sites-modules-remark-when-shriek-equal

Warning! Let $u : \mathcal{C} \to \mathcal{D}$, $g$, $\mathcal{O}_\mathcal{D}$, and $\mathcal{O}_\mathcal{C}$ be as in Lemma \ref{lemma-lower-shriek-modules}. In general it is {\bf not} the case that the diagram $$ \xymatrix{ \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[r]_{g_!} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal{D}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}) \ar[r]^{g^{Ab}_!} & \textit{Ab}(\mathcal{D}) } $$ commutes (here $g^{Ab}_!$ is the one from Lemma \ref{lemma-g-shriek-adjoint}). There is a transformation of functors $$ g_!^{Ab} \circ forget \longrightarrow forget \circ g_! $$ From the proof of Lemma \ref{lemma-lower-shriek-modules} we see that this is an isomorphism if and only if $g^{Ab}_!j_{U!}\mathcal{O}_U \to g_!j_{U!}\mathcal{O}_U$ is an isomorphism for all objects $U$ of $\mathcal{C}$. Since we have $g_!j_{U!}\mathcal{O}_U = j_{u(U)!}\mathcal{O}_{u(U)}$ this holds if and only if $$ g^{Ab}_!j_{U!}\mathcal{O}_U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)} $$ is an isomorphism for all objects $U$ of $\mathcal{C}$. Note that for such a $U$ we obtain a commutative diagram $$ \xymatrix{ \mathcal{C}/U \ar[r]_-{u'} \ar[d]_{j_U} & \mathcal{D}/u(U) \ar[d]^{j_{u(U)}} \\ \mathcal{C} \ar[r]^u & \mathcal{D} } $$ of cocontinuous functors of sites, see Sites, Lemma \ref{sites-lemma-localize-cocontinuous} and therefore $g^{Ab}_!j_{U!} = j_{u(U)!}(g')^{Ab}_!$ where $g' : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{D}/u(U))$ is the morphism of topoi induced by the cocontinuous functor $u'$. Hence we see that $g_! = g^{Ab}_!$ if the canonical map \begin{equation} \label{equation-compare-on-localizations} (g')^{Ab}_!\mathcal{O}_U \longrightarrow \mathcal{O}_{u(U)} \end{equation} is an isomorphism for all objects $U$ of $\mathcal{C}$.


derham-remark-relative-cup-product

Let $p : X \to S$ be a morphism of schemes. Then we can think of $\Omega^\bullet_{X/S}$ as a sheaf of differential graded $p^{-1}\mathcal{O}_S$-algebras, see Differential Graded Sheaves, Definition \ref{sdga-definition-dga}. In particular, the discussion in Differential Graded Sheaves, Section \ref{sdga-section-misc} applies. For example, this means that for any commutative diagram $$ \xymatrix{ X \ar[d]_p \ar[r]_f & Y \ar[d]^q \\ S \ar[r]^h & T } $$ of schemes there is a canonical relative cup product $$ \mu : Rf_*\Omega^\bullet_{X/S} \otimes_{q^{-1}\mathcal{O}_T}^\mathbf{L} Rf_*\Omega^\bullet_{X/S} \longrightarrow Rf_*\Omega^\bullet_{X/S} $$ in $D(Y, q^{-1}\mathcal{O}_T)$ which is associative and which on cohomology reproduces the cup product discussed above.


derham-remark-cup-product-as-a-map

Let $f : X \to S$ be a morphism of schemes. Let $\xi \in H_{dR}^n(X/S)$. According to the discussion Differential Graded Sheaves, Section \ref{sdga-section-misc} there exists a canonical morphism $$ \xi' : \Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}[n] $$ in $D(f^{-1}\mathcal{O}_S)$ uniquely characterized by (1) and (2) of the following list of properties: \begin{enumerate} \item $\xi'$ can be lifted to a map in the derived category of right differential graded $\Omega^\bullet_{X/S}$-modules, and \item $\xi'(1) = \xi$ in $H^0(X, \Omega^\bullet_{X/S}[n]) = H^n_{dR}(X/S)$, \item the map $\xi'$ sends $\eta \in H^m_{dR}(X/S)$ to $\xi \cup \eta$ in $H^{n + m}_{dR}(X/S)$, \item the construction of $\xi'$ commutes with restrictions to opens: for $U \subset X$ open the restriction $\xi'|_U$ is the map corresponding to the image $\xi|_U \in H^n_{dR}(U/S)$, \item for any diagram as in Remark \ref{remark-relative-cup-product} we obtain a commutative diagram $$ \xymatrix{ Rf_*\Omega^\bullet_{X/S} \otimes_{q^{-1}\mathcal{O}_T}^\mathbf{L} Rf_*\Omega^\bullet_{X/S} \ar[d]_{\xi' \otimes \text{id}} \ar[r]_-\mu & Rf_*\Omega^\bullet_{X/S} \ar[d]^{\xi'} \\ Rf_*\Omega^\bullet_{X/S}[n] \otimes_{q^{-1}\mathcal{O}_T}^\mathbf{L} Rf_*\Omega^\bullet_{X/S} \ar[r]^-\mu & Rf_*\Omega^\bullet_{X/S}[n] } $$ in $D(Y, q^{-1}\mathcal{O}_T)$. \end{enumerate}


derham-remark-truncations

Here is a reformulation of the calculations above in more abstract terms. Let $p : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. If we view $\text{d}\log$ as a map $$ \mathcal{O}_X^*[-1] \to \sigma_{\geq 1}\Omega^\bullet_{X/S} $$ then using $\Pic(X) = H^1(X, \mathcal{O}_X^*)$ as above we find a cohomology class $$ \gamma_1(\mathcal{L}) \in H^2(X, \sigma_{\geq 1}\Omega^\bullet_{X/S}) $$ The image of $\gamma_1(\mathcal{L})$ under the map $\sigma_{\geq 1}\Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}$ recovers $c_1^{dR}(\mathcal{L})$. In particular we see that $c_1^{dR}(\mathcal{L}) \in F^1H^2_{dR}(X/S)$, see Section \ref{section-hodge-filtration}. The image of $\gamma_1(\mathcal{L})$ under the map $\sigma_{\geq 1}\Omega^\bullet_{X/S} \to \Omega^1_{X/S}[-1]$ recovers $c_1^{Hodge}(\mathcal{L})$. Taking the cup product (see Section \ref{section-hodge-filtration}) we obtain $$ \xi = \gamma_1(\mathcal{L}_1) \cup \ldots \cup \gamma_1(\mathcal{L}_a) \in H^{2a}(X, \sigma_{\geq a}\Omega^\bullet_{X/S}) $$ The commutative diagrams in Section \ref{section-hodge-filtration} show that $\xi$ is mapped to $c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_a)$ in $H^{2a}_{dR}(X/S)$ by the map $\sigma_{\geq a}\Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}$. Also, it follows $c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_a)$ is contained in $F^a H^{2a}_{dR}(X/S)$. Similarly, the map $\sigma_{\geq a}\Omega^\bullet_{X/S} \to \Omega^a_{X/S}[-a]$ sends $\xi$ to $c_1^{Hodge}(\mathcal{L}_1) \cup \ldots \cup c_1^{Hodge}(\mathcal{L}_a)$ in $H^a(X, \Omega^a_{X/S})$.


derham-remark-log-forms

Let $p : X \to S$ be a morphism of schemes. For $i > 0$ denote $\Omega^i_{X/S, log} \subset \Omega^i_{X/S}$ the abelian subsheaf generated by local sections of the form $$ \text{d}\log(u_1) \wedge \ldots \wedge \text{d}\log(u_i) $$ where $u_1, \ldots, u_n$ are invertible local sections of $\mathcal{O}_X$. For $i = 0$ the subsheaf $\Omega^0_{X/S, log} \subset \mathcal{O}_X$ is the image of $\mathbf{Z} \to \mathcal{O}_X$. For every $i \geq 0$ we have a map of complexes $$ \Omega^i_{X/S, log}[-i] \longrightarrow \Omega^\bullet_{X/S} $$ because the derivative of a logarithmic form is zero. Moreover, wedging logarithmic forms gives another, hence we find bilinear maps $$ \wedge :  \Omega^i_{X/S, log} \times \Omega^j_{X/S, log} \longrightarrow \Omega^{i + j}_{X/S, log} $$ compatible with (\ref{equation-wedge}) and the maps above. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Using the map of abelian sheaves $\text{d}\log : \mathcal{O}_X^* \to \Omega^1_{X/S, log}$ and the identification $\Pic(X) = H^1(X, \mathcal{O}_X^*)$ we find a canonical cohomology class $$ \tilde \gamma_1(\mathcal{L}) \in H^1(X, \Omega^1_{X/S, log}) $$ These classes have the following properties \begin{enumerate} \item the image of $\tilde \gamma_1(\mathcal{L})$ under the canonical map $\Omega^1_{X/S, log}[-1] \to \sigma_{\geq 1}\Omega^\bullet_{X/S}$ sends $\tilde \gamma_1(\mathcal{L})$ to the class $\gamma_1(\mathcal{L}) \in  H^2(X, \sigma_{\geq 1}\Omega^\bullet_{X/S})$ of Remark \ref{remark-truncations}, \item the image of $\tilde \gamma_1(\mathcal{L})$ under the canonical map $\Omega^1_{X/S, log}[-1] \to \Omega^\bullet_{X/S}$ sends $\tilde \gamma_1(\mathcal{L})$ to $c_1^{dR}(\mathcal{L})$ in $H^2_{dR}(X/S)$, \item the image of $\tilde \gamma_1(\mathcal{L})$ under the canonical map $\Omega^1_{X/S, log} \to \Omega^1_{X/S}$ sends $\tilde \gamma_1(\mathcal{L})$ to $c_1^{Hodge}(\mathcal{L})$ in $H^1(X, \Omega^1_{X/S})$, \item the construction of these classes is compatible with pullbacks, \item add more here. \end{enumerate}


derham-remark-de-rham-complex-graded

Let $G$ be an abelian monoid written additively with neutral element $0$. Let $R \to A$ be a ring map and assume $A$ comes with a grading $A = \bigoplus_{g \in G} A_g$ by $R$-modules such that $R$ maps into $A_0$ and $A_g \cdot A_{g'} \subset A_{g + g'}$. Then the module of differentials comes with a grading $$ \Omega_{A/R} = \bigoplus\nolimits_{g \in G} \Omega_{A/R, g} $$ where $\Omega_{A/R, g}$ is the $R$-submodule of $\Omega_{A/R}$ generated by $a_0 \text{d}a_1$ with $a_i \in A_{g_i}$ such that $g = g_0 + g_1$. Similarly, we obtain $$ \Omega^p_{A/R} = \bigoplus\nolimits_{g \in G} \Omega^p_{A/R, g} $$ where $\Omega^p_{A/R, g}$ is the $R$-submodule of $\Omega^p_{A/R}$ generated by $a_0 \text{d}a_1 \wedge \ldots \wedge \text{d}a_p$ with $a_i \in A_{g_i}$ such that $g = g_0 + g_1 + \ldots + g_p$. Of course the differentials preserve the grading and the wedge product is compatible with the gradings in the obvious manner.


