Base Change
Base-change¶
Example: Reduced scheme but not geometrically reduced¶
Let \(k = \mathbf{F}_p(T)\) and \(K = \mathbf{F}_p(T^{1/p})\). The scheme \(\operatorname{Spec}(K)\) is reduced (it is a single point with no nilpotents), but it is not geometrically reduced: the base change \(\operatorname{Spec}(K \otimes_k K)\) carries a nonzero nilpotent.
The extension \(K/k\) is purely inseparable of degree \(p\): the element \(T^{1/p}\) satisfies the minimal polynomial \(X^p - T \in k[X]\), which is irreducible over \(k\) by Eisenstein's criterion at the prime \(T\) in \(\mathbf{F}_p[T]\) (then passing to the fraction field).
The tensor product decomposes as
since \(X^p - T = (X - T^{1/p})^p\) in characteristic \(p\). The element
corresponds to the residue class of \(X - T^{1/p}\) in the quotient. This element is nonzero (since \(X - T^{1/p}\) is not divisible by \((X - T^{1/p})^p\)), but
because \(T \in k\) and \(1 \otimes t = t \otimes 1\) for all \(t \in k\). So \(\alpha\) is a nonzero nilpotent, and \(\operatorname{Spec}(K \otimes_k K)\) is not reduced.
The obstruction to geometric reducedness is precisely that \(k^{1/p} \not\subseteq k\): purely inseparable extensions, invisible over the ground field, create nilpotents upon self-tensor.
Push-forward and pull-back¶
Example: Hartshorne \(\operatorname{III}.12.4\)¶
Let \(Y\) be an integral scheme of finite type over an algebraically closed field \(k\), and let \(f : X \to Y\) be a flat projective morphism whose fibres are integral schemes. If \(L\) and \(M\) are line bundles on \(X\) with \(L_y \cong M_y\) for every \(y \in Y\), then there exists a line bundle \(N\) on \(Y\) such that
Set \(\mathscr{F} = L \otimes M^{-1}\). By hypothesis, \(\mathscr{F}_y \cong \mathcal{O}_{X_y}\) for every \(y \in Y\), so \(h^0(X_y, \mathscr{F}_y) = 1\) for all \(y\).
The direct image is a line bundle. By Grauert's theorem (Hartshorne III.12.9), since \(f\) is a flat projective morphism between Noetherian schemes, \(\mathscr{F}\) is coherent and flat over \(Y\), and \(y \mapsto h^0(X_y, \mathscr{F}_y) = 1\) is constant, the direct image \(f_* \mathscr{F}\) is locally free of rank \(1\). Integrality of the fibres is essential here: since each \(X_y\) is connected and reduced, \(H^0(X_y, \mathcal{O}_{X_y}) = k(y)\), ensuring the rank is exactly \(1\) rather than higher. Set \(N = f_* \mathscr{F}\).
The evaluation map is an isomorphism. By cohomology and base change (applicable since \(h^0\) is constant and \(Y\) is integral), the natural map \(f_* \mathscr{F} \otimes k(y) \to H^0(X_y, \mathscr{F}_y)\) is an isomorphism for each \(y\). Since \(\mathscr{F}_y \cong \mathcal{O}_{X_y}\) is globally generated, the evaluation map
is surjective on each fibre. Both sides are line bundles on \(X\) (the left side because \(f^* N\) is a line bundle when \(N\) is), and a surjection between line bundles is necessarily an isomorphism. Therefore \(\mathscr{F} \cong f^* N\), giving \(L \cong M \otimes f^* N\).
Remark: \(f_*\mathcal{O}_{X}\cong \mathcal{O}_{Y}\)¶
Under the same hypotheses -- \(f : X \to Y\) flat projective with integral fibres, \(Y\) integral of finite type over an algebraically closed field -- we have \(f_* \mathcal{O}_X \cong \mathcal{O}_Y\).
