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
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
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 20.37.1 vanish.
If $U' \subset U$ is an open subscheme, then there is a canonical map
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 étale 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 of topological $k$-vector spaces functorial for $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$.
\[ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(U, K), k) = H^{-i}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(K, \omega _{U/k}^\bullet )) \]
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
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
by Lemma 48.27.1. By Lemma 48.2.4 part (1) the canonical map
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
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 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).
\[ \xymatrix{ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(U, K), k) \ar[rr] & & H^{-i}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(K, \omega _{U/k}^\bullet )) \\ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(U', K|_{U'}), k) \ar[rr] \ar[u] & & H^{-i}_ c(U', R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{U'}}(K, \omega _{U'/k}^\bullet )) \ar[u] } \]
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
and
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
and
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
commute because the construction of the horizontal arrows in Cohomology, Lemma 20.42.9 is functorial in all three entries. Hence we finally come down to the assertion that the diagrams
commute. This is true because the duality isomorphism
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 is given by those global sections of $\mathcal{F}$ which vanish in an open neighbourhood of $Z$.
\[ H^0_ c(U, K|_ U) \subset H^0(X, \mathcal{F}) \]
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
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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