derham-remark-gauss-manin

In Lemma \ref{lemma-spectral-sequence-smooth} consider the cohomology sheaves $$ \mathcal{H}^q_{dR}(X/Y) = H^q(Rf_*\Omega^\bullet_{X/Y})) $$ If $f$ is proper in addition to being smooth and $S$ is a scheme over $\mathbf{Q}$ then $\mathcal{H}^q_{dR}(X/Y)$ is finite locally free (insert future reference here). If we only assume $\mathcal{H}^q_{dR}(X/Y)$ are flat $\mathcal{O}_Y$-modules, then we obtain (tiny argument omitted) $$ E_1^{p, q} = \Omega^p_{Y/S} \otimes_{\mathcal{O}_Y} \mathcal{H}^q_{dR}(X/Y) $$ and the differentials in the spectral sequence are maps $$ d_1^{p, q} : \Omega^p_{Y/S} \otimes_{\mathcal{O}_Y} \mathcal{H}^q_{dR}(X/Y) \longrightarrow \Omega^{p + 1}_{Y/S} \otimes_{\mathcal{O}_Y} \mathcal{H}^q_{dR}(X/Y) $$ In particular, for $p = 0$ we obtain a map $d_1^{0, q} : \mathcal{H}^q_{dR}(X/Y) \to \Omega^1_{Y/S} \otimes_{\mathcal{O}_Y} \mathcal{H}^q_{dR}(X/Y)$ which turns out to be an integrable connection $\nabla$ (insert future reference here) and the complex $$ \mathcal{H}^q_{dR}(X/Y) \to \Omega^1_{Y/S} \otimes_{\mathcal{O}_Y} \mathcal{H}^q_{dR}(X/Y) \to \Omega^2_{Y/S} \otimes_{\mathcal{O}_Y} \mathcal{H}^q_{dR}(X/Y) \to \ldots $$ with differentials given by $d_1^{\bullet, q}$ is the de Rham complex of $\nabla$. The connection $\nabla$ is known as the {\it Gauss-Manin connection}.


derham-remark-projective-space-bundle-formula

In the situation of Proposition \ref{proposition-projective-space-bundle-formula} we get moreover that the map $$ \tilde \xi : \bigoplus\nolimits_{t = 0, \ldots, r - 1} \Omega^\bullet_{X/S}[-2t] \longrightarrow Rp_*\Omega^\bullet_{P/S} $$ is an isomorphism in $D(X, (X \to S)^{-1}\mathcal{O}_X)$ as follows immediately from the application of Proposition \ref{proposition-global-generation-on-fibres}. Note that the arrow for $t = 0$ is simply the canonical map $c_{P/X} : \Omega^\bullet_{X/S} \to Rp_*\Omega^\bullet_{P/S}$ of Section \ref{section-de-rham-complex}. In fact, we can pin down this map further in this particular case. Namely, consider the canonical map $$ \xi' : \Omega^\bullet_{P/S} \to \Omega^\bullet_{P/S}[2] $$ of Remark \ref{remark-cup-product-as-a-map} corresponding to $c_1^{dR}(\mathcal{O}_P(1))$. Then $$ \xi'[2(t - 1)] \circ \ldots \circ \xi'[2] \circ \xi' :  \Omega^\bullet_{P/S} \to \Omega^\bullet_{P/S}[2t] $$ is the map of Remark \ref{remark-cup-product-as-a-map} corresponding to $c_1^{dR}(\mathcal{O}_P(1))^t$. Tracing through the choices made in the proof of Proposition \ref{proposition-global-generation-on-fibres} we find the value $$ \tilde \xi|_{\Omega^\bullet_{X/S}[-2t]} = Rp_*\xi'[-2] \circ \ldots \circ Rp_*\xi'[-2(t - 1)] \circ Rp_*\xi'[-2t] \circ c_{P/X}[-2t] $$ for the restriction of our isomorphism to the summand $\Omega^\bullet_{X/S}[-2t]$. This has the following simple consequence we will use below: let $$ M = \bigoplus\nolimits_{t = 1, \ldots, r - 1} \Omega^\bullet_{X/S}[-2t] \quad\text{and}\quad K = \bigoplus\nolimits_{t = 0, \ldots, r - 2} \Omega^\bullet_{X/S}[-2t] $$ viewed as subcomplexes of the source of the arrow $\tilde \xi$. It follows formally from the discussion above that $$ c_{P/X} \oplus \tilde \xi|_M : \Omega^\bullet_{X/S} \oplus M \longrightarrow Rp_*\Omega^\bullet_{P/S} $$ is an isomorphism and that the diagram $$ \xymatrix{ K \ar[d]_{\tilde \xi|_K} \ar[r]_{\text{id}} & M[2] \ar[d]^{(\tilde \xi|_M)[2]} \\ Rp_*\Omega^\bullet_{P/S} \ar[r]^{Rp_*\xi'} & Rp_*\Omega^\bullet_{P/S}[2] } $$ commutes where $\text{id} : K \to M[2]$ identifies the summand corresponding to $t$ in the deomposition of $K$ to the summand corresponding to $t + 1$ in the decomposition of $M$.


derham-remark-check-log-completion-1

Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite type over $S$. Let $Y \subset X$ be an effective Cartier divisor. If the map $$ \mathcal{O}_{X, y}^\wedge \longrightarrow \mathcal{O}_{Y, y}^\wedge $$ has a section for all $y \in Y$, then the de Rham complex of log poles is defined for $Y \subset X$ over $S$. If we ever need this result we will formulate a precise statement and add a proof here.


derham-remark-check-log-completion-2

Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite type over $S$. Let $Y \subset X$ be an effective Cartier divisor. If for every $y \in Y$ we can find a diagram of schemes over $S$ $$ X \xleftarrow{\varphi} U \xrightarrow{\psi} V $$ with $\varphi$ \'etale and $\psi|_{\varphi^{-1}(Y)} : \varphi^{-1}(Y) \to V$ \'etale, then the de Rham complex of log poles is defined for $Y \subset X$ over $S$. A special case is when the pair $(X, Y)$ \'etale locally looks like $(V \times \mathbf{A}^1, V \times \{0\})$. If we ever need this result we will formulate a precise statement and add a proof here.


derham-remark-local-description

Let $A$ be a ring. Let $P = A[x_1, \ldots, x_n]$. Let $f_1, \ldots, f_n \in P$ and set $B = P/(f_1, \ldots, f_n)$. Assume $A \to B$ is quasi-finite. Then $B$ is a relative global complete intersection over $A$ (Algebra, Definition \ref{algebra-definition-relative-global-complete-intersection}) and $(f_1, \ldots, f_n)/(f_1, \ldots, f_n)^2$ is free with generators the classes $\overline{f}_i$ by Algebra, Lemma \ref{algebra-lemma-relative-global-complete-intersection-conormal}. Consider the following diagram $$ \xymatrix{ \Omega_{A/\mathbf{Z}} \otimes_A B \ar[r] & \Omega_{P/\mathbf{Z}} \otimes_P B \ar[r] & \Omega_{P/A} \otimes_P B \\ & (f_1, \ldots, f_n)/(f_1, \ldots, f_n)^2 \ar[u] \ar@{=}[r] & (f_1, \ldots, f_n)/(f_1, \ldots, f_n)^2 \ar[u] } $$ The right column represents $\NL_{B/A}$ in $D(B)$ hence has cohomology $\Omega_{B/A}$ in degree $0$. The top row is the split short exact sequence $0 \to \Omega_{A/\mathbf{Z}} \otimes_A B \to \Omega_{P/\mathbf{Z}} \otimes_P B \to \Omega_{P/A} \otimes_P B \to 0$. The middle column has cohomology $\Omega_{B/\mathbf{Z}}$ in degree $0$ by Algebra, Lemma \ref{algebra-lemma-differential-seq}. Thus by Lemma \ref{lemma-funny-map} we obtain canonical $B$-module maps $$ \Omega^p_{B/\mathbf{Z}} \longrightarrow \Omega^p_{A/\mathbf{Z}} \otimes_A \det(\NL_{B/A}) $$ whose composition with $\Omega^p_{A/\mathbf{Z}} \to \Omega^p_{B/\mathbf{Z}}$ is multiplication by $\delta(\NL_{B/A})$.


derham-remark-splitting-principle

The analogues of Weil Cohomology Theories, Lemmas \ref{weil-lemma-splitting-principle} (splitting principle) and \ref{weil-lemma-chern-classes-E-tensor-L} (chern classes of tensor products) hold for de Rham Chern classes on quasi-compact and quasi-separated schemes. This is clear as we've shown in the proof of Lemma \ref{lemma-chern-classes} that all the axioms of Weil Cohomology Theories, Section \ref{weil-section-chern} are satisfied.


derham-remark-hodge-cohomology-is-weil

In exactly the same manner as above one can show that Hodge cohomology $X \mapsto H_{Hodge}^*(X/k)$ equipped with $c_1^{Hodge}$ determines a Weil cohomology theory. If we ever need this, we will precisely formulate and prove this here. This leads to the following amusing consequence: If the betti numbers of a Weil cohomology theory are independent of the chosen Weil cohomology theory (over our field $k$ of characteristic $0$), then the Hodge-to-de Rham spectral sequence degenerates at $E_1$! Of course, the degeneration of the Hodge-to-de Rham spectral sequence is known (see for example \cite{Deligne-Illusie} for a marvelous algebraic proof), but it is by no means an easy result! This suggests that proving the independence of betti numbers is a hard problem as well and as far as we know is still an open problem. See Weil Cohomology Theories, Remark \ref{weil-remark-betti-numbers-in-some-sense} for a related question.


derham-remark-gysin-equations

Let $X \to S$ be a morphism of schemes. Let $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$. Let $Z \subset X$ be the closed subscheme cut out by $f_1, \ldots, f_c$. Below we will study the {\it gysin map} \begin{equation} \label{equation-gysin} \gamma^p_{f_1, \ldots, f_c} : \Omega^p_{Z/S} \longrightarrow \mathcal{H}_Z^c(\Omega^{p + c}_{X/S}) \end{equation} defined as follows. Given a local section $\omega$ of $\Omega^p_{Z/S}$ which is the restriction of a section $\tilde \omega$ of $\Omega^p_{X/S}$ we set $$ \gamma^p_{f_1, \ldots, f_c}(\omega) = c_{f_1, \ldots, f_c}(\tilde \omega|_Z) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_c $$ where $c_{f_1, \ldots, f_c} : \Omega^p_{X/S} \otimes \mathcal{O}_Z \to \mathcal{H}_Z^c(\Omega^p_{X/S})$ is the map constructed in Derived Categories of Schemes, Remark \ref{perfect-remark-supported-map-c-equations}. This is well defined: given $\omega$ we can change our choice of $\tilde \omega$ by elements of the form $\sum f_i \omega'_i + \sum \text{d}(f_i) \wedge \omega''_i$ which are mapped to zero by the construction.


derham-remark-how-to-use

Let $X \to S$, $i : Z \to X$, and $c \geq 0$ be as in Lemma \ref{lemma-gysin-global}. Let $p \geq 0$ and assume that $\mathcal{H}^i_Z(\Omega^{p + c}_{X/S}) = 0$ for $i = 0, \ldots, c - 1$. This vanishing holds if $X \to S$ is smooth and $Z \to X$ is a Koszul regular immersion, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-supported-vanishing}. Then we obtain a map $$ \gamma^{p, q} : H^q(Z, \Omega^p_{Z/S}) \longrightarrow H^{q + c}(X, \Omega^{p + c}_{X/S}) $$ by first using $\gamma^p : \Omega^p_{Z/S} \to \mathcal{H}^c_Z(\Omega^{p + c}_{X/S})$ to map into $$ H^q(Z, \mathcal{H}^c_Z(\Omega^{p + c}_{X/S})) = H^q(Z, R\mathcal{H}_Z(\Omega^{p + c}_{X/S})[c]) = H^q(X, i_*R\mathcal{H}_Z(\Omega^{p + c}_{X/S})[c]) $$ and then using the adjunction map $i_*R\mathcal{H}_Z(\Omega^{p + c}_{X/S}) \to \Omega^{p + c}_{X/S}$ to continue on to the desired Hodge cohomology module.