Apply the preceding argument with \(\mathscr{F} = \mathcal{O}_X\). Each fibre \(X_y\) is integral, so \(H^0(X_y, \mathcal{O}_{X_y}) \cong k(y)\), giving \(h^0(X_y, \mathcal{O}_{X_y}) = 1\). By Grauert's theorem, \(f_* \mathcal{O}_X\) is locally free of rank 1. The structure map \(f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X\) is a morphism of \(\mathcal{O}_Y\)-modules sending \(1 \mapsto 1\). At each stalk, this is an injection of free rank-1 modules \(\mathcal{O}_{Y,y} \hookrightarrow (f_* \mathcal{O}_X)_y\). An injection between free modules of the same rank over a local ring is an isomorphism (the cokernel is finitely generated with zero localization, hence zero by Nakayama). Therefore \(f_* \mathcal{O}_X \cong \mathcal{O}_Y\).
Example: Fibres being integral is necessary; flatness matters¶
The conclusion \(f_* \mathcal{O}_X \cong \mathcal{O}_Y\) can fail without flatness, even when fibres are connected. Consider the morphism
induced by \(k[y] \hookrightarrow k[x,y]/(x^2, xy)\), \(y \mapsto y\).
Structure of \(A = k[x,y]/(x^2, xy)\). As a \(k[y]\)-module, \(A \cong k[y] \oplus k \cdot x\), where \(y\) acts on the second summand by \(y \cdot x = 0\) (from the relation \(xy = 0\)). The morphism \(f\) is finite (hence projective) with target a regular Noetherian scheme.
Non-flatness. The element \(x \in A\) is annihilated by \(y \in k[y]\), so \(A\) has \(k[y]\)-torsion. Over the PID \(k[y]\), torsion-free is equivalent to flat, so \(f\) is not flat.
The push-forward. We have \(f_* \mathcal{O}_X = \widetilde{A}\) on \(Y = \operatorname{Spec}(k[y])\). Since \(A = k[y] \oplus k \cdot x\) as a \(k[y]\)-module, the push-forward has rank 2 at the generic point and is not even locally free (it has torsion at \(y = 0\)). In particular, \(f_* \mathcal{O}_X \not\cong \mathcal{O}_Y\).
Flatness is essential for Grauert's theorem and cohomology-and-base-change to apply. A non-flat morphism can have connected fibres yet fail \(f_* \mathcal{O}_X \cong \mathcal{O}_Y\) because the push-forward carries torsion.
Example: Push-forward of a coherent sheaf might not be coherent¶
Let \(f : X = \operatorname{Spec}(k(x)) \to Y = \operatorname{Spec}(k)\) be the morphism induced by \(k \hookrightarrow k(x)\). Then \(f_* \mathcal{O}_X\) is the quasi-coherent sheaf on \(\operatorname{Spec}(k)\) associated to the \(k\)-vector space \(k(x)\). A coherent sheaf on \(\operatorname{Spec}(k)\) corresponds to a finite-dimensional \(k\)-vector space, but \(k(x)\) is infinite-dimensional over \(k\) (the elements \(1, x, x^2, \ldots\) are linearly independent). Therefore \(f_* \mathcal{O}_X\) is not coherent.
The morphism \(f\) is not proper: \(\operatorname{Spec}(k(x))\) is not even of finite type over \(k\) (the ring \(k(x)\) is not a finitely generated \(k\)-algebra). By the proper direct image theorem (EGA III), if \(f : X \to Y\) is proper and \(\mathscr{F}\) is coherent on \(X\), then \(R^i f_* \mathscr{F}\) is coherent on \(Y\) for all \(i\). This example shows the properness hypothesis is necessary: even \(f_* \mathcal{O}_X\) can fail coherence when \(f\) is not proper.
Example: Push-forward of a quasi-coherent sheaf might not be quasi-coherent¶
Let \(f : X = \coprod_{i=1}^{\infty} \operatorname{Spec}(\mathbb{Z}) \to S = \operatorname{Spec}(\mathbb{Z})\) be the fold map (identity on each component). We show that \(f_* \mathcal{O}_X\) is not quasi-coherent on \(S\).
