The Stacks project

48.33 Duality for compactly supported cohomology

Let $k$ be a field. Let $U$ be a separated scheme of finite type over $k$. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. Let us define the compactly supported cohomology $H^ i_ c(U, K)$ of $K$ as follows. Choose an open immersion $j : U \to X$ into a scheme proper over $k$ and a Deligne system $(K_ n)$ for $j : U \to X$ whose restriction to $U$ is constant with value $K$. Then we set

\[ H^ i_ c(U, K) = \mathop{\mathrm{lim}}\nolimits H^ i(X, K_ n) \]

We view this as a topological $k$-vector space using the limit topology (see More on Algebra, Section 15.36). There are several points to make here.

First, this definition is independent of the choice of $X$ and $(K_ n)$. Namely, if $p : U \to \mathop{\mathrm{Spec}}(k)$ denotes the structure morphism, then we already know that $Rp_!K = (R\Gamma (X, K_ n))$ is well defined up to pro-isomorphism in $D(k)$ hence so is the limit defining $H^ i_ c(U, K)$.

Second, it may seem more natural to use the expression

\[ H^ i(R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, K_ n)) = R\Gamma (X, R\mathop{\mathrm{lim}}\nolimits K_ n) \]

but this would give the same answer: since the $k$-vector spaces $H^ j(X, K_ n)$ are finite dimensional, these inverse systems satisfy Mittag-Leffler and hence $R^1\mathop{\mathrm{lim}}\nolimits $ terms of Cohomology, Lemma 20.35.1 vanish.

If $U' \subset U$ is an open subscheme, then there is a canonical map

\[ H^ i_ c(U', K|_{U'}) \longrightarrow H^ i_ c(U, K) \]

functorial for $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. See for example Remark 48.32.7. In fact, using Remark 48.32.8 we see that more generally such a map exists for an ├ętale morphism $U' \to U$ of separated schemes of finite type over $k$.

If $V$ is a $k$-vector space then we put a topology on $\mathop{\mathrm{Hom}}\nolimits _ k(V, k)$ as follows: write $V = \bigcup V_ i$ as the filtered union of its finite dimensional $k$-subvector spaces and use the limit topology on $\mathop{\mathrm{Hom}}\nolimits _ k(V, k) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ k(V_ i, k)$. If $\dim _ k V < \infty $ then the topology on $\mathop{\mathrm{Hom}}\nolimits _ k(V, k)$ is discrete. More generally, if $V = \mathop{\mathrm{colim}}\nolimits _ n V_ n$ is written as a directed colimit of finite dimensional vector spaces, then $\mathop{\mathrm{Hom}}\nolimits _ k(V, k) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ k(V_ n, k)$ as topological vector spaces.

Lemma 48.33.1. Let $p : U \to \mathop{\mathrm{Spec}}(k)$ be separated of finite type where $k$ is a field. Let $\omega _{U/k}^\bullet = p^!\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}$. There are canonical isomorphisms

\[ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(U, K), k) = H^{-i}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(K, \omega _{U/k}^\bullet )) \]

of topological $k$-vector spaces functorial for $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$.

Proof. Choose a compactification $j : U \to X$ over $k$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent ideal sheaf with $V(\mathcal{I}) = X \setminus U$. By Derived Categories of Schemes, Proposition 36.11.2 we may choose $M \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ with $K = M|_ U$. We have

\[ H^ i(U, K) = \mathop{\mathrm{Ext}}\nolimits ^ i_ U(\mathcal{O}_ U, M|_ U) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{I}^ n, M) = \mathop{\mathrm{colim}}\nolimits H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M)) \]

by Lemma 48.30.1. Since $\mathcal{I}^ n$ is a coherent $\mathcal{O}_ X$-module, we have $\mathcal{I}^ n$ in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, hence $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.5.

Let $\omega _{X/k}^\bullet = q^!\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}$ where $q : X \to \mathop{\mathrm{Spec}}(k)$ is the structure morphism, see Section 48.27. We find that

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ k( & H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M)), k) \\ & = \mathop{\mathrm{Ext}}\nolimits ^{-i}_ X(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M), \omega _{X/k}^\bullet ) \\ & = H^{-i}(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}( \mathcal{I}^ n, M), \omega _{X/k}^\bullet )) \end{align*}

by Lemma 48.27.1. By Lemma 48.2.4 part (1) the canonical map

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(M, \omega _{X/k}^\bullet ) \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I}^ n \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}( \mathcal{I}^ n, M), \omega _{X/k}^\bullet ) \]

is an isomorphism. Observe that $\omega ^\bullet _{U/k} = \omega ^\bullet _{X/k}|_ U$ because $p^!$ is constructed as $q^!$ composed with restriction to $U$. Hence $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(M, \omega _{X/k}^\bullet )$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ which restricts to $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(K, \omega _{U/k}^\bullet )$ on $U$. Hence by Lemma 48.30.11 we conclude that

\[ \mathop{\mathrm{lim}}\nolimits H^{-i}(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(M, \omega _{X/k}^\bullet ) \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I}^ n) \]

is an avatar for the right hand side of the equality of the lemma. Combining all the isomorphisms obtained in this manner we get the isomorphism of the lemma. $\square$

Lemma 48.33.2. With notation as in Lemma 48.33.1 suppose $U' \subset U$ is an open subscheme. Then the diagram

\[ \xymatrix{ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(U, K), k) \ar[rr] & & H^{-i}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(K, \omega _{U/k}^\bullet )) \\ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(U', K|_{U'}), k) \ar[rr] \ar[u] & & H^{-i}_ c(U', R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{U'}}(K, \omega _{U'/k}^\bullet )) \ar[u] } \]

is commutative. Here the horizontal arrows are the isomorphisms of Lemma 48.33.1, the vertical arrow on the left is the contragredient to the restriction map $H^ i(U, K) \to H^ i(U', K|_{U'})$, and the right vertical arrow is Remark 48.32.7 (see discussion before the lemma).

