Kähler Differentials and Cotangent Sheaves

edit_note

This post is being revised. The text below is the version as of 2026-08-11, so some statements may be out of date and references to this post from other posts may not line up.

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

We now define the notion of a differential. Since analytic limits do not exist on an algebraic variety or a scheme, we must adopt a purely algebraic approach; the starting point is the Kähler differential module constructed in commutative algebra. In this post, we attach this as the cotangent sheaf of a scheme morphism, pass through the tangent sheaf and Zariski tangent space, carry out computations in affine and projective space, and finally organize the definition of the canonical sheaf and the statement of Serre duality.

Kähler Differentials and Cotangent Sheaves

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

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

for any \(B\)-module \(M\). ([Commutative Algebra] §Differentials, ⁋Lemma 2) That is, \(\Omega_{B/A}\) is the \(B\)-module generated by elements \(\dd{b}\) with the Leibniz rule \(\dd{(xy)}=x\dd{y}+y\dd{x}\) and \(A\)-linearity as relations; intuitively, one may think of \(\Omega_{B/A}\) as viewing elements of \(A\) as constants and elements of \(B\) as functions, then imposing the Leibniz rule accordingly.

Since the above \(B\)-module arises from the \(A\)-algebra structure of \(B\), or in other words from the ring homomorphism \(A\rightarrow B\), we may regard it 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 the quasi-coherent sheaf \(\widetilde{\Omega_{B/A}}\) on \(\Spec B\), and for a general scheme morphism \(\varphi:X\rightarrow S\) we glue these to form \(\Omega_{X/S}\). (§Morphisms of Schemes, ⁋Proposition 1) Thus, following the above intuition, we may think of this as fixing the base \(S\) direction and treating only the fiber direction of \(\varphi:X\rightarrow S\) as functions when taking differentials.

Before beginning the main story, we first recast the algebraic tools into the form we will use. The Kähler differential module enjoys the following two key exact sequences.

Proposition 1 (Cotangent exact sequence) Let an \(A\)-algebra \(B\) and a \(B\)-algebra \(C\) be given. Viewing \(C\) as an \(A\)-algebra via the composition \(A \rightarrow B \rightarrow C\), 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 from [Commutative Algebra] §Differentials, ⁋Proposition 8 by setting \(E=B\), \(E'=C\), and taking the base ring to be \(A\).

The remaining exact sequence is as follows.

Proposition 2 (Conormal exact sequence) For an ideal \(\mathfrak{a}\) of an \(A\)-algebra \(B\), 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

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

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

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

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

1. For any \(g\in B\), the canonical map yields 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 yields an isomorphism \(\Omega_{B/A}\cong\Omega_{B/A_h}\).

Proof

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

For the second claim, apply Proposition 1 (Cotangent exact sequence) to the \(A\)-algebra \(A_h\) and the \(A_h\)-algebra \(B\) to obtain the exact sequence of \(B\)-modules

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

Now 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 we have \(\dd{a}=a\cdot \dd{(1)}=0\) for any \(a\in A\); therefore \(\Omega_{A/A}=0\). Applying the first claim to \(B=A\) and \(g=h\) gives \(\Omega_{A_h/A}\cong(\Omega_{A/A})_h=0\), so the first term of the above sequence vanishes and the remaining map \(\Omega_{B/A} \rightarrow \Omega_{B/A_h}\) is an isomorphism.

Now we must glue the local models \(\widetilde{\Omega_{B/A}}\) defined for each pair of affine open subsets \(U=\Spec B\subseteq X\) and \(V=\Spec A\subseteq S\) with \(\varphi(U)\subseteq V\). This form of gluing was already encountered in §Fiber Products, ⁋Theorem 8 when constructing fiber products, where we likewise reduced on the factor side and the base side before gluing the pieces.