For any open \(U \subseteq S\), we have \(f_* \mathcal{O}_X(U) = \prod_{i=1}^{\infty} \mathcal{O}_S(U)\). Quasi-coherence on \(S = \operatorname{Spec}(\mathbb{Z})\) requires \(\mathscr{F}(D(a)) \cong \mathscr{F}(S) \otimes_\mathbb{Z} \mathbb{Z}[1/a]\) for every \(a\). Testing at \(a = 2\): the global sections are \(f_* \mathcal{O}_X(S) = \prod_{i=1}^{\infty} \mathbb{Z}\) and the sections over \(D(2)\) are \(f_* \mathcal{O}_X(D(2)) = \prod_{i=1}^{\infty} \mathbb{Z}[1/2]\). If \(f_* \mathcal{O}_X\) were quasi-coherent, the natural map
would be an isomorphism. Every element of the left-hand side can be written as \((a_i)_i \otimes 2^{-M}\) for some fixed \(M\), giving entries \(a_i / 2^M\) with a uniform bound on the power of \(2\) in the denominator. But the right-hand side contains sequences like \((1/2, 1/4, 1/8, \ldots, 1/2^n, \ldots)\) where the denominators grow without bound. Such a sequence is not in the image, so the map is not surjective.
A second example. On \(\mathbb{A}^1_k = \operatorname{Spec}(k[t])\), define a sheaf by
This is a sheaf (restriction maps and gluing are easily verified). The global sections vanish since \(0 \in \mathbb{A}^1_k\), yet \(\mathscr{F}\) is not the zero sheaf (it has nonzero sections on opens avoiding the origin). A quasi-coherent sheaf with zero global sections on an affine scheme must be zero, so \(\mathscr{F}\) is not quasi-coherent.
Quasi-coherence of \(f_* \mathscr{F}\) requires conditions on \(f\): the standard sufficient condition is that \(f\) be quasi-compact and quasi-separated. The infinite coproduct morphism fails quasi-compactness, and the resulting infinite product of rings does not commute with localization.
Example: Push-forward might have different cohomology¶
Let \(f : \mathbb{P}^1_k \to \operatorname{Spec}(k)\) be the structure morphism and \(\mathscr{F} = \mathcal{O}_{\mathbb{P}^1_k}(-2)\). We compute \(H^1\) via two different routes and find they disagree.
Via \(f_*\). Since \(H^0(\mathbb{P}^1_k, \mathcal{O}(-2)) = 0\) (a line bundle of negative degree on \(\mathbb{P}^1\) has no nonzero global sections), the push-forward is \(f_* \mathcal{O}(-2) = 0\) as a sheaf on \(\operatorname{Spec}(k)\). Therefore
Directly on \(\mathbb{P}^1_k\). The canonical bundle is \(\omega_{\mathbb{P}^1} \cong \mathcal{O}(-2)\). By Serre duality,
Alternatively, from the Cech computation with the standard cover \(U_0 = D_+(x)\), \(U_1 = D_+(y)\): the group \(H^1(\mathbb{P}^1_k, \mathcal{O}(-2))\) is the \(k\)-vector space spanned by the class of \(x^{-1}y^{-1}\), which is \(1\)-dimensional.
Why the discrepancy. For an affine morphism \(f\), the Leray spectral sequence degenerates and gives \(H^i(X, \mathscr{F}) \cong H^i(Y, f_* \mathscr{F})\) for all \(i\). The structure morphism \(\mathbb{P}^1_k \to \operatorname{Spec}(k)\) is projective but not affine, so the spectral sequence does not degenerate. The higher direct image \(R^1 f_* \mathcal{O}(-2) \cong k\) accounts for the discrepancy.
Remark: Push-forward from a Noetherian scheme¶
Two standard preservation results for quasi-coherent sheaves under push-forward:
If \(f : X \to Y\) is a morphism with \(X\) Noetherian (or more generally, \(f\) quasi-compact and quasi-separated), then \(f_* \mathscr{F}\) is quasi-coherent on \(Y\) for any quasi-coherent \(\mathscr{F}\) on \(X\).
If moreover \(X\) is separated and Noetherian, then Cech cohomology agrees with derived functor cohomology: for any quasi-coherent sheaf \(\mathscr{F}\) on \(X\) and any open affine cover \(\mathfrak{U}\) of \(X\),
for all \(p \geq 0\). The key input is Leray's theorem: on a separated Noetherian scheme, intersections of affine opens are affine (by separatedness), so affine opens form a Leray cover for quasi-coherent sheaves (since quasi-coherent sheaves are acyclic on affine schemes by Serre's theorem).
Example: Proper push-forward vs. finite flat push-forward¶
Several fundamental results govern when direct images preserve finiteness. We state the key facts and the role of each hypothesis.
