Sheaf Operations Exactness Ringed Spaces and Module Pullback

Let $\mathcal{F}$ be a sheaf on $X$, let $V={\bigcup }_i{V}_i$ be an open cover in $Y$, and let $s_i\in (f_{\ast }\mathcal{F})(V_i)=\mathcal{F}(f^{-1}(V_i))$ be compatible on overlaps. By [F1], the sets $f^{-1}(V_i)$ cover $f^{-1}(V)$, so [L1] gives a unique section $s\in \mathcal{F}(f^{-1}(V))$ restricting to every $s_i$. This section is exactly an element of $(f_{\ast }\mathcal{F})(V)$, so $f_{\ast }\mathcal{F}$ is a sheaf.

If $\mathcal{F}$ is a sheaf of groups, rings, or modules, then each section set of $f_{\ast }\mathcal{F}$ is literally the corresponding section set of $\mathcal{F}$ over a preimage open set. Hence the algebraic operations are inherited objectwise, and the restriction maps are the same homomorphisms as before. By [L2] and step 1.1, $f_{\ast }\mathcal{F}$ is a sheaf of the same kind.

Steps 1.1 and 2.1 prove both assertions.

Let $\alpha :f^{-1}\mathcal{G}\to \mathcal{F}$ be a sheaf morphism. By [F2] and [L1], $\alpha$ corresponds uniquely to a presheaf morphism $\tilde{\alpha }:f_{\mathrm{p}}\mathcal{G}\to \mathcal{F}$.

Conversely, let $\beta :\mathcal{G}\to f_{\ast }\mathcal{F}$ be a sheaf morphism. For an open $U\subseteq X$, represent an element of $(f_{\mathrm{p}}\mathcal{G})(U)$ by a pair $(V,s)$ with $f(U)\subseteq V$ and $s\in \mathcal{G}(V)$, and define ${\tilde{\beta }}_U([V,s]):={\beta }_V(s){\mathrm{\mid }}_U\in \mathcal{F}(U),$ where the restriction is taken from $\mathcal{F}(f^{-1}(V))$ to $\mathcal{F}(U)$. If $(V,s)$ and $(V^{\mathrm{\prime }},s^{\mathrm{\prime }})$ represent the same colimit class, then on some smaller open neighbourhood $W\subseteq V\cap V^{\mathrm{\prime }}$ of $f(U)$ one has $s{\mathrm{\mid }}_W=s^{\mathrm{\prime }}{\mathrm{\mid }}_W$, so ${\beta }_V(s){\mathrm{\mid }}_U={\beta }_{V^{\mathrm{\prime }}}(s^{\mathrm{\prime }}){\mathrm{\mid }}_U$. Thus $\tilde{\beta }$ is well defined and natural in $U$.

For an open set $V\subseteq Y$ and a section $s\in \mathcal{G}(V)$, let $[V,s]$ denote its class in $(f_{\mathrm{p}}\mathcal{G})(f^{-1}(V))$, and define ${\beta }_V(s):={\tilde{\alpha }}_{f^{-1}(V)}([V,s])\in \mathcal{F}(f^{-1}(V))=(f_{\ast }\mathcal{F})(V)\mathrm{.}$ If $V^{\mathrm{\prime }}\subseteq V$, naturality of $\tilde{\alpha }$ shows ${\beta }_{V^{\mathrm{\prime }}}(s{\mathrm{\mid }}_{V^{\mathrm{\prime }}})={\beta }_V(s){\mathrm{\mid }}_{V^{\mathrm{\prime }}}$, so the ${\beta }_V$ define a morphism $\beta :\mathcal{G}\to f_{\ast }\mathcal{F}$.

By [F2] and [L1], the presheaf morphism $\tilde{\beta }:f_{\mathrm{p}}\mathcal{G}\to \mathcal{F}$ factors uniquely through a sheaf morphism $\alpha :f^{-1}\mathcal{G}\to \mathcal{F}$.