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

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

so by §Quasi-coherent Sheaves, ⁋Proposition 5 we have \(\widetilde{\Omega_{B/A}}\vert_{D(g)}\cong \widetilde{\Omega_{B_g/A}}\). Reducing on the base side is similar: replacing \(V\) by a principal open \(D(h)=\Spec A_h\) (\(h\in A\)) with \(\varphi(U)\subseteq D(h)\), the image of \(h\) is invertible since it does not belong to any prime ideal of \(B\), so by the second claim of Proposition 3 we have \(\Omega_{B/A}\cong\Omega_{B/A_h}\) 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 principal on both sides to identify the local models canonically, 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 scheme morphism \(\varphi:X \rightarrow S\), we define the cotangent sheaf or sheaf of relative differentials \(\Omega_{X/S}\) on \(X\) as the \(\mathcal{O}_X\)-module obtained by setting

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

for each pair of affine open subsets \(U=\Spec B\subseteq X\) and \(V=\Spec A\subseteq S\) with \(\varphi(U)\subseteq V\). (§Quasi-coherent Sheaves, ⁋Definition 4)

What this \(\Omega_{X/S}\) measures becomes apparent when we view \(\varphi\) as a family parametrized by \(S\) and perform a base change. The key point 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 to \(b\otimes a'\mapsto a'D_0(b)\) by the Leibniz rule, so this restriction yields \(\Der_{A'}(B', M)\cong\Der_A(B, M)\). But by [Commutative Algebra] §Differentials, ⁋Lemma 2 the left-hand side of this isomorphism is \(\Hom_{B'}(\Omega_{B'/A'}, M)\), and the right-hand side is

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

([Algebraic Structures] §Change of Base Ring, ⁋Proposition 6); hence the two \(B'\)-modules are canonically isomorphic, as they represent the same functor. ([Category Theory] §Representable Functors, ⁋Proposition 8)

Since both sides are determined by their values on affine opens and the identification comes from the canonical one arising from the universal derivation, this isomorphism glues to the scheme level. That is, for a morphism \(S' \rightarrow S\), letting \(X'=X\times_SS'\) with 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 \(S'=\Spec\kappa(s)\), since \(X'\) is the fiber \(X_s\) over \(s\) (§Fiber Products, ⁋Definition 12), letting \(\iota:X_s \rightarrow X\) be the canonical morphism we obtain

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

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

When the family is nontrivial, \(\Omega_{X/S}\) records even the geometry of the fibers. For \(A=\mathbb{K}[t]\) and \(B=\mathbb{K}[t, \x, \y]/(\x\y-t)\), consider \(\varphi:X=\Spec B \rightarrow S=\Spec A\). The relation gives \(t=\x\y\), so \(B\cong \mathbb{K}[\x, \y]\) and \(X\) itself is the affine plane. However, to see this properly one must look at how \(\varphi\) makes \(X\) into a family. Specifically, when \(a\in \mathbb{K}\) is nonzero the fiber \(X_a\) over \(t=a\) is the hyperbola \(\x\y=a\), but the fiber \(X_0\) over \(t=0\) degenerates into the two lines \(\x\y=0\).

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

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

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

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

We can now see how this relation resolves on each fiber. First, over \(t=a\) the element \(\x\) is invertible, so the relation becomes \(\dd{\y}=-(\y/\x)\dd{\x}\), and \(\Omega_{X_a/\mathbb{K}}\) is the free module of rank \(1\) with basis \(\dd{\x}\). On the other hand, on the fiber \(X_0\) at the origin the situation changes: for a point \(q\) on the \(x\)-axis (not the origin) the maximal ideal \(\mathfrak{m}_q\) does not contain \(\x\), and conversely for a point \(q\) on the \(y\)-axis the maximal ideal \(\mathfrak{m}_q\) does not contain \(\y\); in both cases the relation is nontrivial 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 above relation vanishes and \(\Omega_{X/S}\otimes\kappa(p)\) becomes the \(2\)-dimensional vector space generated by \(\dd{\x}\) and \(\dd{\y}\). Thus the dimension of the fiber \(\Omega_{X/S}\otimes\kappa(q)\) jumps from \(1\) to \(2\) only at the intersection point of the two lines, which is information invisible from the fact that \(\Omega_{X/\mathbb{K}}\), taking only differentials over \(\mathbb{K}\) on \(X\), is a free module of rank \(2\).