(a) Proper direct image theorem (EGA III). If \(f : X \to Y\) is a proper morphism of Noetherian schemes and \(\mathscr{F}\) is coherent on \(X\), then \(R^i f_* \mathscr{F}\) is coherent on \(Y\) for all \(i \geq 0\). The proof reduces to the projective case via Chow's lemma, then uses the explicit computation of cohomology of coherent sheaves on projective space.
(b) Properness is essential. The morphism \(\operatorname{Spec}(k(x)) \to \operatorname{Spec}(k)\) is not proper (\(k(x)\) is not a finitely generated \(k\)-algebra), and \(f_* \mathcal{O}_X = k(x)\) is infinite-dimensional over \(k\), hence not coherent.
(c) Finite \(\Rightarrow\) locally free iff flat. For a finite morphism \(f : X \to Y\) of Noetherian schemes, \(f_* \mathcal{O}_X\) is locally free if and only if \(f\) is flat. Forward: finiteness implies \(f_* \mathcal{O}_X\) is coherent, and a coherent flat module over a Noetherian local ring is free (by the local criterion plus Nakayama). Converse: for an affine morphism, flatness of \(f\) is equivalent to flatness of \(f_* \mathcal{O}_X\) as an \(\mathcal{O}_Y\)-module.
(d) Automatic flatness over regular curves. Any surjective morphism \(f : X \to Y\) with \(Y\) a regular one-dimensional scheme is automatically flat. Over a DVR, every torsion-free module is flat, and surjectivity ensures the push-forward has nonzero generic fibre (hence is torsion-free).
(e) Miracle flatness theorem. If \(f : X \to Y\) is a morphism of finite type between Noetherian local rings with \(Y\) regular, \(X\) Cohen--Macaulay, and \(\dim X = \dim Y + \dim(\text{fibre})\), then \(f\) is flat. This provides a powerful geometric criterion: correct fiber dimension plus Cohen--Macaulayness forces flatness over a regular base.
Example: Push-forward of a trivial bundle might not be trivial¶
Let \(f : \mathbb{P}^1 \to \mathbb{P}^1\) be the degree-2 map \([x : y] \mapsto [x^2 : y^2]\). Since \(f\) is finite and flat of degree 2, the push-forward \(f_* \mathcal{O}_{\mathbb{P}^1}\) is locally free of rank 2. By Grothendieck's splitting theorem, every vector bundle on \(\mathbb{P}^1\) decomposes as a direct sum of line bundles:
for some \(a, b \in \mathbb{Z}\). We determine \(a\) and \(b\) using the projection formula: for any \(n \in \mathbb{Z}\),
since \(f^* \mathcal{O}(1) \cong \mathcal{O}(2)\) (pulling back along a degree-2 map doubles the degree). The left side gives \(h^0(\mathcal{O}(n+a)) + h^0(\mathcal{O}(n+b))\).
| \(n\) | RHS: \(h^0(\mathcal{O}(2n))\) | LHS: \(h^0(\mathcal{O}(n+a)) + h^0(\mathcal{O}(n+b))\) | Constraint |
|---|---|---|---|
| \(-1\) | \(h^0(\mathcal{O}(-2)) = 0\) | \(h^0(\mathcal{O}(a-1)) + h^0(\mathcal{O}(b-1)) = 0\) | \(a \leq 0\) and \(b \leq 0\) |
| \(0\) | \(h^0(\mathcal{O}(0)) = 1\) | \(h^0(\mathcal{O}(a)) + h^0(\mathcal{O}(b)) = 1\) | Exactly one of \(a, b\) is \(0\) |
| \(1\) | \(h^0(\mathcal{O}(2)) = 3\) | \(h^0(\mathcal{O}(1+a)) + h^0(\mathcal{O}(1+b)) = 3\) | Determines the other |
From \(n = -1\): both \(a, b \leq 0\). From \(n = 0\): exactly one equals \(0\); say \(a = 0\), then \(h^0(\mathcal{O}(b)) = 0\) forces \(b < 0\). From \(n = 1\): \(2 + h^0(\mathcal{O}(1 + b)) = 3\), so \(h^0(\mathcal{O}(1 + b)) = 1\), giving \(b = -1\). Therefore
The trivial bundle \(\mathcal{O}_{\mathbb{P}^1}\) acquires a nontrivial summand under push-forward. Grothendieck's splitting theorem reduces the identification to a finite computation of global sections, which the projection formula translates to cohomology on the source.