trace-remark-may-be-confusing

It may or may not be the case that $F^f_U$ equals $\pi_U$.


trace-remark-compute-degree-lifting

The computation of the degrees can be done by lifting (in some obvious sense) to characteristic 0 and considering the situation with complex coefficients. This method almost never works, since lifting is in general impossible for schemes which are not projective space.


trace-remarks-derived-categories

Notes on derived categories. \begin{enumerate} \item There are some set-theoretical problems when $\mathcal{A}$ is somewhat arbitrary, which we will happily disregard. \item The categories $K(A)$ and $D(A)$ are endowed with the structure of a triangulated category. \item The categories $\text{Comp}(\mathcal{A})$ and $K(\mathcal{A})$ can also be defined when $\mathcal{A}$ is an additive category. \end{enumerate}


trace-remark-cohomology-of-derived-functor

In these cases, it is true that $R^iF(K^\bullet) = H^i(RF(K^\bullet))$, where the left hand side is defined to be $i$th homology of the complex $F(K^\bullet)$.


trace-remark-variant

A variant of this lemma is to consider a Noetherian scheme $X$ and the category $D_{perf}(\mathcal{O}_X)$ of complexes which are locally quasi-isomorphic to a finite complex of finite locally free $\mathcal{O}_X$-modules. Objects $K$ of $D_{perf}(\mathcal{O}_X)$ can be characterized by having coherent cohomology sheaves and bounded tor dimension.


trace-remark-content-trivial-trace

Let us try to illustrate the content of the formula of Lemma \ref{lemma-weak-trace}. Suppose that $\Lambda$, viewed as a trivial $\Gamma$-module, admits a finite resolution $ 0\to P_r\to \ldots \to P_1 \to P_0\to \Lambda\to 0 $ by some $\Lambda[\Gamma]$-modules $P_i$ which are finite and projective as $\Lambda[G]$-modules. In that case $$ H_*\left(\left(P_\bullet\right)_G\right) = \text{Tor}_*^{\Lambda[G]}\left(\Lambda, \Lambda\right) = H_*(G, \Lambda) $$ and $$ \text{Tr}_\Lambda^{Z_\gamma}\left(\gamma, P_\bullet\right) =\frac{1}{\# Z_\gamma}\text{Tr}_\Lambda(\gamma, P_\bullet)=\frac{1}{\# Z_\gamma}\text{Tr}(\gamma, \Lambda) = \frac{1}{\# Z_\gamma}. $$ Therefore, Lemma \ref{lemma-weak-trace} says $$ \text{Tr}_\Lambda (1 , P_G) = \text{Tr}\left(1 |_{H_*(G, \Lambda)}\right) = {\sum_{\gamma\mapsto 1}}'\frac{1}{\# Z_\gamma}. $$ This can be interpreted as a point count on the stack $BG$. If $\Lambda = \mathbf{F}_\ell$ with $\ell$ prime to $\# G$, then $H_*(G, \Lambda)$ is $\mathbf{F}_\ell$ in degree 0 (and $0$ in other degrees) and the formula reads $$ 1 = \sum\nolimits_{ \frac{\sigma\text{-conjugacy}}{\text{classes}\langle\gamma\rangle} } \frac{1}{\# Z_\gamma} \mod \ell. $$ This is in some sense a ``trivial'' trace formula for $G$. Later we will see that (\ref{equation-trace-formula}) can in some cases be viewed as a highly nontrivial trace formula for a certain type of group, see Section \ref{section-abstract-trace-formula}.


trace-remark-on-trace-formula-again

Remarks on Theorem \ref{theorem-trace-formula-again}. \begin{enumerate} \item This formula holds in any dimension. By a d\'evissage lemma (which uses proper base change etc.) it reduces to the current statement -- in that generality. \item The complex $R\Gamma_c(X_{\bar k}, K)$ is defined by choosing an open immersion $j : X \hookrightarrow \bar X$ with $\bar X$ projective over $k$ of dimension at most 1 and setting $$ R\Gamma_c(X_{\bar k}, K) := R\Gamma(\bar X_{\bar k}, j_!K). $$ This is independent of the choice of $\bar X$ follows from (insert reference here). We define $H^i_c(X_{\bar k}, K)$ to be the $i$th cohomology group of $R\Gamma_c(X_{\bar k}, K)$. \end{enumerate}


trace-remark-stronger

Even though all we did are reductions and mostly algebra, the trace formula Theorem \ref{theorem-trace-formula-again} is much stronger than Weil's geometric trace formula (Theorem \ref{theorem-weil-trace-formula}) because it applies to coefficient systems (sheaves), not merely constant coefficients.


trace-remark-stalk-l-adic-sheaf

If $\mathcal{F} = \left\{\mathcal{F}_n\right\}_{n\geq 1}$ is a $\mathbf{Z}_\ell$-sheaf on $X$ and $\bar x$ is a geometric point then $M_n = \left\{\mathcal{F}_{n, \bar x}\right\}$ is an inverse system of finite $\mathbf{Z}/\ell^n\mathbf{Z}$-modules such that $M_{n+1}\to M_n$ is surjective and $M_n = M_{n+1}/\ell^n M_{n+1}$. It follows that $$ M = \lim_n M_n = \lim \mathcal{F}_{n, \bar x} $$ is a finite $\mathbf{Z}_\ell$-module. Indeed, $M/\ell M= M_1$ is finite over $\mathbf{F}_\ell$, so by Nakayama $M$ is finite over $\mathbf{Z}_\ell$. Therefore, $M\cong \mathbf{Z}_\ell^{\oplus r} \oplus \oplus_{i = 1}^t \mathbf{Z}_\ell/\ell^{e_i}\mathbf{Z}_\ell$ for some $r, t\geq 0$, $e_i\geq 1$. The module $M = \mathcal{F}_{\bar x}$ is called the {\it stalk} of $\mathcal{F}$ at $\bar x$.


trace-remark-torsion-stalks

Since a $\mathbf{Z}_\ell$-sheaf is only defined on a Noetherian scheme, it is torsion if and only if its stalks are torsion.


trace-remark-T

Intuitively, $T$ should be thought of as $T = t^f$ where $p^f = \# k$. The definitions are then independent of the size of the ground field.


trace-remark-which-is-harder

Since we have only developed some theory of traces and not of determinants, Theorem \ref{theorem-A} is harder to prove than Theorem \ref{theorem-B}. We will only prove the latter, for the former see \cite{SGA4.5}. Observe also that there is no version of this theorem more general for $\mathbf{Z}_\ell$ coefficients since there is no $\ell$-torsion.


trace-remark-projective

Thus we conclude that if $X$ is also projective then we have functorially in the representation $\rho$ the identifications $$ H^0(X_{\overline{k}}, \mathcal{F}_\rho) = M^{\pi_1(X_{\overline{k}}, \overline\eta)} $$ and $$ H_c^2(X_{\overline{k}}, \mathcal{F}_\rho) = M_{\pi_1(X_{\overline{k}}, \overline \eta)}(-1) $$ Of course if $X$ is not projective, then $H^0_c(X_{\overline{k}}, \mathcal{F}_\rho) = 0$.


trace-remark-poincare-groups

By the proposition and Trivial duality then you get $$ H^{2-i}_c(X_{\overline{k}}, \mathcal{F}_\rho) \times H^i(X_{\overline{k}}, \mathcal{F}_\rho^\wedge(1)) \to \mathbf{Q}/\mathbf{Z} $$ a perfect pairing. If $X$ is projective then this is Poincare duality.


trace-remark-abstract-trace-formula

Here are some observations concerning this notion. \begin{enumerate} \item If modeling projective curves then we can use cohomology and we don't need factor $q^n$. \item The only examples I know are $\Gamma = \pi_1(X, \overline \eta)$ where $X$ is smooth, geometrically irreducible and $K(\pi, 1)$ over finite field. In this case $q = (\# k)^{\dim X}$. Modulo the proposition, we proved this for curves in this course. \item Given the integer $q$ then the sets $S_d$ are uniquely determined. (You can multiple $q$ by an integer $m$ and then replace $S_d$ by $m^d$ copies of $S_d$ without changing the formula.) \end{enumerate}


trace-remark-lafforgue

We now have, thanks to Lafforgue and many other mathematicians, complete theorems like this two above for $\text{GL}_n$ and allowing ramification! In other words, the full global Langlands correspondence for $\text{GL}_n$ is known for function fields of curves over finite fields. At the same time this does not mean there aren't a lot of interesting questions left to answer about the fundamental groups of curves over finite fields, as we shall see below.


sheaves-remark-confusion

There is always a bit of confusion as to whether it is necessary to say something about the set of sections of a sheaf over the empty set $\emptyset \subset X$. It is necessary, and we already did if you read the definition right. Namely, note that the empty set is covered by the empty open covering, and hence the ``collection of sections $s_i$'' from the definition above actually form an element of the empty product which is the final object of the category the sheaf has values in. In other words, if you read the definition right you automatically deduce that $\mathcal{F}(\emptyset) = \textit{a final object}$, which in the case of a sheaf of sets is a singleton. If you do not like this argument, then you can just require that $\mathcal{F}(\emptyset) = \{*\}$. \medskip\noindent In particular, this condition will then ensure that if $U, V \subset X$ are open and {\it disjoint} then $$ \mathcal{F}(U \cup V) = \mathcal{F}(U) \times \mathcal{F}(V). $$ (Because the fibre product over a final object is a product.)


sheaves-remark-j-shriek-not-exact

Let $j : U \to X$ be an open immersion of topological spaces as above. Let $x \in X$, $x \not \in U$. Let $\mathcal{F}$ be a sheaf of sets on $U$. Then $j_!\mathcal{F}_x = \emptyset$ by Lemma \ref{lemma-j-shriek}. Hence $j_!$ does not transform a final object of $\Sh(U)$ into a final object of $\Sh(X)$ unless $U = X$. According to our conventions in Categories, Section \ref{categories-section-exact-functor} this means that the functor $j_!$ is not left exact as a functor between the categories of sheaves of sets. It will be shown later that $j_!$ on abelian sheaves is exact, see Modules, Lemma \ref{modules-lemma-j-shriek-exact}.


sheaves-remark-i-star-not-exact

Let $i : Z \to X$ be a closed immersion of topological spaces as above. Let $x \in X$, $x \not \in Z$. Let $\mathcal{F}$ be a sheaf of sets on $Z$. Then $(i_*\mathcal{F})_x = \{ * \}$ by Lemma \ref{lemma-stalks-closed-pushforward}. Hence if $\mathcal{F} = * \amalg *$, where $*$ is the singleton sheaf, then $i_*\mathcal{F}_x = \{*\} \not = i_*(*)_x \amalg i_*(*)_x$ because the latter is a two point set. According to our conventions in Categories, Section \ref{categories-section-exact-functor} this means that the functor $i_*$ is not right exact as a functor between the categories of sheaves of sets. In particular, it cannot have a right adjoint, see Categories, Lemma \ref{categories-lemma-exact-adjoint}. \medskip\noindent On the other hand, we will see later (see Modules, Lemma \ref{modules-lemma-i-star-right-adjoint}) that $i_*$ on abelian sheaves is exact, and does have a right adjoint, namely the functor that associates to an abelian sheaf on $X$ the sheaf of sections supported in $Z$.