The above definition can be used directly for computation, but it requires a choice of chart. A coordinate-independent description was already obtained at the algebraic level, because [Commutative Algebra] §Differentials, ⁋Proposition 4 gives a canonical isomorphism \(\mathfrak{I}/\mathfrak{I}^2\cong\Omega_{B/A}\) for the kernel \(\mathfrak{I}\) of the multiplication \(m:B\otimes_AB \rightarrow B\). Translating this geometrically, \(\Spec(B\otimes_AB)\) is the fiber product of \(\Spec B\) over \(\Spec A\) (§Fiber Products, ⁋Lemma 2), and what corresponds to \(m\) is the diagonal morphism \(\Delta\); hence \(\mathfrak{I}\) is the ideal of \(\Delta\) and \(\mathfrak{I}/\mathfrak{I}^2\) is its conormal module. In other words, we have read off differentials in the first-order neighborhood of \(\Delta\), and transferring this to schemes yields a description of \(\Omega_{X/S}\) without choosing a chart.

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, so its image defines a closed subscheme of \(X\times_SX\). Let \(\mathcal{I}\) be the ideal sheaf of this closed subscheme; then there is an isomorphism with the conormal sheaf

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

Here \(\Delta^\ast\) denotes the pullback. (§Quasi-coherent Sheaves, ⁋Definition 14)

Proof

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

We now show that \(\mathfrak{a}/\mathfrak{a}^2\cong \Omega_{B/A}\) as \(B\)-modules. To this end, consider first 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 both sides are additive in each argument, so for any \(u, v\in B\otimes_AB\) we have \(\theta(uv)=m(u)\theta(v)+m(v)\theta(u)\). In particular, if \(u, v\in\mathfrak{a}=\ker m\) then the right-hand side vanishes, so \(\theta(\mathfrak{a}^2)=0\), and therefore \(\theta\) descends to a map \(\mathfrak{a}/\mathfrak{a}^2 \rightarrow \Omega_{B/A}\) which sends \(b\otimes 1-1\otimes b\) to \(\dd{b}\).

Conversely, define \(\delta:B \rightarrow \mathfrak{a}/\mathfrak{a}^2\) by \(\delta(b)=(b\otimes 1-1\otimes b)+\mathfrak{a}^2\); then for any \(b, b'\in B\), inside \(B\otimes_AB\) 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. Hence by the universal property we obtain a map \(\Omega_{B/A} \rightarrow \mathfrak{a}/\mathfrak{a}^2\), and since \(\mathfrak{a}\) is generated by elements of the form \(b\otimes 1-1\otimes b\) while \(\Omega_{B/A}\) is generated by the \(\dd{b}\), the fact that this is the inverse of \(\theta\) is checked on generators.

On the other hand, letting \(R=B\otimes_AB\), the map \(\Delta\) is a closed immersion coming from the surjection \(m:R \rightarrow R/\mathfrak{a}\cong B\), so on the affine side what the pullback \(\Delta^\ast\) does is apply \(-\otimes_RB\) to an \(R\)-module. However, since \(\mathfrak{a}\) annihilates \(\mathfrak{a}/\mathfrak{a}^2\), its \(R\)-module structure is already given through \(R/\mathfrak{a}=B\), so \((\mathfrak{a}/\mathfrak{a}^2)\otimes_RB\cong \mathfrak{a}/\mathfrak{a}^2\), and therefore we obtain the isomorphism

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

Here the leftmost \(\widetilde{(-)}\) is the associated sheaf on \(\Spec R\), and the remaining two are the associated sheaves on \(\Spec B\).