Remark: Good things happen sometimes¶
In favorable situations, push-forwards of structure sheaves of divisors decompose simply. Let \(k\) be algebraically closed, \(f : X \to Y\) a finite morphism between smooth projective curves, and \(D = \sum n_i p_i\) an effective divisor on \(X\). Define \(f_* D = \sum n_i f(p_i)\) as a divisor on \(Y\). Then
Here \(\mathcal{O}_D = \mathcal{O}_X / \mathcal{O}_X(-D)\) is the structure sheaf of the subscheme defined by \(D\), a skyscraper sheaf supported at the points \(p_i\). Since \(f\) is finite, \(f_*\) on skyscraper sheaves simply relocates stalks: the stalk of \(f_* \mathcal{O}_D\) at \(q \in Y\) is \(\bigoplus_{f(p_i) = q} \mathcal{O}_{X, p_i} / \mathfrak{m}_{p_i}^{n_i} \cong \bigoplus_{f(p_i) = q} k[t]/(t^{n_i})\). Over an algebraically closed field with distinct image points, this coincides with \(\mathcal{O}_{f_* D}\) at \(q\).
Example: \(f^{*}f_{*}\mathscr{F}\) and \(f_{*}f^{*}\mathscr{G}\)¶
For a morphism \(f : X \to Y\), the functors \((f^*, f_*)\) form an adjoint pair, with natural transformations
We make these explicit in two settings.
Affine case. Let \(f : \operatorname{Spec}(A) \to \operatorname{Spec}(B)\) correspond to a ring map \(\varphi : B \to A\), \(M\) an \(A\)-module, and \(N\) a \(B\)-module. Write \(M_B\) for \(M\) viewed as a \(B\)-module via \(\varphi\).
The counit is induced by the multiplication map
This is surjective (every \(m \in M\) equals \(1 \otimes m\)), but typically not injective: the tensor product is over \(B\), not \(A\), so distinct simple tensors can collapse. For example, if \(A = k[x]\), \(B = k\), and \(M = k[x]\), then \(A \otimes_B M_B = k[x] \otimes_k k[x]\), which has dimension \(> \dim_k k[x]\) in each graded piece.
The unit is
This is injective when \(\varphi : B \to A\) is faithfully flat.
Case \(Y = \operatorname{Spec}(k)\). For a coherent sheaf \(\mathscr{F}\) on \(X\), the push-forward \(f_* \mathscr{F} = H^0(X, \mathscr{F})\) is a \(k\)-vector space, and
The counit \(\mathcal{O}_X \otimes_k H^0(X, \mathscr{F}) \to \mathscr{F}\) is the evaluation map \(g \otimes s \mapsto g \cdot s\). This is surjective precisely when \(\mathscr{F}\) is globally generated.
For a finite-dimensional \(k\)-vector space \(V\), the unit \(V \to V \otimes_k H^0(X, \mathcal{O}_X)\) sends \(v \mapsto v \otimes 1\), embedding \(V\) as the "constant" elements.
The counit measures how far \(\mathscr{F}\) is from being generated by its global sections (pushed forward to the base), while the unit measures how far \(\mathscr{G}\) is from being determined by its pullback data.
Remark: Decomposition of \(f^{*}f_{*}\mathscr{F}\) and \(f_{*}f^{*}\mathscr{G}\)¶
When \(f : X \to Y\) is a finite etale Galois cover with Galois group \(G\) (so \(G\) acts on \(X\) over \(Y\) and \(Y \cong X/G\)), the adjunction maps yield explicit decompositions.
Pull-back of push-forward. For a coherent sheaf \(\mathscr{F}\) on \(X\):
This follows from the fibre product decomposition \(X \times_Y X \cong \coprod_{g \in G} X\) (a defining property of Galois covers) and flat base change:
Push-forward of pull-back. For a coherent sheaf \(\mathscr{G}\) on \(Y\):
By the projection formula, \(f_* f^* \mathscr{G} \cong \mathscr{G} \otimes f_* \mathcal{O}_X\). For an etale Galois cover, \(f_* \mathcal{O}_X\) is a locally free \(\mathcal{O}_Y\)-module of rank \(|G|\), decomposing (after passage to representations) according to the regular representation of \(G\).