The constructions of steps 2.1 and 2.2 are inverse, because both are recovered from the same formula on representative classes $[V,s]$. Therefore ${\mathrm{Hom}}_X(f^{-1}\mathcal{G},\mathcal{F})\cong {\mathrm{Hom}}_Y(\mathcal{G},f_{\ast }\mathcal{F})$ naturally in both sheaves.

By [F1] and [L1], it is enough to identify the stalk $(f_{\mathrm{p}}\mathcal{G}{)}_x$.

If $t\in \mathcal{G}(V)$ with $f(x)\in V$, then $x\in f^{-1}(V)$ and $t$ defines a class $[V,t]\in (f_{\mathrm{p}}\mathcal{G})(f^{-1}(V))$. Sending the germ $t_{f(x)}$ to the germ of $[V,t]$ at $x$ gives a map $\mathrm{\Phi }:{\mathcal{G}}_{f(x)}⟶(f_{\mathrm{p}}\mathcal{G}{)}_x\mathrm{.}$ If two representatives agree on a smaller neighbourhood of $f(x)$, then their induced sections agree on the inverse image of that smaller neighbourhood, so $\mathrm{\Phi }$ is well defined.

Conversely, represent a germ in $(f_{\mathrm{p}}\mathcal{G}{)}_x$ by a section $[V,t]\in (f_{\mathrm{p}}\mathcal{G})(U)$ with $x\in U$ and $f(U)\subseteq V$. Since $f(x)\in V$, the section $t$ has a germ $t_{f(x)}\in {\mathcal{G}}_{f(x)}$. This depends only on the original germ at $x$, because equality of germs in $(f_{\mathrm{p}}\mathcal{G}{)}_x$ means equality after restricting to some smaller neighbourhood of $x$, hence after restricting $t$ and $t^{\mathrm{\prime }}$ to some common neighbourhood of $f(x)$. Thus there is a map $\mathrm{\Psi }:(f_{\mathrm{p}}\mathcal{G}{)}_x⟶{\mathcal{G}}_{f(x)}\mathrm{.}$

The maps $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are inverse on representatives, so $(f_{\mathrm{p}}\mathcal{G}{)}_x\cong {\mathcal{G}}_{f(x)}$. Step 1.1 then gives $(f^{-1}\mathcal{G}{)}_x\cong {\mathcal{G}}_{f(x)}$.

Let ${\mathfrak{m}}_{f(x)}$ and ${\mathfrak{m}}_x$ be the maximal ideals of the two stalks. By [F2], the composite ${\mathcal{O}}_{Y,f(x)}\stackrel{f_x^{\mathrm{♯}}}{\to }{\mathcal{O}}_{X,x}⟶{\mathcal{O}}_{X,x}\mathrm{/}{\mathfrak{m}}_x$ kills ${\mathfrak{m}}_{f(x)}$, so it factors through the quotient ${\mathcal{O}}_{Y,f(x)}\mathrm{/}{\mathfrak{m}}_{f(x)}\mathrm{.}$

By [F1], the source and target quotients in step 1.1 are exactly the residue fields $\kappa (f(x))$ and $\kappa (x)$. Therefore step 1.1 is the desired field homomorphism $\kappa (f(x))\to \kappa (x)\mathrm{.}$

By [F1] and [L1], it is enough to identify the stalk of the tensor-product presheaf $U⟼\mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)\mathrm{.}$

For every neighbourhood $U$ of $x$, the germ maps $\mathcal{F}(U)\to {\mathcal{F}}_x$, $\mathcal{G}(U)\to {\mathcal{G}}_x$, and ${\mathcal{O}}_X(U)\to {\mathcal{O}}_{X,x}$ induce an ${\mathcal{O}}_X(U)$-balanced pairing into ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x$. Hence there is a homomorphism $\mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)⟶{\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x,$ and these maps are compatible with restriction. Therefore they induce a map $\mathrm{\Phi }:(\mathcal{F}{\otimes }_{p,{\mathcal{O}}_X}\mathcal{G}{)}_x⟶{\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x\mathrm{.}$