For 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 again gives the kernel of the multiplication \(B\otimes_AB \rightarrow B\), so the above computation applies directly and yields \(\Delta^\ast(\mathcal{I}/\mathcal{I}^2)\vert_U\cong \widetilde{\Omega_{B/A}}=\Omega_{X/S}\vert_U\). This identification is constructed canonically from the universal derivation, so the local pieces agree when charts are restricted or changed, and hence they glue into a single global isomorphism.

As seen in the proof above, \(\mathcal{I}/\mathcal{I}^2\) is a sheaf on \(\Delta(X)\), and since \(\Delta\) identifies \(X\) with its image, it is pulled back to a sheaf on \(X\) via \(\Delta^\ast\); when we actually compute, we will calculate \(\widetilde{\Omega_{B/A}}\) on affine opens according to Definition 4, but the above proposition gives a coordinate-independent description of this sheaf.

The two exact sequences from the previous section also carry over immediately to the sheaf level by exactness of the associated sheaf functor. (§Quasi-coherent Sheaves, ⁋Proposition 6) For a composition of scheme morphisms \(X \rightarrow S' \rightarrow S\) with first morphism \(\psi: X \rightarrow S'\), translating Proposition 1 (Cotangent exact sequence) to associated sheaves 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 immersion \(\iota:Z\hookrightarrow Y\) is given by an ideal sheaf \(\mathcal{J}\), translating 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 \(Y\) to \(Z\) along \(\iota\), and the first term \(\mathcal{J}/\mathcal{J}^2\) must also be read as a sheaf on \(Z\). This is possible because §Closed Subschemes, ⁋Definition 5 gives \(\mathcal{O}_Y/\mathcal{J}\cong \iota_\ast \mathcal{O}_Z\), and \(\mathcal{J}\) annihilates the quotient \(\mathcal{J}/\mathcal{J}^2\) so that its \(\mathcal{O}_Y\)-module structure is already given through \(\mathcal{O}_Y/\mathcal{J}\). Indeed, for an affine open subset \(\Spec B\subseteq Y\), an affine \(\Spec A\subseteq S\) containing its image, and \(\mathfrak{a}=\mathcal{J}(\Spec B)\) and \(C=B/\mathfrak{a}\), the three terms become the associated sheaves of \(\mathfrak{a}/\mathfrak{a}^2\), \(\Omega_{B/A}\otimes_BC\), and \(\Omega_{C/A}\), returning to the sequence of Proposition 2 (Conormal exact sequence).

Since these two exact sequences are standard tools for computing sheaves of differentials, let us now translate the algebraic intuition we introduced with them into geometric terms. We have already seen that \(\Omega_{X/S}\) is the differential proceeding by treating only the fiber direction as functions while fixing the base direction. To examine the first exact sequence, suppose first that \(S'\rightarrow S\) sends \(s'\) to \(s\); then \(X_{s'}\) always lies inside \(X_s\). That is, in this situation the directions measured over \(S'\) are narrower, so \(\Omega_{X/S'}\) is given as a quotient of \(\Omega_{X/S}\), and the part that is erased 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) that \(Z\) possesses inside \(Y\). When \(S=\Spec A\), \(Y=\Spec A[\x_1,\ldots, \x_n]\), and \(\mathcal{J}\) is generated by \(f_1,\ldots, f_r\), the sheaf \(\iota^\ast\Omega_{Y/S}\) has basis \(\dd{\x_1},\ldots, \dd{\x_n}\) and \(\bar{d}\) sends \(f_j\mapsto\sum_i(\partial f_j/\partial \x_i)\dd{\x_i}\), so the matrix expressing this morphism in these bases is precisely the Jacobian \((\partial f_j/\partial \x_i)\). That is, differentials on \(Z\) are obtained from those on the ambient space by quotienting out the part generated by the differentials of the equations, namely the directions perpendicular to \(Z\), and dualizing this at a point returns to the description of [Algebraic Varieties] §Tangent Spaces and Smoothness, ⁋Proposition 2.