Example: Push-forward behavior by class of morphism¶
Push-forward sheaves behave very differently depending on the class of the morphism. The following hierarchy summarizes the situation, with progressively weaker hypotheses yielding weaker conclusions.
(1) Finite flat. The push-forward \(f_* \mathcal{O}_X\) is locally free of rank \(\deg(f)\). All higher direct images vanish: \(R^i f_* \mathscr{F} = 0\) for \(i > 0\) and any quasi-coherent \(\mathscr{F}\), since finite morphisms are affine and affine morphisms have trivial higher direct images. The projection formula \(f_*(\mathscr{F} \otimes f^* \mathscr{G}) \cong f_* \mathscr{F} \otimes \mathscr{G}\) holds.
(2) Finite (not necessarily flat). The functor \(f_*\) is exact (\(R^i f_* = 0\) for \(i > 0\)) since \(f\) is affine. However, \(f_* \mathcal{O}_X\) need not be locally free; it is locally free if and only if \(f\) is flat.
(3) Flat projective. Grauert's theorem and cohomology-and-base-change apply. The sheaves \(R^i f_* \mathscr{F}\) are coherent for coherent \(\mathscr{F}\). When \(h^i(X_y, \mathscr{F}_y)\) is constant in \(y\), the sheaf \(R^i f_* \mathscr{F}\) is locally free and its formation commutes with base change.
(4) Proper birational. If \(Y\) is normal, Zariski's main theorem gives \(f_* \mathcal{O}_X = \mathcal{O}_Y\). Higher direct images \(R^i f_*\) encode information about the exceptional locus and are central to the study of resolutions of singularities.
The hierarchy finite flat \(\subset\) finite \(\subset\) proper reflects decreasing control: flatness ensures local freeness, finiteness ensures exactness, and properness ensures only coherence. Serre's criterion characterizes the affine case precisely: \(f\) is affine if and only if \(R^i f_* \mathscr{F} = 0\) for all quasi-coherent \(\mathscr{F}\) and all \(i > 0\).
Example: What prevents \(f_{*}\mathcal{O}_{X}\) from being \(\mathcal{O}_{Y}\)¶
Two important situations guarantee \(f_* \mathcal{O}_X \cong \mathcal{O}_Y\).
(a) When \(f_* \mathcal{O}_X\) is invertible. The structure map \(f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X\) is a morphism of \(\mathcal{O}_Y\)-modules with \(f^\#(1) = 1\). If \(f_* \mathcal{O}_X\) is invertible, then locally \(f^\#\) is a map \(\mathcal{O}_{Y,y} \to \mathcal{O}_{Y,y}\) (after trivialization) sending \(1\) to \(1\), hence \(f^\#\) maps the generator to the generator. An injection between free rank-1 modules whose image contains a generator is an isomorphism. Therefore \(f_* \mathcal{O}_X \cong \mathcal{O}_Y\).
The rigidity here is purely ring-theoretic: the condition \(1 \mapsto 1\) pins down the map once the target is known to be rank 1.
(b) Proper birational morphism to a normal target. Let \(U \subseteq Y\) be the dense open over which \(f\) is an isomorphism (existing by birationality). For any open \(V \subseteq Y\), the map \(\mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}(V))\) is injective (it restricts to an isomorphism over the dense open \(V \cap U\)). For surjectivity: a section \(s \in \mathcal{O}_X(f^{-1}(V))\) restricts to a regular function on \(f^{-1}(V \cap U) \cong V \cap U\), defining a rational function on \(V\) that is regular on the dense open \(V \cap U\). Normality of \(Y\) provides the S2 condition (algebraic Hartogs' lemma): a rational function on a normal scheme that is regular in codimension 1 extends to a regular function everywhere. Therefore \(s\) extends to \(\mathcal{O}_Y(V)\), giving \(f_* \mathcal{O}_X \cong \mathcal{O}_Y\).
Example: Pullback of a locally free sheaf need not reflect local freeness¶
Consider \(f : X = \operatorname{Spec}(\mathbb{Q}) \to Y = \operatorname{Spec}(\mathbb{Z})\) and the \(\mathbb{Z}\)-module \(M = \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}\), defining a coherent sheaf \(\mathscr{F} = \widetilde{M}\) on \(Y\).