Conversely, if $s_x\in {\mathcal{F}}_x$ and $t_x\in {\mathcal{G}}_x$ are represented by sections $s\in \mathcal{F}(U)$ and $t\in \mathcal{G}(U)$ on the same neighbourhood $U$ of $x$, define $\mathrm{\Psi }(s_x\otimes t_x):=(s\otimes t{)}_x\mathrm{.}$ If the representatives are changed on a smaller neighbourhood, the represented germ of $s\otimes t$ is unchanged there, so $\mathrm{\Psi }$ is well defined on simple tensors and extends linearly to a homomorphism $\mathrm{\Psi }:{\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x⟶(\mathcal{F}{\otimes }_{p,{\mathcal{O}}_X}\mathcal{G}{)}_x\mathrm{.}$

On a simple tensor represented on one neighbourhood, $\mathrm{\Phi }$ and $\mathrm{\Psi }$ plainly undo each other. Since both sides are generated by simple tensors, they are inverse isomorphisms. Combining this with step 1.1 gives the stated isomorphism for the sheaf tensor product.

Let $\alpha :f^{\ast }\mathcal{G}\to \mathcal{F}$ be an ${\mathcal{O}}_X$-linear morphism. By [F1] and [L3], $\alpha$ is equivalent to a morphism from the tensor-product presheaf $U⟼{\mathcal{O}}_X(U){\otimes }_{(f^{-1}{\mathcal{O}}_Y)(U)}(f^{-1}\mathcal{G})(U)$ into the sheaf $\mathcal{F}$.

Conversely, let $\beta :\mathcal{G}\to f_{\ast }\mathcal{F}$ be an ${\mathcal{O}}_Y$-linear morphism. By [L2], it corresponds to an $f^{-1}{\mathcal{O}}_Y$-linear morphism $\tilde{\beta }:f^{-1}\mathcal{G}\to \mathcal{F}$. Applying [L1] on each open set gives a morphism from the tensor-product presheaf of [F1] into $\mathcal{F}$, and then [L3] factors it uniquely through the sheafification $f^{\ast }\mathcal{G}$. This yields an ${\mathcal{O}}_X$-linear morphism $\alpha :f^{\ast }\mathcal{G}\to \mathcal{F}$.

Applying [L1] on each open set $U\subseteq X$ converts the map of step 1.1 into an $(f^{-1}{\mathcal{O}}_Y)(U)$-linear map $(f^{-1}\mathcal{G})(U)\to \mathcal{F}(U)$, compatible with restriction. Hence step 1.1 is equivalent to an $f^{-1}{\mathcal{O}}_Y$-linear sheaf morphism $\tilde{\alpha }:f^{-1}\mathcal{G}\to \mathcal{F}\mathrm{.}$

By [L2], the underlying sheaf morphism $\tilde{\alpha }$ corresponds to a sheaf morphism $\beta :\mathcal{G}\to f_{\ast }\mathcal{F}$. Because the maps in step 2.1 are $(f^{-1}{\mathcal{O}}_Y)(U)$-linear, the corresponding components ${\beta }_V$ are ${\mathcal{O}}_Y(V)$-linear. Thus $\beta$ is a morphism of ${\mathcal{O}}_Y$-modules.

The constructions in steps 3.1 and 1.2 are inverse because each stage is built from an adjunction with the standard mutually inverse formulas. Therefore ${\mathrm{Hom}}_{{\mathcal{O}}_X}(f^{\ast }\mathcal{G},\mathcal{F})\cong {\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{G},f_{\ast }\mathcal{F})$ naturally in both variables.

In both categories, sums and zero morphisms are defined sectionwise, the zero sheaf is a zero object, and finite direct sums are obtained by taking direct sums on every open set. Hence both categories are additive.

By [F2], every morphism has a kernel and a cokernel. The same item also gives the image sheaf by sheafifying the objectwise image presheaf, and the coimage is the cokernel of the kernel inclusion.

Let $\phi :\mathcal{F}\to \mathcal{G}$ be a morphism in either category. At a point $x\in X$, the stalks of the kernel, cokernel, image, and coimage from step 2.1 are the usual kernel, cokernel, image, and coimage of the stalk map ${\phi }_x$, because kernels are objectwise and [L2] identifies stalks after sheafification. By [L1], module categories are abelian, and the abelian-group case is the special case of modules over $\mathbb{Z}$, so the canonical map $\mathrm{coim}({\phi }_x)\to \mathrm{im}({\phi }_x)$ is an isomorphism for every $x$.