sheaves-remark-closed-immersion-spaces

We have not discussed the relationship between closed immersions and ringed spaces. This is because the notion of a closed immersion of ringed spaces is best discussed in the setting of quasi-coherent sheaves, see Modules, Section \ref{modules-section-closed-immersion}.


descent-remark-standard-covering

Let $R$ be a ring. Let $f_1, \ldots, f_n\in R$ generate the unit ideal. The ring $A = \prod_i R_{f_i}$ is a faithfully flat $R$-algebra. We remark that the cosimplicial ring $(A/R)_\bullet$ has the following ring in degree $n$: $$ \prod\nolimits_{i_0, \ldots, i_n} R_{f_{i_0}\ldots f_{i_n}} $$ Hence the results above recover Algebra, Lemmas \ref{algebra-lemma-standard-covering}, \ref{algebra-lemma-cover-module} and \ref{algebra-lemma-glue-modules}. But the results above actually say more because of exactness in higher degrees. Namely, it implies that {\v C}ech cohomology of quasi-coherent sheaves on affines is trivial. Thus we get a second proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-cech-cohomology-quasi-coherent-trivial}.


descent-remark-homotopy-equivalent-cosimplicial-algebras

Let $R$ be a ring. Let $A_\bullet$ be a cosimplicial $R$-algebra. In this setting a descent datum corresponds to an cosimplicial $A_\bullet$-module $M_\bullet$ with the property that for every $n, m \geq 0$ and every $\varphi : [n] \to [m]$ the map $M(\varphi) : M_n \to M_m$ induces an isomorphism $$ M_n \otimes_{A_n, A(\varphi)} A_m \longrightarrow M_m. $$ Let us call such a cosimplicial module a {\it cartesian module}. In this setting, the proof of Proposition \ref{proposition-descent-module} can be split in the following steps \begin{enumerate} \item If $R \to R'$ and $R \to A$ are faithfully flat, then descent data for $A/R$ are effective if descent data for $(R' \otimes_R A)/R'$ are effective. \item Let $A$ be an $R$-algebra. Descent data for $A/R$ correspond to cartesian $(A/R)_\bullet$-modules. \item If $R \to A$ has a section then $(A/R)_\bullet$ is homotopy equivalent to $R$, the constant cosimplicial $R$-algebra with value $R$. \item If $A_\bullet \to B_\bullet$ is a homotopy equivalence of cosimplicial $R$-algebras then the functor $M_\bullet \mapsto M_\bullet \otimes_{A_\bullet} B_\bullet$ induces an equivalence of categories between cartesian $A_\bullet$-modules and cartesian $B_\bullet$-modules. \end{enumerate} For (1) see Lemma \ref{lemma-descent-descends}. Part (2) uses Lemma \ref{lemma-descent-datum-cosimplicial}. Part (3) we have seen in the proof of Lemma \ref{lemma-with-section-exact} (it relies on Simplicial, Lemma \ref{simplicial-lemma-push-outs-simplicial-object-w-section}). Moreover, part (4) is a triviality if you think about it right!


descent-remark-reflects

Any functor $F : \mathcal{A} \to \mathcal{B}$ of abelian categories which is exact and takes nonzero objects to nonzero objects reflects injections and surjections. Namely, exactness implies that $F$ preserves kernels and cokernels (compare with Homology, Section \ref{homology-section-functors}). For example, if $f : R \to S$ is a  faithfully flat ring homomorphism, then $\bullet \otimes_R S: \text{Mod}_R \to \text{Mod}_S$ has these properties.


descent-remark-adjunction

We will use frequently the standard adjunction between $\Hom$ and tensor  product, in the form of the natural isomorphism of contravariant functors \begin{equation} \label{equation-adjunction} C(\bullet_1 \otimes_R \bullet_2) \cong \Hom_R(\bullet_1, C(\bullet_2)):  \text{Mod}_R \times \text{Mod}_R \to \text{Mod}_R \end{equation} taking $f: M_1 \otimes_R M_2 \to \mathbf{Q}/\mathbf{Z}$ to the map $m_1 \mapsto  (m_2 \mapsto f(m_1 \otimes m_2))$. See Algebra, Lemma \ref{algebra-lemma-hom-from-tensor-product-variant}. A corollary of this observation is that if $$ \xymatrix@C=9pc{ C(M) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N) \ar[r] & C(P) } $$ is a split coequalizer diagram in $\text{Mod}_R$, then so is $$ \xymatrix@C=9pc{ C(M \otimes_R Q) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N \otimes_R Q) \ar[r] & C(P  \otimes_R Q) } $$ for any $Q \in \text{Mod}_R$.


descent-remark-functorial-splitting

Let $f: M \to N$ be a universally injective morphism in $\text{Mod}_R$. By  choosing a splitting $g$ of $C(f)$, we may construct a functorial splitting of $C(1_P \otimes f)$  for each $P \in \text{Mod}_R$. Namely, by (\ref{equation-adjunction}) this amounts to splitting $\Hom_R(P,  C(f))$  functorially in $P$, and this is achieved by the map $g \circ \bullet$.


descent-remark-descent-lemma

If $f$ is a split injection in $\text{Mod}_R$, one can simplify the argument by  splitting $f$ directly, without using $C$. Things are even simpler if $f$ is faithfully flat; in this  case, the conclusion of Lemma \ref{lemma-descent-lemma}  is immediate because tensoring over $R$ with $S$ preserves all equalizers.


descent-remark-when-locally-split

It would make things easier to have a faithfully flat ring homomorphism $g: R \to T$ for which $T \to S \otimes_R T$ has some  extra structure. For instance, if one could ensure that $T \to S \otimes_R T$ is split in  $\textit{Rings}$, then it would follow that every property of a module or algebra which is stable  under base extension and which descends along faithfully flat morphisms also descends along  universally injective morphisms. An obvious guess would be to find $g$ for which $T$ is not only faithfully flat  but also injective in $\text{Mod}_R$, but even for $R = \mathbf{Z}$ no such homomorphism can exist.


descent-remark-locally-free-descends

Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology. Namely, suppose that $R$ is a ring and that $M$ is a projective $R$-module which is a countable direct sum $M = \bigoplus L_n$ of rank 1 locally free modules, but not locally free, see Examples, Lemma \ref{examples-lemma-projective-not-locally-free}. Then $M$ becomes free on making the faithfully flat base change $$ R \longrightarrow \bigoplus\nolimits_{m \geq 1} \bigoplus\nolimits_{(i_1, \ldots, i_m) \in \mathbf{Z}^{\oplus m}} L_1^{\otimes i_1} \otimes_R \ldots \otimes_R L_m^{\otimes i_m} $$ But we don't know what happens for fppf coverings. In other words, we don't know the answer to the following question: Suppose $A \to B$ is a faithfully flat ring map of finite presentation. Let $M$ be an $A$-module such that $M \otimes_A B$ is free. Is $M$ a locally free $A$-module? It turns out that if $A$ is Noetherian, then the answer is yes. This follows from the results of \cite{Bass}. But in general we don't know the answer. If you know the answer, or have a reference, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.


descent-remark-Zariski-site-space

In Topologies, Lemma \ref{topologies-lemma-Zariski-usual} we have seen that the small Zariski site of a scheme $S$ is equivalent to $S$ as a topological space in the sense that the categories of sheaves are naturally equivalent. Now that $S_{Zar}$ is also endowed with a structure sheaf $\mathcal{O}$ we see that sheaves of modules on the ringed site $(S_{Zar}, \mathcal{O})$ agree with sheaves of modules on the ringed space $(S, \mathcal{O}_S)$.


descent-remark-change-topologies-ringed

Let $f : T \to S$ be a morphism of schemes. Each of the morphisms of sites $f_{sites}$ listed in Topologies, Section \ref{topologies-section-change-topologies} becomes a morphism of ringed sites. Namely, each of these morphisms of sites $f_{sites} : (\Sch/T)_\tau \to (\Sch/S)_{\tau'}$, or $f_{sites} : (\Sch/S)_\tau \to S_{\tau'}$ is given by the continuous functor $S'/S \mapsto T \times_S S'/S$. Hence, given $S'/S$ we let $$ f_{sites}^\sharp : \mathcal{O}(S'/S) \longrightarrow f_{sites, *}\mathcal{O}(S'/S) = \mathcal{O}(S \times_S S'/T) $$ be the usual map $\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times_S S')$. Similarly, the morphism $i_f : \Sh(T_\tau) \to \Sh((\Sch/S)_\tau)$ for $\tau \in \{Zar, \etale\}$, see Topologies, Lemmas \ref{topologies-lemma-put-in-T} and \ref{topologies-lemma-put-in-T-etale}, becomes a morphism of ringed topoi because $i_f^{-1}\mathcal{O} = \mathcal{O}$. Here are some special cases: \begin{enumerate} \item The morphism of big sites $f_{big} : (\Sch/X)_{fppf} \to (\Sch/Y)_{fppf}$, becomes a morphism of ringed sites $$ (f_{big}, f_{big}^\sharp) : ((\Sch/X)_{fppf}, \mathcal{O}_X) \longrightarrow ((\Sch/Y)_{fppf}, \mathcal{O}_Y) $$ as in Modules on Sites, Definition \ref{sites-modules-definition-ringed-site}. Similarly for the big syntomic, smooth, \'etale and Zariski sites. \item The morphism of small sites $f_{small} : X_\etale \to Y_\etale$ becomes a morphism of ringed sites $$ (f_{small}, f_{small}^\sharp) : (X_\etale, \mathcal{O}_X) \longrightarrow (Y_\etale, \mathcal{O}_Y) $$ as in Modules on Sites, Definition \ref{sites-modules-definition-ringed-site}. Similarly for the small Zariski site. \end{enumerate}


descent-remark-change-topologies-ringed-sites

Remark \ref{remark-change-topologies-ringed} and Lemma \ref{lemma-compare-sites} have the following applications: \begin{enumerate} \item Let $S$ be a scheme. The construction $\mathcal{F} \mapsto \mathcal{F}^a$ is the pullback under the morphism of ringed sites $\text{id}_{\tau, Zar} : ((\Sch/S)_\tau, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$ or the morphism $\text{id}_{small, \etale, Zar} : (S_\etale, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$. \item Let $f : X \to Y$ be a morphism of schemes. For any of the morphisms $f_{sites}$ of ringed sites of Remark \ref{remark-change-topologies-ringed} we have $$ (f^*\mathcal{F})^a = f_{sites}^*\mathcal{F}^a. $$ This follows from (1) and the fact that pullbacks are compatible with compositions of morphisms of ringed sites, see Modules on Sites, Lemma \ref{sites-modules-lemma-push-pull-composition-modules}. \end{enumerate}


descent-remark-smooth-permanence

With the assumptions (1) and $p$ smooth in Lemma \ref{lemma-smooth-permanence} it is not automatically the case that $X \to Y$ is smooth. A counter example is $S = \Spec(k)$, $X = \Spec(k[s])$, $Y = \Spec(k[t])$ and $f$ given by $t \mapsto s^2$. But see also Lemma \ref{lemma-syntomic-permanence} for some information on the structure of $f$.


descent-remark-descending-properties-standard

In Lemma \ref{lemma-descending-properties} above if $\tau = smooth$ then in condition (3) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\tau = syntomic$ or $\tau = \etale$.