The pullback is free. We compute
Since \(\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Q} \cong \mathbb{Q}\) and \(\mathbb{Z}/2\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Q} = 0\) (torsion modules tensor to zero with \(\mathbb{Q}\): for any \(\bar{a} \otimes q\), we have \(\bar{a} \otimes q = \bar{a} \otimes 2 \cdot (q/2) = 2\bar{a} \otimes (q/2) = 0\)), we get \(f^* \mathscr{F} \cong \widetilde{\mathbb{Q}}\), free of rank 1.
The original sheaf is not locally free. At the prime \((2)\), the stalk \(M_{(2)} = \mathbb{Z}_{(2)} \oplus \mathbb{Z}/2\mathbb{Z}\) has torsion, so it is not free over \(\mathbb{Z}_{(2)}\). A locally free sheaf on \(\operatorname{Spec}(\mathbb{Z})\) has torsion-free stalks at every prime, so \(\mathscr{F}\) is not locally free.
The role of faithful flatness. The morphism \(\operatorname{Spec}(\mathbb{Q}) \to \operatorname{Spec}(\mathbb{Z})\) is flat (localization is flat) but not faithfully flat: it is not surjective, since the closed points \((p)\) are not in the image. Faithful flatness is needed to descend properties like local freeness: if \(f\) is faithfully flat and \(f^* \mathscr{F}\) is locally free, then \(\mathscr{F}\) is locally free. Without faithful flatness, the pullback can kill torsion and make a non-free module appear free.
Example: Push-forward of a vector bundle along a projective morphism need not be locally free, even with constant \(h^0\)¶
Let \(Y = \operatorname{Spec}(k[x]/(x^2))\) be the spectrum of the dual numbers and \(f : X = \mathbb{P}^1_Y \to Y\) the projection. We construct a vector bundle \(E\) on \(X\), flat over \(Y\), such that \(f_* E\) is not locally free -- even though \(h^0(X_y, E_y)\) is constant. This shows the hypothesis that \(Y\) be integral (or at least reduced) in the cohomology-and-base-change theorem is essential.
Construction via extensions. The Euler exact sequence on \(\mathbb{P}^1_Y\) and Serre duality give
Choose the extension class \(x \in k[x]/(x^2)\) (nonzero but nilpotent) to get
The middle term \(E\) is locally free of rank 2 on \(\mathbb{P}^1_Y\) (an extension of locally free sheaves on a smooth scheme is locally free) and flat over \(Y\).
Computing \(f_* E\). Apply \(f_*\) to the short exact sequence. The relevant portion of the long exact cohomology sequence is
where \(\delta\) is multiplication by the extension class \(x\). Both \(H^0\) and \(H^1\) are isomorphic to \(k[x]/(x^2)\), and the connecting map is
Therefore \(H^0(\mathbb{P}^1_Y, E) = \ker(\delta) = (x) \subset k[x]/(x^2)\). As a \(k[x]/(x^2)\)-module, \((x) \cong k\) (since \(x \cdot x = 0\)). This has length 1, while a free module of rank 1 has length 2. So \(f_* E \cong \widetilde{k}\) is not locally free on \(Y\).
Fiber analysis. The unique point \(y\) of \(Y\) has fiber \(X_y = \mathbb{P}^1_k\), and \(E_y\) fits into \(0 \to \mathcal{O}(-2) \to E_y \to \mathcal{O} \to 0\) with extension class \(x|_{y} = 0 \in k\), so \(E_y \cong \mathcal{O} \oplus \mathcal{O}(-2)\) and \(h^0(X_y, E_y) = 1\). The function \(h^0\) is constant (trivially, as there is one point), yet \(f_* E\) is not free.
Over a non-reduced base, the extension class can be nilpotent but nonzero, creating a non-split extension whose fiber over the closed point looks split. The cohomology-and-base-change theorem in its strong form (constancy of \(h^i\) implies local freeness) requires integrality or at least reducedness of the base.
Remark: \(\operatorname{Ext}^{1}\) and connecting maps in extensions¶
Given an exact sequence of coherent sheaves on a scheme \(X\),
with extension class \(e \in \operatorname{Ext}^1(\mathscr{G}, \mathscr{F})\), the connecting homomorphisms in the long exact cohomology sequence are governed by \(e\) via the Yoneda product.