The sheaf morphism $\mathrm{coim}(\phi )\to \mathrm{im}(\phi )$ is therefore an isomorphism on every stalk, so [L3] makes it an isomorphism globally. Together with steps 1.1 and 2.1, [F1] shows that both categories are abelian.

Assume the sheaf sequence is exact. At each middle term ${\mathcal{F}}_i$, [F1] says that the canonical map $\mathrm{im}(d_{i-1})\to \ker (d_i)$ is an isomorphism. Using [F2] for the stalk construction and then [L1] and [L2], its stalk at $x$ is the canonical map $\mathrm{im}((d_{i-1}{)}_x)⟶\ker ((d_i{)}_x),$ so the stalk sequence is exact at $x$. Since $x$ was arbitrary, the whole stalk sequence is exact.

Conversely, assume every stalk sequence is exact. For each middle term, consider the canonical morphism $\mathrm{im}(d_{i-1})\to \ker (d_i)$. Using [F2] for the stalk construction and then [L1] and [L2], its stalk at $x$ is $\mathrm{im}((d_{i-1}{)}_x)⟶\ker ((d_i{)}_x),$ which is an isomorphism by the stalkwise exactness hypothesis. Therefore [L3] implies that $\mathrm{im}(d_{i-1})\to \ker (d_i)$ is an isomorphism of sheaves.

By [F1], step 1.2 is exactly the assertion that the original sequence is exact. Together with step 1.1, this proves the equivalence.

Suppose $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ is exact. By [L1], the kernel of the map $\mathrm{\Gamma }(X,\mathcal{F})\to \mathrm{\Gamma }(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ is the section group of the kernel sheaf at $X$. By [L2], that kernel sheaf is the image of ${\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}$, so its sections over $X$ are exactly the image of $\mathrm{\Gamma }(X,{\mathcal{F}}^{\mathrm{\prime }})\to \mathrm{\Gamma }(X,\mathcal{F})$. Together with [F1], this is the displayed left exactness.

For the failure of surjectivity, let $X=S^1$, let $\mathcal{C}$ be the sheaf of continuous real-valued functions on $S^1$, and let $\underset{‾}{\mathbb{Z}}$ be the subsheaf of locally constant integer-valued functions. Let $\mathcal{Q}$ be the quotient sheaf obtained by sheafifying the presheaf $V\mapsto \mathcal{C}(V)\mathrm{/}\underset{‾}{\mathbb{Z}}(V)$. Every germ of a section of $\mathcal{Q}$ is represented locally by a continuous real-valued function, so the canonical map $\mathcal{C}\to \mathcal{Q}$ is locally surjective and hence an epimorphism of sheaves.

Cover the circle by the arcs $U_0=S^1\setminus \{(-1,0)\}$ and $U_1=S^1\setminus \{(1,0)\}$. Choose continuous angle functions ${\theta }_0:U_0\to (-1\mathrm{/}2,1\mathrm{/}2)$ and ${\theta }_1:U_1\to (0,1)$ with $e^{2\pi i{\theta }_i(z)}=z$. On one connected component of $U_0\cap U_1$ the difference ${\theta }_1-{\theta }_0$ is $0$, and on the other it is $1$, so the classes of ${\theta }_0$ and ${\theta }_1$ agree in the quotient presheaf and glue to a global section $q\in \mathrm{\Gamma }(X,\mathcal{Q})$. If $q$ had a lift $g\in \mathrm{\Gamma }(X,\mathcal{C})$, then $g-{\theta }_i$ would be an integer-valued continuous function on the connected arc $U_i$, hence a constant $n_i$. On the two components of $U_0\cap U_1$ this would force $n_0-n_1$ to be both $0$ and $1$, a contradiction. Thus $\mathrm{\Gamma }(X,\mathcal{C})\to \mathrm{\Gamma }(X,\mathcal{Q})$ is not surjective.