descent-remark-descending-properties-morphisms-standard

(This is a repeat of Remark \ref{remark-descending-properties-standard} above.) In Lemma \ref{lemma-descending-properties-morphisms} above if $\tau = smooth$ then in condition (3) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\tau = syntomic$ or $\tau = \etale$.


descent-remark-properties-morphisms-local-source-standard

(This is a repeat of Remarks \ref{remark-descending-properties-standard} and \ref{remark-descending-properties-morphisms-standard} above.) In Lemma \ref{lemma-properties-morphisms-local-source} above if $\tau = smooth$ then in condition (4) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\tau = syntomic$ or $\tau = \etale$.


descent-remark-list-local-source-target

Using Lemma \ref{lemma-etale-local-source-target} and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are \'etale local on the source-and-target. In each case we list the lemma which implies the property is \'etale local on the source and the lemma which implies the property is \'etale local on the target. In each case the third assumption of Lemma \ref{lemma-etale-local-source-target} is trivial to check, and we omit it. Here is the list: \begin{enumerate} \item flat, see Lemmas \ref{lemma-flat-fpqc-local-source} and \ref{lemma-descending-property-flat}, \item locally of finite presentation, see Lemmas \ref{lemma-locally-finite-presentation-fppf-local-source} and \ref{lemma-descending-property-locally-finite-presentation}, \item locally finite type, see Lemmas \ref{lemma-locally-finite-type-fppf-local-source} and \ref{lemma-descending-property-locally-finite-type}, \item universally open, see Lemmas \ref{lemma-universally-open-fppf-local-source} and \ref{lemma-descending-property-universally-open}, \item syntomic, see Lemmas \ref{lemma-syntomic-syntomic-local-source} and \ref{lemma-descending-property-syntomic}, \item smooth, see Lemmas \ref{lemma-smooth-smooth-local-source} and \ref{lemma-descending-property-smooth}, \item \'etale, see Lemmas \ref{lemma-etale-etale-local-source} and \ref{lemma-descending-property-etale}, \item locally quasi-finite, see Lemmas \ref{lemma-locally-quasi-finite-etale-local-source} and \ref{lemma-descending-property-quasi-finite}, \item unramified, see Lemmas \ref{lemma-unramified-etale-local-source} and \ref{lemma-descending-property-unramified}, \item G-unramified, see Lemmas \ref{lemma-unramified-etale-local-source} and \ref{lemma-descending-property-unramified}, and \item add more here as needed. \end{enumerate}


descent-remark-compare-definitions

At this point we have three possible definitions of what it means for a property $\mathcal{P}$ of morphisms to be ``\'etale local on the source and target'': \begin{enumerate} \item[(ST)] $\mathcal{P}$ is \'etale local on the source and $\mathcal{P}$ is \'etale local on the target, \item[(DM)] (the definition in the paper \cite[Page 100]{DM} by Deligne and Mumford) for every diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with surjective \'etale vertical arrows we have $\mathcal{P}(h) \Leftrightarrow \mathcal{P}(f)$, and \item[(SP)] $\mathcal{P}$ is \'etale local on the source-and-target. \end{enumerate} In this section we have seen that (SP) $\Rightarrow$ (DM) $\Rightarrow$ (ST). The Examples \ref{example-silly-one} and \ref{example-silly-two} show that neither implication can be reversed. Finally, Lemma \ref{lemma-etale-local-source-target} shows that the difference disappears when looking at properties of morphisms which are stable under postcomposing with open immersions, which in practice will always be the case.


descent-remark-easier

Let $X \to S$ be a morphism of schemes. Let $(V/X, \varphi)$ be a descent datum relative to $X \to S$. We may think of the isomorphism $\varphi$ as an isomorphism $$ (X \times_S X) \times_{\text{pr}_0, X} V \longrightarrow (X \times_S X) \times_{\text{pr}_1, X} V $$ of schemes over $X \times_S X$. So loosely speaking one may think of $\varphi$ as a map $\varphi : \text{pr}_0^*V \to \text{pr}_1^*V$\footnote{Unfortunately, we have chosen the ``wrong'' direction for our arrow here. In Definitions \ref{definition-descent-datum} and \ref{definition-descent-datum-for-family-of-morphisms} we should have the opposite direction to what was done in Definition \ref{definition-descent-datum-quasi-coherent} by the general principle that ``functions'' and ``spaces'' are dual.}. The cocycle condition then says that $\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.


descent-remark-easier-family

Let $S$ be a scheme. Let $\{X_i \to S\}_{i \in I}$ be a family of morphisms with target $S$. Let $(V_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to S\}$. We may think of the isomorphisms $\varphi_{ij}$ as isomorphisms $$ (X_i \times_S X_j) \times_{\text{pr}_0, X_i} V_i \longrightarrow (X_i \times_S X_j) \times_{\text{pr}_1, X_j} V_j $$ of schemes over $X_i \times_S X_j$. So loosely speaking one may think of $\varphi_{ij}$ as an isomorphism $\text{pr}_0^*V_i \to \text{pr}_1^*V_j$ over $X_i \times_S X_j$. The cocycle condition then says that $\text{pr}_{02}^*\varphi_{ik} = \text{pr}_{12}^*\varphi_{jk} \circ \text{pr}_{01}^*\varphi_{ij}$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.


descent-remark-morphisms-of-schemes-satisfy-fpqc-descent

Lemma \ref{lemma-refine-coverings-fully-faithful} says that morphisms of schemes satisfy fpqc descent. In other words, given a scheme $S$ and schemes $X$, $Y$ over $S$ the functor $$ (\Sch/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \Mor_T(X_T, Y_T) $$ satisfies the sheaf condition for the fpqc topology. The simplest case of this is the following. Suppose that $T \to S$ is a surjective flat morphism of affines. Let $\psi_0 : X_T \to Y_T$ be a morphism of schemes over $T$ which is compatible with the canonical descent data. Then there exists a unique morphism $\psi : X \to Y$ whose base change to $T$ is $\psi_0$. In fact this special case follows in a straightforward manner from Lemma \ref{lemma-fully-faithful}. And, in turn, that lemma is a formal consequence of the following two facts: (a) the base change functor by a faithfully flat morphism is faithful, see Lemma \ref{lemma-ff-base-change-faithful} and (b) a scheme satisfies the sheaf condition for the fpqc topology, see Lemma \ref{lemma-fpqc-universal-effective-epimorphisms}.


descent-remark-what-product-means

In the statement of Lemma \ref{lemma-descent-data-sheaves} the condition that $h_{S_i} \times F$ is representable is equivalent to the condition that the restriction of $F$ to $(\Sch/S_i)_\tau$ is representable.


simplicial-remark-relations

By abuse of notation we sometimes write $d_i : U_n \to U_{n - 1}$ instead of $d^n_i$, and similarly for $s_i : U_n \to U_{n + 1}$. The relations among the morphisms $d^n_i$ and $s^n_i$ may be expressed as follows: \begin{enumerate} \item If $i < j$, then $d_i \circ d_j = d_{j - 1} \circ d_i$. \item If $i < j$, then $d_i \circ s_j = s_{j - 1} \circ d_i$. \item We have $\text{id} = d_j \circ s_j = d_{j + 1} \circ s_j$. \item If $i > j + 1$, then $d_i \circ s_j = s_j \circ d_{i - 1}$. \item If $i \leq j$, then $s_i \circ s_j = s_{j + 1} \circ s_i$. \end{enumerate} This means that whenever the compositions on both the left and the right are defined then the corresponding equality should hold.


simplicial-remark-relations-cosimplicial

By abuse of notation we sometimes write $\delta_i : U_{n - 1} \to U_n$ instead of $\delta^n_i$, and similarly for $\sigma_i : U_{n + 1} \to U_n$. The relations among the morphisms $\delta^n_i$ and $\sigma^n_i$ may be expressed as follows: \begin{enumerate} \item If $i < j$, then $\delta_j \circ \delta_i = \delta_i \circ \delta_{j - 1}$. \item If $i < j$, then $\sigma_j \circ \delta_i = \delta_i \circ \sigma_{j - 1}$. \item We have $\text{id} = \sigma_j \circ \delta_j = \sigma_j \circ \delta_{j + 1}$. \item If $i > j + 1$, then $\sigma_j \circ \delta_i = \delta_{i - 1} \circ \sigma_j$. \item If $i \leq j$, then $\sigma_j \circ \sigma_i = \sigma_i \circ \sigma_{j + 1}$. \end{enumerate} This means that whenever the compositions on both the left and the right are defined then the corresponding equality should hold.


simplicial-remark-explicit-face-degeneracy

Let $U$, and $U_{n + 1}$ be as in Lemma \ref{lemma-work-out}. On $T$-valued points we can easily describe the face and degeneracy maps of $\tilde U$. Explicitly, the maps $d^{n + 1}_i : U_{n + 1} \to U_n$ are given by $$ (f_0, \ldots, f_{n + 1}) \longmapsto f_i. $$ And the maps $s^n_j : U_n \to U_{n + 1}$ are given by \begin{eqnarray*} f & \longmapsto & ( s^{n - 1}_{j - 1} \circ d^{n - 1}_0 \circ f, \\ & & s^{n - 1}_{j - 1} \circ d^{n - 1}_1 \circ f, \\ & & \ldots\\ & & s^{n - 1}_{j - 1} \circ d^{n - 1}_{j - 1} \circ f, \\ & & f, \\ & & f, \\ & & s^{n - 1}_j \circ d^{n - 1}_{j + 1} \circ f, \\ & & s^{n - 1}_j \circ d^{n - 1}_{j + 2} \circ f, \\ & & \ldots\\ & & s^{n - 1}_j \circ d^{n - 1}_n \circ f ) \end{eqnarray*} where we leave it to the reader to verify that the RHS is an element of the displayed set of Lemma \ref{lemma-work-out}. For $n = 0$ there is one map, namely $f \mapsto (f, f)$. For $n = 1$ there are two maps, namely $f \mapsto (f, f, s_0d_1f)$ and $f \mapsto (s_0d_0f, f, f)$. For $n = 2$ there are three maps, namely $f \mapsto (f, f, s_0d_1f, s_0d_2f)$, $f \mapsto (s_0d_0f, f, f, s_1d_2f)$, and $f \mapsto (s_1d_0f, s_1d_1f, f, f)$. And so on and so forth.


simplicial-remark-cosk-simplicial-sets

The construction of Lemma \ref{lemma-work-out} above in the case of simplicial sets is the following. Given an $n$-truncated simplicial set $U$, we make a canonical $(n + 1)$-truncated simplicial set $\tilde U$ as follows. We add a set of $(n + 1)$-simplices $U_{n + 1}$ by the formula of the lemma. Namely, an element of $U_{n + 1}$ is a numbered collection of $(f_0, \ldots, f_{n + 1})$ of $n$-simplices, with the property that they glue as they would in a $(n + 1)$-simplex. In other words, the $i$th face of $f_j$ is the $(j-1)$st face of $f_i$ for $i < j$. Geometrically it is obvious how to define the face and degeneracy maps for $\tilde U$. If $V$ is an $(n + 1)$-truncated simplicial set, then its $(n + 1)$-simplices give rise to compatible collections of $n$-simplices $(f_0, \ldots, f_{n + 1})$ with $f_i \in V_n$. Hence there is a natural map $\Mor(\text{sk}_nV, U) \to \Mor(V, \tilde U)$ which is inverse to the canonical restriction mapping the other way. \medskip\noindent Also, it is enough to do the combinatorics of the construction in the case of truncated simplicial sets. Namely, for any object $T$ of the category $\mathcal{C}$, and any $n$-truncated simplicial object $U$ of $\mathcal{C}$ we can consider the $n$-truncated simplicial set $\Mor(T, U)$. We may apply the construction to this, and take its set of $(n + 1)$-simplices, and require this to be representable. This is a good way to think about the result of Lemma \ref{lemma-work-out}.