Tangent Sheaf and Zariski Tangent Space

Taking the dual of the cotangent sheaf yields the sheaf of tangent vectors. This is the object corresponding to the tangent bundle on a variety.

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

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

(§Quasi-coherent Sheaves, ⁋Definition 2)

To see what this definition does, let us again look at the affine case. For \(X=\Spec B\) and \(S=\Spec A\), Definition 4 gives \(\Omega_{X/S}=\widetilde{\Omega_{B/A}}\). The global sections of the \(\sHom\) constructed from this are \(\mathcal{O}_X\)-module homomorphisms, so a global section of \(\mathcal{T}_{X/S}\) is \(\Hom_{\mathcal{O}_X}(\widetilde{\Omega_{B/A}}, \widetilde B)\cong\Hom_B(\Omega_{B/A}, B)\) (§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 a principal open \(D(g)\) gives \(\Der_A(B_g, B_g)\) by the first assertion of Proposition 3, so \(\mathcal{T}_{X/S}\) is the sheaf obtained by locally collecting the \(A\)-derivations of \(B\), and gluing these gives the general case as well.

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

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

Since \(\Omega_{B/A}\) is generated by elements of the form \(\dd{b}\), this pairing is determined by the above formula alone; fixing \(D\) and varying \(b\) yields the operation of differentiating functions in the direction determined by \(D\), and conversely fixing \(b\) and varying \(D\) gives the function that assigns to each direction the rate of change of \(b\). This is the same structure as on a differentiable manifold, where a vector field assigns to a function its derivative and a \(1\)-form takes a vector field and outputs a function. The fact that this pairing gives dual bases is seen immediately when \(B=A[\x_1,\ldots, \x_n]\). In this case \(\Omega_{B/A}\) is the 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 its values \(D(\x_i)\) at the \(\x_i\) by \(A\)-linearity and the Leibniz rule; writing these using the dual basis gives \(D=\sum_iD(\x_i)\partial/\partial \x_i\).

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

For this reason, care is needed even when computing the tangent space at a point. For a point \(x\) of a \(\mathbb{K}\)-scheme \(X\), letting \(\kappa(x)\) be the residue field (§Schemes, ⁋Definition 5), we can consider the fiber \(\Omega_{X/\mathbb{K}}\otimes_{\mathcal{O}_X}\kappa(x)\) of the cotangent sheaf. If we consider the fiber

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

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

Therefore the correct definition among these two is the following.

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

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

This definition agrees with the description 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 yields that the canonical map \(\mathfrak{m}_x/\mathfrak{m}_x^2 \rightarrow \Omega_{X/\mathbb{K}}\otimes\kappa(x)\) is surjective; this sequence has no reason to be exact on the left, so injectivity must come from elsewhere. It is the hypothesis of being \(\mathbb{K}\)-rational that provides it: 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 inducing an inverse to the above map, and therefore \(\Omega_{X/\mathbb{K}}\otimes \kappa(x)\cong \mathfrak{m}_x/\mathfrak{m}_x^2\) 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 \(\dim_{\kappa(x)}T_xX\) at a point equals the local dimension \(\dim \mathcal{O}_{X,x}\) at that point is the criterion for whether the point is nonsingular, and if \(\mathcal{O}_{X,x}\) is Noetherian then in general \(\dim_{\kappa(x)}T_xX\geq \dim \mathcal{O}_{X,x}\).

Sheaves of Differentials on Affine and Projective Spaces

First, our simplest example is as follows.

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

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

with basis \(\dd{\x_1},\ldots, \dd{\x_n}\).