Connecting map as cup product. For \(\alpha \in H^i(X, \mathscr{G})\), the connecting homomorphism \(\delta^i : H^i(X, \mathscr{G}) \to H^{i+1}(X, \mathscr{F})\) is
When \(i = 0\), the map \(\delta^0 : H^0(X, \mathscr{G}) \to H^1(X, \mathscr{F})\) sends a global section \(s\) of \(\mathscr{G}\) to the obstruction class for lifting \(s\) to a global section of \(\mathscr{E}\).
Functoriality. The association of extension classes to short exact sequences is functorial in both arguments: a morphism \(\varphi : \mathscr{F} \to \mathscr{F}'\) sends \(e\) to the pushout class \(\varphi_*(e) \in \operatorname{Ext}^1(\mathscr{G}, \mathscr{F}')\), and a morphism \(\psi : \mathscr{G}' \to \mathscr{G}\) sends \(e\) to the pullback class \(\psi^*(e) \in \operatorname{Ext}^1(\mathscr{G}', \mathscr{F})\).
Splitting criterion. The extension splits (\(\mathscr{E} \cong \mathscr{F} \oplus \mathscr{G}\)) if and only if \(e = 0\).
In ecag-0021, the connecting map \(\delta^0\) was multiplication by the extension class \(x \in \operatorname{Ext}^1(\mathcal{O}, \mathcal{O}(-2)) \cong k[x]/(x^2)\), and \(\ker(\delta^0)\) determined \(H^0(X, E)\).
Example: Pulling back a line bundle along a translation on an elliptic curve¶
Let \(E\) be an elliptic curve over an algebraically closed field \(k\) with origin \(O\), and let \(\tau_a : E \to E\) denote translation by \(a \in E\). For any point \(p \in E\),
and \(\mathcal{O}_E(p) \cong \mathcal{O}_E(q)\) if and only if \(p = q\). In particular, \(\tau_a^* \mathcal{O}_E(p) \cong \mathcal{O}_E(p)\) for all \(p\) if and only if \(a = O\).
Pullback shifts the divisor. The divisor of \(\mathcal{O}_E(p)\) is the point \(p\). Pulling back along \(\tau_a\) replaces \(p\) by \(\tau_a^{-1}(p) = p - a\) (in the group law of \(E\)), giving \(\tau_a^* \mathcal{O}_E(p) \cong \mathcal{O}_E(p - a)\).
Degree-1 line bundles classify points. On an elliptic curve, two degree-1 line bundles \(\mathcal{O}_E(p)\) and \(\mathcal{O}_E(q)\) are isomorphic if and only if \(p \sim q\) (linear equivalence). By Riemann--Roch, \(h^0(\mathcal{O}_E(p)) = \deg(p) = 1\) for any point \(p\), so the linear system \(|p|\) consists of the single effective divisor \(p\). Therefore \(p \sim q\) implies \(p = q\): the Abel--Jacobi map \(E \to \operatorname{Pic}^1(E)\), \(p \mapsto \mathcal{O}_E(p)\), is injective (and in fact an isomorphism).
It follows that \(\tau_a^* \mathcal{O}_E(p) \cong \mathcal{O}_E(p)\) requires \(p - a = p\), i.e., \(a = O\).
Equivariant Picard group. For a smooth variety \(X\) with a finite automorphism group \(G\), the group \(\operatorname{Pic}^G(X)\) of \(G\)-linearized line bundles embeds into \(\operatorname{Pic}(X)\) via the forgetful map. By Hilbert's Theorem 90, \(H^1(G, \mathcal{O}(X)^*) = 0\), which guarantees this forgetful map is injective: a \(G\)-equivariant structure on a line bundle, if it exists, is unique up to isomorphism. One can also construct an averaging map \(\operatorname{Pic}(X) \to \operatorname{Pic}^G(X)\) sending \(\mathcal{L}\) to \(\bigotimes_{g \in G} g^* \mathcal{L}\), which provides a partial inverse (up to torsion issues depending on \(|G|\) and the characteristic).
Unlike topological bundles, algebraic line bundles are sensitive to the precise algebraic structure of automorphisms. On an elliptic curve, the Abel--Jacobi map identifies degree-1 line bundles with points of the curve, so translation acts nontrivially on \(\operatorname{Pic}^1(E)\).