simplicial-remark-inductive-coskeleton

{\it Inductive construction of coskeleta.} Suppose that $\mathcal{C}$ is a category with finite limits. Suppose that $U$ is an $m$-truncated simplicial object in $\mathcal{C}$. Then we can inductively construct $n$-truncated objects $U^n$ as follows: \begin{enumerate} \item To start, set $U^m = U$. \item Given $U^n$ for $n \geq m$ set $U^{n + 1} = \tilde U^n$, where $\tilde U^n$ is constructed from $U^n$ as in Lemma \ref{lemma-work-out}. \end{enumerate} Since the construction of Lemma \ref{lemma-work-out} has the property that it leaves the $n$-skeleton of $U^n$ unchanged, we can then define $\text{cosk}_m U$ to be the simplicial object with $(\text{cosk}_m U)_n = U^n_n = U^{n + 1}_n = \ldots$. And it follows formally from Lemma \ref{lemma-work-out} that $U^n$ satisfies the formula $$ \Mor_{\text{Simp}_n(\mathcal{C})}(V, U^n) = \Mor_{\text{Simp}_m(\mathcal{C})}(\text{sk}_mV, U) $$ for all $n \geq m$. It also then follows formally from this that $$ \Mor_{\text{Simp}(\mathcal{C})}(V, \text{cosk}_mU) = \Mor_{\text{Simp}_m(\mathcal{C})}(\text{sk}_mV, U) $$ with $\text{cosk}_mU$ chosen as above.


simplicial-remark-existence-cosk

We do not need all finite limits in order to be able to define the coskeleton functors. Here are some remarks \begin{enumerate} \item We have seen in Examples \ref{example-cosk0} that if $\mathcal{C}$ has products of pairs of objects then $\text{cosk}_0$ exists. \item For $k > 0$ the functor $\text{cosk}_k$ exists if $\mathcal{C}$ has finite connected limits. \end{enumerate} This is clear from the inductive procedure of constructing coskeleta (Remarks \ref{remark-cosk-simplicial-sets} and \ref{remark-inductive-coskeleton}) but it also follows from the fact that the categories $(\Delta/[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ used in Lemma \ref{lemma-existence-cosk} are connected. Observe that we do not need the categories for $n \leq k$ by Lemma \ref{lemma-trivial-cosk} or Lemma \ref{lemma-recover-cosk}. (As $k$ gets higher the categories $(\Delta/[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ are more and more connected in a topological sense.)


simplicial-remark-augmentation

Let $\mathcal{C}$ be a category with fibre products. Let $V$ be a simplicial object. Let $\epsilon : V \to X$ be an augmentation. Let $U$ be the simplicial object whose $n$th term is the $(n + 1)$st fibred product of $V_0$ over $X$. By a simple combination of Lemmas \ref{lemma-augmentation-howto} and \ref{lemma-cosk-minus-one} we  obtain a canonical morphism $V \to U$.


simplicial-remark-sk-literature

In some texts the composite functor $$ \text{Simp}(\mathcal{C}) \xrightarrow{\text{sk}_m} \text{Simp}_m(\mathcal{C}) \xrightarrow{i_{m!}} \text{Simp}(\mathcal{C}) $$ is denoted $\text{sk}_m$. This makes sense for simplicial sets, because then Lemma \ref{lemma-n-skeleton-sets} says that $i_{m!} \text{sk}_m V$ is just the sub simplicial set of $V$ consisting of all $i$-simplices of $V$, $i \leq m$ and their degeneracies. In those texts it is also customary to denote the composition $$ \text{Simp}(\mathcal{C}) \xrightarrow{\text{sk}_m} \text{Simp}_m(\mathcal{C}) \xrightarrow{\text{cosk}_m} \text{Simp}(\mathcal{C}) $$ by $\text{cosk}_m$.


simplicial-remark-degenerate-subcomplex

In the situation of Lemma \ref{lemma-decompose-associated-complexes} the subcomplex $D(U) \subset s(U)$ can also be defined as the subcomplex with terms $$ D(U)_n = \Im\left( \bigoplus\nolimits_{\varphi : [n] \to [m]\text{ surjective}, \ m < n} U_m \xrightarrow{\bigoplus U(\varphi)} U_n\right) $$ Namely, since $U_m$ is the direct sum of the subobject $N(U_m)$ and the images of $N(U_k)$ for surjections $[m] \to [k]$ with $k < m$ this is clearly the same as the definition of $D(U)_n$ given in the proof of Lemma \ref{lemma-decompose-associated-complexes}. Thus we see that if $U$ is a simplicial abelian group, then elements of $D(U)_n$ are exactly the sums of degenerate $n$-simplices.


simplicial-remark-homotopy-better

Let $\mathcal{C}$ be any category (no assumptions whatsoever). Let $U$ and $V$ be simplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{C}$. A {\it homotopy from $a$ to $b$} is given by morphisms\footnote{In the literature, often the maps $h_{n + 1, i} \circ s_i : U_n \to V_{n + 1}$ are used instead of the maps $h_{n, i}$. Of course the relations these maps satisfy are different from the ones in Lemma \ref{lemma-relations-homotopy}.} $h_{n, i} : U_n \to V_n$, for $n \geq 0$, $i = 0, \ldots, n + 1$ satisfying the relations of Lemma \ref{lemma-relations-homotopy}. As in Definition \ref{definition-homotopy} we say the morphisms $a$ and $b$ are {\it homotopic} if there exists a sequence of morphisms $a = a_0, a_1, \ldots, a_n = b$ from $U$ to $V$ such that for each $i = 1, \ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy from $a_i$ to $a_{i - 1}$. Clearly, if $F : \mathcal{C} \to \mathcal{C}'$ is any functor and $\{h_{n, i}\}$ is a homotopy from $a$ to $b$, then $\{F(h_{n, i})\}$ is a homotopy from $F(a)$ to $F(b)$. Similarly, if $a$ and $b$ are homotopic, then $F(a)$ and $F(b)$ are homotopic. Since the lemma says that the newer notion is the same as the old one in case finite coproduct exist, we deduce in particular that functors preserve the original notion whenever both categories have finite coproducts.


simplicial-remark-homotopy-pre-post-compose

Let $\mathcal{C}$ be any category. Suppose two morphisms $a, a' : U \to V$ of simplicial objects are homotopic. Then for any morphism $b : V \to W$ the two maps $b \circ a, b \circ a' : U \to W$ are homotopic. Similarly, for any morphism $c : X \to U$ the two maps $a \circ c, a' \circ c : X \to V$ are homotopic. In fact the maps $b \circ a \circ c, b \circ a' \circ c : X \to W$ are homotopic. Namely, if the maps $h_{n, i} : U_n \to V_n$ define a homotopy from $a$ to $a'$ then the maps $b \circ h_{n, i} \circ c$ define a homotopy from $b \circ a \circ c$ to $b \circ a' \circ c$. In this way we see that we obtain a new category $\text{hSimp}(\mathcal{C})$ with the same objects as $\text{Simp}(\mathcal{C})$ but whose morphisms are homotopy classes of of morphisms of $\text{Simp}(\mathcal{C})$. Thus there is a canonical functor $$ \text{Simp}(\mathcal{C}) \longrightarrow \text{hSimp}(\mathcal{C}) $$ which is essentially surjective and surjective on sets of morphisms.


simplicial-remark-homotopy-cosimplicial-better

Let $\mathcal{C}$ be any category (no assumptions whatsoever). Let $U$ and $V$ be cosimplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{C}$. A {\it homotopy from $a$ to $b$} is given by morphisms $h_{n, \alpha} : U_n \to V_n$, for $n \geq 0$, $\alpha \in \Delta[1]_n$ satisfying (\ref{equation-property-homotopy-cosimplicial}) for all morphisms $f$ of $\Delta$ and such that $a_n = h_{n, 0 : [n] \to [1]}$ and $b_n = h_{n, 1 : [n] \to [1]}$ for all $n \geq 0$. As in Definition \ref{definition-homotopy-cosimplicial} we say the morphisms $a$ and $b$ are {\it homotopic} if there exists a sequence of morphisms $a = a_0, a_1, \ldots, a_n = b$ from $U$ to $V$ such that for each $i = 1, \ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy from $a_i$ to $a_{i - 1}$. Clearly, if $F : \mathcal{C} \to \mathcal{C}'$ is any functor and $\{h_{n, i}\}$ is a homotopy from $a$ to $b$, then $\{F(h_{n, i})\}$ is a homotopy from $F(a)$ to $F(b)$. Similarly, if $a$ and $b$ are homotopic, then $F(a)$ and $F(b)$ are homotopic. This new notion is the same as the old one in case finite products exist. We deduce in particular that functors preserve the original notion whenever both categories have finite products.


discriminant-remark-relative-dualizing-for-quasi-finite

Let $f : Y \to X$ be a locally quasi-finite morphism of locally Noetherian schemes. It is clear from Lemma \ref{lemma-localize-dualizing} that there is a unique coherent $\mathcal{O}_Y$-module $\omega_{Y/X}$ on $Y$ such that for every pair of affine opens $\Spec(B) = V \subset Y$, $\Spec(A) = U \subset X$ with $f(V) \subset U$ there is a canonical isomorphism $$ H^0(V, \omega_{Y/X}) = \omega_{B/A} $$ and where these isomorphisms are compatible with restriction maps.


discriminant-remark-relative-dualizing-for-flat-quasi-finite

Let $f : Y \to X$ be a flat locally quasi-finite morphism of locally Noetherian schemes. Let $\omega_{Y/X}$ be as in Remark \ref{remark-relative-dualizing-for-quasi-finite}. It is clear from the uniqueness, existence, and compatibility with localization of trace elements (Lemmas \ref{lemma-trace-unique}, \ref{lemma-dualizing-tau}, and \ref{lemma-trace-base-change}) that there exists a global section $$ \tau_{Y/X} \in \Gamma(Y, \omega_{Y/X}) $$ such that for every pair of affine opens $\Spec(B) = V \subset Y$, $\Spec(A) = U \subset X$ with $f(V) \subset U$ that element $\tau_{Y/X}$ maps to $\tau_{B/A}$ under the canonical isomorphism $H^0(V, \omega_{Y/X}) = \omega_{B/A}$.


discriminant-remark-construction-pairing

Let $A \to B$ be a quasi-finite homomorphism of Noetherian rings. Let $J$ be the annihilator of $\Ker(B \otimes_A B \to B)$. There is a canonical $B$-bilinear pairing \begin{equation} \label{equation-pairing-noether} \omega_{B/A} \times J \longrightarrow B \end{equation} defined as follows. Choose a factorization $A \to B' \to B$ with $A \to B'$ finite and $B' \to B$ inducing an open immersion of spectra. Let $J'$ be the annihilator of $\Ker(B' \otimes_A B' \to B')$. We first define $$ \Hom_A(B', A) \times J' \longrightarrow B',\quad (\lambda, \sum b_i \otimes c_i) \longmapsto \sum \lambda(b_i)c_i $$ This is $B'$-bilinear exactly because for $\xi \in J'$ and $b \in B'$ we have $(b \otimes 1)\xi = (1 \otimes b)\xi$. By Lemma \ref{lemma-noether-different-localization} and the fact that $\omega_{B/A} = \Hom_A(B', A) \otimes_{B'} B$ we can extend this to a $B$-bilinear pairing as displayed above.