Proof

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

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

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

holds for any polynomial \(f\). Hence \(\Omega_{B/A}\) is generated by \(\dd{\x_1},\ldots, \dd{\x_n}\). To show that they are linearly independent over \(B\), we use 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 \(\partial_j:\Omega_{B/A} \rightarrow B\) with \(\partial_j(\dd{\x_i})=\delta_{ij}\), so applying \(\partial_j\) to \(\sum_i b_i \dd{\x_i}=0\) yields \(b_j=0\). Therefore \(\dd{\x_1},\ldots, \dd{\x_n}\) form a free basis and \(\Omega_{B/A}\cong B^{\oplus n}\).

The same result holds even when the base is not affine. Any scheme \(S\) is uniquely a scheme over \(\Spec \mathbb{Z}\), so we can define the relative affine space over \(S\) by 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 \(\Spec A\) of \(S\) this gives \(\mathbb{A}^n_A\). (§Fiber Products, ⁋Example 9) Now, by Definition 4, the cotangent sheaf is determined by the local model on each affine open of the base, so applying the above proposition on each chart yields \(\Omega_{\mathbb{A}^n_S/S}\cong \mathcal{O}_{\mathbb{A}^n_S}^{\oplus n}\).

Thus, on affine space the cotangent sheaf is a trivial bundle whose basis is given by the differentials of the coordinate functions. The situation becomes more interesting when we pass to projective space: the cotangent sheaf of \(\mathbb{P}^n\) is no longer free, but is expressed by a short exact sequence of twisting sheaves, namely the Euler exact sequence.

Theorem 10 (Euler exact sequence) For the projective space \(\mathbb{P}^n_A=\Proj A[\x_0,\ldots, \x_n]\) over a ring \(A\) (§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 notational convenience we write \(\mathbb{P}^n=\mathbb{P}^n_A\) and work on the standard affine open \(U_i=D_+(\x_i)\cong \Spec A[\x_0,\ldots, \x_n]_{(\x_i)}\). This coordinate ring is the polynomial ring in \(n\) variables \(\y^{(i)}_j=\x_j/\x_i\) (\(j\neq i\)), so \(U_i\) is the affine space \(\mathbb{A}^n_A\) over \(A\), and hence by Proposition 9, \(\Omega_{\mathbb{P}^n/A}\vert_{U_i}\) is a free sheaf of rank \(n\) with basis \(\dd{\y}^{(i)}_j\) (\(j\neq i\)).

First, let us define the 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 way we obtain the morphism

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

which is surjective because on each \(U_i\), the element \(\x_i^{-1}e_i\) maps 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 its kernel consists of those \((s_0,\ldots, s_n)\) satisfying \(\sum_j s_j(\x_j/\x_i)=0\). Here the coefficient of the \(i\)-th component is \(\x_i/\x_i=1\), so \(s_i\) is uniquely determined by the other components, while the remaining \(s_j\) are free. That is, the kernel is a free module of rank \(n\) where the \(s_j\) with \(j\neq i\) vary freely.

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

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

Sending the right-hand side via \((\ast)\) gives

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

so its image lies in the kernel we just computed. Moreover, both sides are free modules of rank \(n\) indexed by \(j\neq i\), and the basis maps to the basis, so the image of this morphism coincides exactly with the kernel of \((\ast)\). These locally defined morphisms agree on \(U_i\cap U_k\): indeed, using the two identities

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

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

As in Proposition 9, the base can be lifted to an arbitrary scheme. For \(\mathbb{P}^n_S=\mathbb{P}^n_\mathbb{Z}\times_{\Spec \mathbb{Z}}S\) with projection \(\pi:\mathbb{P}^n_S \rightarrow \mathbb{P}^n_\mathbb{Z}\), 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 carry over unchanged. In particular, when the base is affine this agrees with the twisting sheaf of §Sheaf Cohomology of Schemes, ⁋Definition 5, and by Proposition 5 (Base change) we have \(\Omega_{\mathbb{P}^n_S/S}\cong\pi^\ast\Omega_{\mathbb{P}^n_\mathbb{Z}/\mathbb{Z}}\). On the other hand, applying Theorem 10 (Euler exact sequence) with \(A=\mathbb{Z}\) yields the sequence

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

and we have already verified in the proof above that \(\Omega_{\mathbb{P}^n_\mathbb{Z}/\mathbb{Z}}\) is a free sheaf of rank \(n\) on each \(U_i\), while the other two sheaves are locally free by definition; in particular the last one, \(\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}\), 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). But split exact sequences are preserved by additive functors, and exactness is a local property, so applying \(\pi^\ast\) yields the Euler exact sequence

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

for an arbitrary scheme \(S\).

