transferring this to general \(X\) along a finite surjective morphism; since the detailed proof is beyond the scope of this post, we refer the reader to [Algebraic Varieties] §Serre Duality, §§Serre Duality on Projective Space.

Meanwhile, in the above discussion we required that \(\Omega_{X/\mathbb{K}}\) be locally free; if this does not hold, the rank is not constant, so there is no basis to even choose a top exterior power. This kind of generalization was also already treated in [Algebraic Varieties] §Serre Duality, §§Generalizations of Serre Duality, where one had to lift the isomorphism in the theorem to \(\Ext\) and introduce a separate dualizing sheaf. Moreover, although it is relatively less critical since the cases of our universal interest are \(\mathbb{K}=\mathbb{C}\), it is worth remembering the fact that if \(\mathbb{K}\) is not a perfect field, a problem may arise where the rank of \(\Omega_{X/\mathbb{K}}\) and the dimension of \(X\) differ. In general, the locally free assumption on \(\Omega_{X/S}\) is automatically satisfied if \(\varphi:X\rightarrow S\) is smooth, and examining this is the goal of §Smooth and Étale Morphisms.

---

References

[Har] R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics. Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundations of algebraic geometry. Available online.
[Eis] D. Eisenbud, Commutative algebra: with a view toward algebraic geometry. Springer, 1995.
[Stacks] The Stacks Project Authors, The Stacks Project. Available online.