discriminant-remark-universal-finite-syntomic-smooth-top

Let $\pi_d : Y_d \to X_d$ be as in Example \ref{example-universal-finite-syntomic}. Let $U_d \subset X_d$ be the maximal open over which $V_d = \pi_d^{-1}(U_d)$ is finite syntomic as in Lemma \ref{lemma-universal-finite-syntomic}. Then it is also true that $V_d$ is smooth over $\mathbf{Z}$. (Of course the morphism $V_d \to U_d$ is not smooth when $d \geq 2$.) Arguing as in the proof of Lemma \ref{lemma-universal-finite-syntomic-smooth} this corresponds to the following deformation problem: given a small extension $C' \to C$ and a finite syntomic $C$-algebra $B$ with a section $B \to C$, find a finite syntomic $C'$-algebra $B'$ and a section $B' \to C'$ whose tensor product with $C$ recovers $B \to C$. By Lemma \ref{lemma-syntomic-finite} we may write $B = C[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$ as a relative global complete intersection. After a change of coordinates with may assume $x_1, \ldots, x_n$ are in the kernel of $B \to C$. Then the polynomials $f_i$ have vanishing constant terms. Choose any lifts $f'_i \in C'[x_1, \ldots, x_n]$ of $f_i$ with vanishing constant terms. Then  $B' = C'[x_1, \ldots, x_n]/(f'_1, \ldots, f'_n)$ with section $B' \to C'$ sending $x_i$ to zero works.


discriminant-remark-local-description-delta

Let $Y \to X$ be a locally quasi-finite syntomic morphism of schemes. What does the pair $(\det(\NL_{Y/X}), \delta(\NL_{Y/X}))$ look like locally? Choose affine opens $V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$ with $f(V) \subset U$ and an integer $n$ and $f_1, \ldots, f_n \in A[x_1, \ldots, x_n]$ such that $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$. Then $$ \NL_{B/A} = \left( (f_1, \ldots, f_n)/(f_1, \ldots, f_n)^2 \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} B \text{d} x_i\right) $$ and $(f_1, \ldots, f_n)/(f_1, \ldots, f_n)^2$ is free with generators the classes $\overline{f}_i$. See proof of Lemma \ref{lemma-syntomic-quasi-finite}. Thus $\det(L_{B/A})$ is free on the generator $$ \text{d}x_1 \wedge \ldots \wedge \text{d}x_n \otimes (\overline{f}_1 \wedge \ldots \wedge \overline{f}_n)^{\otimes -1} $$ and the section $\delta(\NL_{B/A})$ is the element $$ \delta(\NL_{B/A}) = \det(\partial f_j/ \partial x_i) \cdot \text{d}x_1 \wedge \ldots \wedge \text{d}x_n \otimes (\overline{f}_1 \wedge \ldots \wedge \overline{f}_n)^{\otimes -1} $$ by definition.


discriminant-remark-different-generalization

We can generalize Definition \ref{definition-different}. Suppose that $f : Y \to X$ is a quasi-finite morphism of Noetherian schemes with the following properties \begin{enumerate} \item the open $V \subset Y$ where $f$ is flat contains $\text{Ass}(\mathcal{O}_Y)$ and $f^{-1}(\text{Ass}(\mathcal{O}_X))$, \item the trace element $\tau_{V/X}$ comes from a section $\tau \in \Gamma(Y, \omega_{Y/X})$. \end{enumerate} Condition (1) implies that $V$ contains the associated points of $\omega_{Y/X}$ by Lemma \ref{lemma-dualizing-associated-primes}. In particular, $\tau$ is unique if it exists (Divisors, Lemma \ref{divisors-lemma-restriction-injective-open-contains-ass}). Given $\tau$ we can define the different $\mathfrak{D}_f$ as the annihilator of $\Coker(\tau : \mathcal{O}_Y \to \omega_{Y/X})$. This agrees with the Dedekind different in many cases (Lemma \ref{lemma-agree-dedekind}). However, for non-flat maps between non-normal rings, this generalization no longer measures ramification of the morphism, see Example \ref{example-no-different}.


discriminant-remark-collect-results-qf-gorenstein

Let $f : Y \to X$ be a quasi-finite Gorenstein morphism of Noetherian schemes. Let $\mathfrak D_f \subset \mathcal{O}_Y$ be the different and let $R \subset Y$ be the closed subscheme cut out by $\mathfrak D_f$. Then we have \begin{enumerate} \item $\mathfrak D_f$ is a locally principal ideal, \item $R$ is a locally principal closed subscheme, \item $\mathfrak D_f$ is affine locally the same as the Noether different, \item formation of $R$ commutes with base change, \item if $f$ is finite, then the norm of $R$ is the discriminant of $f$, and \item if $f$ is \'etale in the associated points of $Y$, then $R$ is an effective Cartier divisor and $\omega_{Y/X} = \mathcal{O}_Y(R)$. \end{enumerate} This follows from Lemmas \ref{lemma-flat-gorenstein-agree-noether}, \ref{lemma-base-change-different}, and \ref{lemma-norm-different-is-discriminant}.


discriminant-remark-collect-results-qf-gorenstein-two

Let $S$ be a Noetherian scheme endowed with a dualizing complex $\omega_S^\bullet$. Let $f : Y \to X$ be a quasi-finite Gorenstein morphism of compactifyable schemes over $S$. Assume moreover $Y$ and $X$ Cohen-Macaulay and $f$ \'etale at the generic points of $Y$. Then we can combine Duality for Schemes, Remark \ref{duality-remark-CM-morphism-compare-dualizing} and Remark \ref{remark-collect-results-qf-gorenstein} to see that we have a canonical isomorphism $$ \omega_Y = f^*\omega_X \otimes_{\mathcal{O}_Y} \omega_{Y/X} = f^*\omega_X \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(R) $$ of $\mathcal{O}_Y$-modules. If further $f$ is finite, then the isomorphism $\mathcal{O}_Y(R) = \omega_{Y/X}$ comes from the global section $\tau_{Y/X} \in H^0(Y, \omega_{Y/X})$ which corresponds via duality to the map $\text{Trace}_f : f_*\mathcal{O}_Y \to \mathcal{O}_X$, see Lemma \ref{lemma-compare-trace}.


limits-remark-limit-preserving

Let $S$ be a scheme. Let us say that a functor $F : (\Sch/S)^{opp} \to \textit{Sets}$ is {\it limit preserving} if for every directed inverse system $\{T_i\}_{i \in I}$ of affine schemes with limit $T$ we have $F(T) = \colim_i F(T_i)$. Let $X$ be a scheme over $S$, and let $h_X : (\Sch/S)^{opp} \to \textit{Sets}$ be its functor of points, see Schemes, Section \ref{schemes-section-representable}. In this terminology Proposition \ref{proposition-characterize-locally-finite-presentation} says that a scheme $X$ is locally of finite presentation over $S$ if and only if $h_X$ is limit preserving.


limits-remark-cannot-do-better

We cannot do better than this if we do not assume more on $S$ and the morphism $f : X \to S$. For example, in general it will not be possible to find a {\it closed} immersion $X \to X'$ as in the lemma. The reason is that this would imply that $f$ is quasi-compact which may not be the case. An example is to take $S$ to be infinite dimensional affine space with $0$ doubled and $X$ to be one of the two infinite dimensional affine spaces.


limits-remark-more-general-modification

The lemma above can be generalized as follows. Let $S$ be a scheme and let $T \subset S$ be a closed subset. Assume there exists a cofinal system of open neighbourhoods $T \subset W_i$ such that (1) $W_i \setminus T$ is quasi-compact and (2) $W_i \subset W_j$ is an affine morphism. Then $W = \lim W_i$ is a scheme which contains $T$ as a closed subscheme. Set $U = X \setminus T$ and $V = W \setminus T$. Then the base change functor $$ \left\{ \begin{matrix} f : X \to S\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to W\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\} $$ is an equivalence of categories. If we ever need this we will change this remark into a lemma and provide a detailed proof.


limits-remark-finite-type-gives-well-defined-system

In Situation \ref{situation-limit-noetherian} Lemmas \ref{lemma-good-diagram}, \ref{lemma-limit-from-good-diagram}, and \ref{lemma-morphism-good-diagram} tell us that the category of schemes quasi-separated and of finite type over $S$ is equivalent to certain types of inverse systems of schemes over $(S_i)_{i \in I}$, namely the ones produced by applying Lemma \ref{lemma-limit-from-good-diagram} to a diagram of the form (\ref{equation-good-diagram}). For example, given $X \to S$ finite type and quasi-separated if we choose two different diagrams $X \to V_1 \to S_{i_1}$ and $X \to V_2 \to S_{i_2}$ as in (\ref{equation-good-diagram}), then applying Lemma \ref{lemma-morphism-good-diagram} to $\text{id}_X$ (in two directions) we see that the corresponding limit descriptions of $X$ are canonically isomorphic (up to shrinking the directed set $I$). And so on and so forth.


spaces-more-morphisms-lemma-flat-case

A flat monomorphism of algebraic spaces is representable by schemes.


spaces-more-morphisms-lemma-flat-surjective-monomorphism

A quasi-compact flat surjective monomorphism of algebraic spaces is an isomorphism.


spaces-more-morphisms-lemma-match-modules-differentials

Let $f : X \to Y$ be a morphism of schemes. Let $f_{small} : X_\etale \to Y_\etale$ be the associated morphism of small \'etale sites, see Descent, Remark \ref{descent-remark-change-topologies-ringed}. Then there is a canonical isomorphism $$ (\Omega_{X/Y})^a = \Omega_{X_\etale/Y_\etale} $$ compatible with universal derivations. Here the first module is the sheaf on $X_\etale$ associated to the quasi-coherent $\mathcal{O}_X$-module $\Omega_{X/Y}$, see Morphisms, Definition \ref{morphisms-definition-sheaf-differentials}, and the second module is the one from Modules on Sites, Definition \ref{sites-modules-definition-module-differentials}.


spaces-more-morphisms-lemma-immersion-differentials

Let $S$ be a scheme. If $X \to Y$ is an immersion of algebraic spaces over $S$ then $\Omega_{X/S}$ is zero.


spaces-more-morphisms-lemma-immersion-universal-thickening

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Then \begin{enumerate} \item $i$ is formally unramified, \item the universal first order thickening of $Z$ over $X$ is the first order infinitesimal neighbourhood of $Z$ in $X$ of Definition \ref{definition-first-order-infinitesimal-neighbourhood}, \item the conormal sheaf of $i$ in the sense of Definition \ref{definition-conormal-sheaf} agrees with the conormal sheaf of $i$ in the sense of Definition \ref{definition-universal-thickening}. \end{enumerate}


spaces-more-morphisms-lemma-transitivity-conormal-unramified

Let $S$ be a scheme. Let $Z \to Y \to X$ be formally unramified morphisms of algebraic spaces over $S$. \begin{enumerate} \item If $Z \subset Z'$ is the universal first order thickening of $Z$ over $X$ and $Y \subset Y'$ is the universal first order thickening of $Y$ over $X$, then there is a morphism $Z' \to Y'$ and $Y \times_{Y'} Z'$ is the universal first order thickening of $Z$ over $Y$. \item There is a canonical exact sequence $$ i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ where the maps come from Lemma \ref{lemma-universal-thickening-functorial} and $i : Z \to Y$ is the first morphism. \end{enumerate}


spaces-more-morphisms-lemma-unramified-flat-formally-etale

An unramified flat morphism is formally \'etale.