Canonical sheaf

When the cotangent sheaf is locally free, its top exterior power becomes a sheaf of rank \(1\), that is, an invertible sheaf. This single invertible sheaf obtained in this way controls a large 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) For schemes, we have already defined the exterior power \(\bigwedge^r\mathcal{F}\) of an \(\mathcal{O}_X\)-module by sheafifying the exterior power on each open set (§Quasi-coherent Sheaves, ⁋Definition 2), and we have seen that this preserves quasi-coherence and that for a locally free sheaf \(\mathcal{E}\) of rank \(n\), the sheaf \(\bigwedge^r\mathcal{E}\) is locally free of rank \(\binom{n}{r}\). In particular, for \(r=n\) the determinant \(\det\mathcal{E}=\bigwedge^n\mathcal{E}\) is an invertible sheaf. (§Quasi-coherent Sheaves, §§Locally free sheaf and invertible sheaf)

Definition 11 For a scheme \(X\) over a field \(\mathbb{K}\), suppose the cotangent sheaf \(\Omega_{X/\mathbb{K}}\) is a locally free sheaf of rank \(n\). Then we define the canonical sheaf \(\omega_X\) of \(X\) as 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 scheme morphism \(\varphi:X \rightarrow S\), if \(\Omega_{X/S}\) 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 actually computing the canonical sheaf is the fact that the determinant decomposes as a tensor product along short exact sequences.

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

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

Proof

First we check the rank. Since \(\mathcal{E}''\) is locally free, each point has an open neighborhood \(U\) such that \(\mathcal{E}''\vert_U\cong\mathcal{O}_U^{\oplus s}\) and \(\mathcal{E}'\vert_U\cong\mathcal{O}_U^{\oplus r}\). A surjection of sheaves only guarantees surjectivity at the level of stalks, so to lift each basis section of \(\mathcal{E}''\vert_U\) to a section of \(\mathcal{E}\) we must shrink \(U\) further. Since the basis is finite, we intersect the open neighborhoods over which each section lifts to obtain such a \(U\) again. On this shrunken \(U\), lifting the basis sections of \(\mathcal{E}''\vert_U\) to \(\mathcal{E}\vert_U\) gives a splitting of the surjection \(\mathcal{E}\vert_U \rightarrow \mathcal{E}''\vert_U\), so \(\mathcal{E}\vert_U\cong\mathcal{E}'\vert_U\oplus\mathcal{O}_U^{\oplus s}\cong\mathcal{O}_U^{\oplus(r+s)}\), and hence \(\mathcal{E}\) is a locally free sheaf of rank \(r+s\).

Now we construct the morphism

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

Given a section \(\alpha\in(\det\mathcal{E}')(V)\) and \(\bar t_1\wedge\cdots\wedge\bar t_s\in(\det\mathcal{E}'')(V)\) on an open set \(V\), we shrink \(V\) sufficiently so that each \(\bar t_i\) lifts to \(t_i\in\mathcal{E}(V)\), and 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 the \(\alpha\) on the right-hand side denotes the section transferred via the map \(\det\mathcal{E}' \rightarrow \bigwedge^r\mathcal{E}\) induced by the inclusion \(\mathcal{E}'\hookrightarrow\mathcal{E}\). This value is independent of the choice of lift. The difference of two lifts is a section of \(\mathcal{E}'\), so changing \(t_i\) to \(t_i+a_i\) (\(a_i\in\mathcal{E}'(V)\)) produces a difference consisting of terms each having at least one \(a_i\) as a factor. But on a local splitting, \(\alpha\) is a multiple of \(f_1\wedge\cdots\wedge f_r\) for a basis \(f_1,\ldots, f_r\) of \(\mathcal{E}'\vert_V\), and each \(a_i\) is an \(\mathcal{O}_V\)-linear combination of the \(f_j\), so such a term vanishes because it contains some \(f_j\) twice. Therefore the locally defined \(\lambda\)’s agree on overlaps and glue to a global morphism.