Proof. We strongly urge the reader to skip this proof. Choose $X$ and $M$ as in the proof of Lemma 48.33.1. We are going to drop the subscript $\mathcal{O}_ X$ from $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ and $\otimes ^\mathbf {L}$. We write

\[ H^ i(U, K) = \mathop{\mathrm{colim}}\nolimits H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{I}^ n, M)) \]

and

\[ H^ i(U', K|_{U'}) = \mathop{\mathrm{colim}}\nolimits H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits ((\mathcal{I}')^ n, M)) \]

as in the proof of Lemma 48.33.1 where we choose $\mathcal{I}' \subset \mathcal{I}$ as in the discussion in Remark 48.31.3 so that the map $H^ i(U, K) \to H^ i(U', K|_{U'})$ is induced by the maps $(\mathcal{I}')^ n \to \mathcal{I}^ n$. We similarly write

\[ H^ i_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, \omega _{U/k}^\bullet )) = \mathop{\mathrm{lim}}\nolimits H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, \omega _{X/k}^\bullet ) \otimes ^\mathbf {L} \mathcal{I}^ n) \]

and

\[ H^ i_ c(U', R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K|_{U'}, \omega _{U'/k}^\bullet )) = \mathop{\mathrm{lim}}\nolimits H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, \omega _{X/k}^\bullet ) \otimes ^\mathbf {L} (\mathcal{I}')^ n) \]

so that the arrow $H^ i_ c(U', R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K|_{U'}, \omega _{U'/k}^\bullet )) \to H^ i_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, \omega _{U/k}^\bullet ))$ is similarly deduced from the maps $(\mathcal{I}')^ n \to \mathcal{I}^ n$. The diagrams

\[ \xymatrix{ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, \omega _{X/k}^\bullet ) \otimes ^\mathbf {L} \mathcal{I}^ n \ar[rr] & & R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{I}^ n, M), \omega _{X/k}^\bullet ) \\ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, \omega _{X/k}^\bullet ) \otimes ^\mathbf {L} (\mathcal{I}')^ n \ar[rr] \ar[u] & & R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits ((\mathcal{I}')^ n, M), \omega _{X/k}^\bullet ) \ar[u] } \]

commute because the construction of the horizontal arrows in Cohomology, Lemma 20.39.9 is functorial in all three entries. Hence we finally come down to the assertion that the diagrams

\[ \xymatrix{ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{I}^ n, M)), k) \ar[r] & H^{-i}(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits ( \mathcal{I}^ n, M), \omega _{X/k}^\bullet )) \\ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits ((\mathcal{I}')^ n, M)), k) \ar[r] \ar[u] & H^{-i}(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits ( (\mathcal{I}')^ n, M), \omega _{X/k}^\bullet )) \ar[u] } \]

commute. This is true because the duality isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(X, L), k) = \mathop{\mathrm{Ext}}\nolimits ^{-i}_ X(L, \omega _{X/k}^\bullet ) = H^{-i}(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \omega _{X/k}^\bullet )) \]

is functorial for $L$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$. $\square$

Lemma 48.33.3. Let $X$ be a proper scheme over a field $k$. Let $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ with $H^ i(K) = 0$ for $i < 0$. Set $\mathcal{F} = H^0(K)$. Let $Z \subset X$ be closed with complement $U = X \setminus U$. Then

\[ H^0_ c(U, K|_ U) \subset H^0(X, \mathcal{F}) \]

is given by those global sections of $\mathcal{F}$ which vanish in an open neighbourhood of $Z$.

Proof. Consider the map $H^0_ c(U, K|_ U) \to H^0_ x(X, K) = H^0(X, K) = H^0(X, \mathcal{F})$ of Remark 48.32.7. To study this we represent $K$ by a bounded complex $\mathcal{F}^\bullet $ with $\mathcal{F}^ i = 0$ for $i < 0$. Then we have by definition

\[ H^0_ c(U, K|_ U) = \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{I}^ n\mathcal{F}^\bullet ) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Ker}}( H^0(X, \mathcal{I}^ n\mathcal{F}^0) \to H^0(X, \mathcal{I}^ n\mathcal{F}^1)) \]

By Artin-Rees (Cohomology of Schemes, Lemma 30.10.3) this is the same as $\mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{I}^ n\mathcal{F})$. Thus the arrow $H^0_ c(U, K|_ U) \to H^0(X, \mathcal{F})$ is injective and the image consists of those global sections of $\mathcal{F}$ which are contained in the subsheaf $\mathcal{I}^ n\mathcal{F}$ for any $n$. The characterization of these as the sections which vanish in a neighbourhood of $Z$ comes from Krull's intersection theorem (Algebra, Lemma 10.51.4) by looking at stalks of $\mathcal{F}$. See discussion in Algebra, Remark 10.51.6 for the case of functions. $\square$


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