Step 1.1 gives left exactness, and step 1.3 gives an epimorphism whose global sections map is not surjective.

Let $\mathrm{\Phi }:j_{!}\mathcal{F}\to \mathcal{G}$ be a morphism on $X$. If $W\subseteq U$ is open, then [F2] gives $(j_{!}\mathcal{F})(W)=\mathcal{F}(W)$, so the components ${\mathrm{\Phi }}_W$ define a morphism $\mathcal{F}\to j^{-1}\mathcal{G}$ on $U$.

Conversely, let $\mathrm{\Psi }:\mathcal{F}\to j^{-1}\mathcal{G}$ be a morphism on $U$. For an open set $V\subseteq X$ and a section $s\in (j_{!}\mathcal{F})(V)$, use $\mathrm{\Psi }$ to obtain a section on $V\cap U$. For each point $x\in V\setminus U$, the support condition in [F2] gives an open neighbourhood $W_x\subseteq V$ with $s{\mathrm{\mid }}_{W_x\cap U}=0$, so we may assign the zero section of $\mathcal{G}$ on $W_x$. These local sections agree on overlaps, so they glue uniquely to a section of $\mathcal{G}(V)$. This defines a morphism $j_{!}\mathcal{F}\to \mathcal{G}$.

Let $x\in X$. If $x\in U$, then the neighbourhoods of $x$ that lie inside $U$ are cofinal among all neighbourhoods of $x$, so $(j_{!}\mathcal{F}{)}_x\cong {\mathcal{F}}_x$. If $x\mathrm{\notin }U$ and $s\in (j_{!}\mathcal{F})(V)$ for some neighbourhood $V$ of $x$, then [F2] gives an open neighbourhood $W_x\subseteq V$ with $s{\mathrm{\mid }}_{W_x\cap U}=0$, so the germ of $s$ at $x$ is zero. Therefore $(j_{!}\mathcal{F}{)}_x=0$.

The constructions in steps 1.1 and 1.2 are inverse, because on opens contained in $U$ they both recover the original morphism on $\mathcal{F}$, and outside $U$ the support condition forces the extension to vanish locally. Therefore $j_{!}$ is left adjoint to $j^{-1}$.

Apply step 1.3 to a short exact sequence on $U$. At points of $U$ the stalk sequence after $j_{!}$ is the original exact stalk sequence, and outside $U$ it is $0\to 0\to 0\to 0$. Hence [L1] implies that $j_{!}$ preserves exact sequences.

Let ${\mathcal{M}}^{\mathrm{\prime }}\to \mathcal{M}\to {\mathcal{M}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be an exact sequence of ${\mathcal{O}}_Y$-modules, and fix $x\in X$ with $y=f(x)$. By [L1], the stalk sequence after applying $f^{-1}$ is ${\mathcal{M}}_y^{\mathrm{\prime }}⟶{\mathcal{M}}_y⟶{\mathcal{M}}_y^{\mathrm{\prime }\mathrm{\prime }}⟶0,$ which is exact because the original sequence is exact at every stalk. Hence $f^{-1}$ is exact on the underlying sheaves.

By [F1] and [L2], the stalk of the pulled-back sequence at $x$ is ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{M}}_y^{\mathrm{\prime }}⟶{\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{M}}_y⟶{\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{M}}_y^{\mathrm{\prime }\mathrm{\prime }}⟶0.$ Step 1.1 gives exactness before tensoring, so [L4] makes this sequence right exact. Since this holds for every $x$, [L3] shows that $f^{\ast }$ is right exact.

Assume every stalk ring map ${\mathcal{O}}_{Y,f(x)}\to {\mathcal{O}}_{X,x}$ is flat. By [F2], for each $x\in X$ the functor ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,f(x)}}-$ is exact on ${\mathcal{O}}_{Y,f(x)}$-modules. Applying this to the exact stalk sequence from step 1.1 shows that the sequence in step 2.1 is exact at every $x$. Therefore [L3] implies that $f^{\ast }$ is exact.