It suffices to check that \(\lambda\) is an isomorphism locally. On the above \(U\), let \(f_1,\ldots, f_r\) be a basis of \(\mathcal{E}'\vert_U\) and let \(g_1,\ldots, g_s\) be lifts of a basis \(\bar g_1,\ldots, \bar g_s\) of \(\mathcal{E}''\vert_U\); then by the splitting, \(f_1,\ldots, f_r, g_1,\ldots, g_s\) form a basis of \(\mathcal{E}\vert_U\). By [Multilinear Algebra] §Tensor Algebra, ⁋Proposition 13, \(\det\mathcal{E}\vert_U\) is a free sheaf of rank \(1\) with basis \(f_1\wedge\cdots\wedge f_r\wedge g_1\wedge\cdots\wedge g_s\), and likewise \((\det\mathcal{E}'\otimes\det\mathcal{E}'')\vert_U\) 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, it is an isomorphism on \(U\), and hence an isomorphism globally.

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

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

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

Applying the same proposition repeatedly to a direct sum, the left-hand side is \(\mathcal{O}(-1)^{\otimes(n+1)}\), and since the transition function of \(\mathcal{O}(d)\) is \((\x_i/\x_j)^d\) (§Sheaf Cohomology of Schemes, ⁋Definition 5 and the discussion following), 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 computation on varieties using the transition functions of \(n\)-forms. ([Algebraic Varieties] §Canonical Line Bundle, ⁋Example 8)

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

Theorem 14 (Serre duality) Let \(\Omega_{X/\mathbb{K}}\) be a locally free sheaf of rank \(n\) for an \(n\)-dimensional integral projective scheme \(X\) over an algebraically closed field \(\mathbb{K}\) (§Closed Subschemes of Projective Space, ⁋Definition 7, §Algebraic Structure of Schemes, §§Reduced and Integral Schemes). Then for any locally free sheaf \(\mathcal{E}\) on \(X\) 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 8)

The proof of this theorem proceeds by showing that the pairing constructed from the trace map and cup product on \(\mathbb{P}^n\) is a perfect pairing, and then transporting this to a general \(X\) via a finite surjective morphism; since the concrete proof lies beyond the scope of this article, we refer the reader to [Algebraic Varieties] §Serre Duality, §§Serre Duality on Projective Spaces.

Meanwhile, in the discussion above we required \(\Omega_{X/\mathbb{K}}\) to be locally free; if this fails, the rank is no longer constant and there is no basis for choosing a top exterior power to begin with. This kind of generalization is already treated in [Algebraic Varieties] §Serre Duality, §§Generalization of Serre Duality, where in this case the isomorphism of the theorem must be lifted to \(\Ext\) and a separate dualizing sheaf must be introduced. Also, although it is relatively less important since the cases we are generally interested in are those with \(\mathbb{K}=\mathbb{C}\), it is worth remembering that if \(\mathbb{K}\) is not a perfect field, the rank of \(\Omega_{X/\mathbb{K}}\) may differ from the dimension of \(X\). In general, the locally free assumption on \(\Omega_{X/S}\) is automatically satisfied if \(\varphi:X\rightarrow S\) is smooth; examining this is the goal of §Smooth Morphisms and Étale Morphisms.

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[Stacks] The Stacks Project Authors, The Stacks Project. Available online.