spaces-more-morphisms-lemma-descending-property-formally-smooth

The property $\mathcal{P}(f) =$``$f$ is formally smooth'' is fpqc local on the base.


spaces-more-morphisms-lemma-NL-formally-etale

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is formally \'etale, \item $H^{-1}(\NL_{X/Y}) = H^0(\NL_{X/Y}) = 0$. \end{enumerate}


spaces-more-morphisms-lemma-NL-smooth

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is smooth, and \item $f$ is locally of finite presentation, $H^{-1}(\NL_{X/Y}) = 0$, and $H^0(\NL_{X/Y}) = \Omega_{X/Y}$ is finite locally free. \end{enumerate}


spaces-more-morphisms-lemma-morphism-between-flat

Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $Z$, \item $X$ is flat over $Z$, \item for every $z \in |Z|$ the fibre of $X$ over $z$ is flat over the fibre of $Y$ over $z$, and \item $Y$ is locally of finite type over $Z$. \end{enumerate} Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$.


spaces-more-morphisms-lemma-flat-morphism-from-CM

Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of locally Noetherian algebraic spaces over $S$. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\mathcal{O}_{Y, f(\overline{x})}$ is Cohen-Macaulay for all $x \in |X|$.


spaces-more-morphisms-lemma-flat-morphism-from-gorenstein

Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of locally Noetherian algebraic spaces over $S$. If $X$ is Gorenstein, then $f$ is Gorenstein and $\mathcal{O}_{Y, f(\overline{x})}$ is Gorenstein for all $x \in |X|$.


spaces-more-morphisms-lemma-base-change-gorenstein

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in |X'|$ be a point with image $x \in |X|$. \begin{enumerate} \item If $f$ is Gorenstein at $x$, then $f' : X' \to Y'$ is Gorenstein at $x'$. \item If $f$ is flat at $x$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \item If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \end{enumerate}


spaces-more-morphisms-lemma-universally-catenary-dimension-function

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\delta : |B| \to \mathbf{Z}$ be a function. Assume $B$ is decent, locally Noetherian, and universally catenary and $\delta$ is a dimension function. If $X$ is a decent algebraic space over $B$ whose structure morphism $f : X \to B$ is locally of finite type we define $\delta_X : |X| \to \mathbf{Z}$ by the rule $$ \delta_X(x) = \delta(f(x)) + \text{transcendence degreeof }x/f(x) $$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre}). Then $\delta_X$ is a dimension function.


spaces-more-morphisms-lemma-etale-splits-off-just-one-quasi-finite-part

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ with image $y \in |Y|$. Assume that \begin{enumerate} \item $f$ is locally of finite type, \item $f$ is separated, and \item $f$ is quasi-finite at $x$. \end{enumerate} Then there exists an \'etale morphism $(U, u) \to (Y, y)$ of pointed algebraic spaces and a decomposition $$ U \times_Y X = W \amalg V $$ into open and closed subspaces such that the morphism $V \to U$ is finite and there exists a point $v \in |V|$ which maps to $x$ in $|X|$ and $u$ in $|U|$.


spaces-more-morphisms-lemma-algebraize-formal-algebraic-space-finite-over-proper

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X \to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \geq 1$ we set $X_n = X \times_S S_n$. Suppose given a commutative diagram $$ \xymatrix{ Y_1 \ar[r] \ar[d] & Y_2 \ar[r] \ar[d] & Y_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots } $$ of algebraic spaces with cartesian squares. Assume that \begin{enumerate} \item $Y_1 \to X_1$ is a finite morphism, and \item $Y_1 \to S_1$ is proper. \end{enumerate} Then there exists a finite morphism of algebraic spaces $Y \to X$ such that $Y_n = Y \times_S S_n$ for all $n \geq 1$. Moreover, $Y$ is proper over $S$.


spaces-more-morphisms-lemma-flat-base-change-regular-immersion

\begin{slogan} Regular immersions are stable under flat base change. \end{slogan} Let $S$ be a scheme. Let $i : Z \to X$ be a Koszul-regular, $H_1$-regular, or quasi-regular immersion of algebraic spaces over $S$. Let $X' \to X$ be a flat morphism of algebraic spaces over $S$. Then the base change $i' : Z \times_X X' \to X'$ is a Koszul-regular, $H_1$-regular, or quasi-regular immersion.


spaces-more-morphisms-lemma-transitivity-conormal-quasi-regular

Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces over $S$. Assume that $Z \to Y$ is $H_1$-regular. Then the canonical sequence of Lemma \ref{lemma-transitivity-conormal} $$ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ is exact and (\'etale) locally split.


spaces-more-morphisms-lemma-composition-regular-immersion

Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$. \begin{enumerate} \item If $i$ and $j$ are Koszul-regular immersions, so is $j \circ i$. \item If $i$ and $j$ are $H_1$-regular immersions, so is $j \circ i$. \item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular immersion, then $j \circ i$ is a quasi-regular immersion. \end{enumerate}


spaces-more-morphisms-lemma-permanence-regular-immersion

Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$. Assume that the sequence $$ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ of Lemma \ref{lemma-transitivity-conormal} is exact and locally split. \begin{enumerate} \item If $j \circ i$ is a quasi-regular immersion, so is $i$. \item If $j \circ i$ is a $H_1$-regular immersion, so is $i$. \item If both $j$ and $j \circ i$ are Koszul-regular immersions, so is $i$. \end{enumerate}


spaces-more-morphisms-lemma-extra-permanence-regular-immersion-noetherian

Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$. Assume $X$ is locally Noetherian. The following are equivalent \begin{enumerate} \item $i$ and $j$ are Koszul regular immersions, \item $i$ and $j \circ i$ are Koszul regular immersions, \item $j \circ i$ is a Koszul regular immersion and the conormal sequence $$ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ is exact and locally split. \end{enumerate}


spaces-more-morphisms-lemma-composition-pseudo-coherent

A composition of pseudo-coherent morphisms is pseudo-coherent.


spaces-more-morphisms-lemma-flat-finite-presentation-pseudo-coherent

A flat morphism which is locally of finite presentation is pseudo-coherent.


spaces-more-morphisms-lemma-permanence-pseudo-coherent

Let $f : X \to Y$ be a morphism of algebraic spaces pseudo-coherent over a base algebraic space $B$. Then $f$ is pseudo-coherent.


spaces-more-morphisms-lemma-flat-base-change-perfect

A flat base change of a perfect morphism is perfect.


spaces-more-morphisms-lemma-composition-perfect

A composition of perfect morphisms is perfect.


spaces-more-morphisms-lemma-lci

Let $S$ be a scheme. Let $f : X \to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Let $P$ be an algebraic space smooth over $Y$. Let $U \to X$ be an \'etale morphism of algebraic spaces and let $i : U \to P$ an immersion of algebraic spaces over $Y$. Picture: $$ \xymatrix{ X \ar[rd] & U \ar[l] \ar[d] \ar[r]_i & P \ar[ld] \\ & Y } $$ Then $i$ is a Koszul-regular immersion of algebraic spaces.


spaces-more-morphisms-lemma-lci-properties

Let $S$ be a scheme. Let $f : X \to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Then \begin{enumerate} \item $f$ is locally of finite presentation, \item $f$ is pseudo-coherent, and \item $f$ is perfect. \end{enumerate}


spaces-more-morphisms-lemma-regular-immersion-lci

Let $S$ be a scheme. A Koszul-regular immersion of algebraic spaces over $S$ is a local complete intersection morphism.


spaces-more-morphisms-lemma-descending-property-lci

The property $\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is fpqc local on the base.


spaces-more-morphisms-lemma-lci-syntomic-local-source

The property $\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is syntomic local on the source.


spaces-more-morphisms-lemma-triangulated

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. The full subcategory of $D(\mathcal{O}_X)$ consisting of $Y$-perfect objects is a saturated\footnote{Derived Categories, Definition \ref{derived-definition-saturated}.} triangulated subcategory.


algebra-lemma-characterize-finitely-presented-module-hom

Let $R$ be a ring. Let $N$ be an $R$-module. The following are equivalent \begin{enumerate} \item $N$ is a finitely presented $R$-module, \item for any filtered colimit $M = \colim M_i$ of $R$-modules the map $\colim \Hom_R(N, M_i) \to \Hom_R(K, M)$ is bijective. \end{enumerate}


algebra-lemma-characterize-local-ring-map

Let $\varphi : R \to S$ be a ring map. Assume $R$ and $S$ are local rings. The following are equivalent: \begin{enumerate} \item $\varphi$ is a local ring map, \item $\varphi(\mathfrak m_R) \subset \mathfrak m_S$, and \item $\varphi^{-1}(\mathfrak m_S) = \mathfrak m_R$. \item For any $x \in R$, if $\varphi(x)$ is invertible in $S$, then $x$ is invertible in $R$. \end{enumerate}


algebra-lemma-surjective-on-spec-units

Let $\varphi : R \to S$ be a ring map such that the induced map $\Spec(S) \to \Spec(R)$ is surjective. Then an element $x \in R$ is a unit if and only if $\varphi(x) \in S$ is a unit.


algebra-lemma-closed-union-connected-components

Let $R$ be a ring. Let $T \subset \Spec(R)$ be a subset of the spectrum. The following are equivalent \begin{enumerate} \item $T$ is closed and is a union of connected components of $\Spec(R)$, \item $T$ is an intersection of open and closed subsets of $\Spec(R)$, and \item $T = V(I)$ where $I \subset R$ is an ideal generated by idempotents. \end{enumerate} Moreover, the ideal in (3) if it exists is unique.


algebra-lemma-primes-principal

Let $R$ be a ring. \begin{enumerate} \item An ideal $I \subset R$ maximal with respect to not being principal is prime. \item If every prime ideal of $R$ is principal, then every ideal of $R$ is principal. \end{enumerate}


algebra-lemma-invert-closed-split

Let $R$ be a ring. Let $S \subset R$ be a multiplicative subset. Assume the image of the map $\Spec(S^{-1}R) \to \Spec(R)$ is closed. If $R$ is Noetherian, or $\Spec(R)$ is a Noetherian topological space, or $S$ is finitely generated as a monoid, then $R \cong S^{-1}R \times R'$ for some ring $R'$.


algebra-lemma-corollary-jacobson

Any finite type algebra over $\mathbf{Z}$ is Jacobson.


algebra-lemma-geometrically-connected

Let $k$ be a field. Let $R$ be a $k$-algebra. The following are equivalent \begin{enumerate} \item for every field extension $k \subset k'$ the spectrum of $R \otimes_k k'$ is connected, and \item for every finite separable field extension $k \subset k'$ the spectrum of $R \otimes_k k'$ is connected. \end{enumerate}


algebra-lemma-characterize-geometrically-integral

Let $k$ be a field. Let $S$ be a $k$-algebra. The following are equivalent \begin{enumerate} \item $S$ is geometrically integral over $k$, \item for every finite extension $k'/k$ of fields the ring $S \otimes_k k'$ is a domain, \item $S \otimes_k \overline{k}$ is a domain where $\overline{k}$ is the algebraic closure of $k$. \end{enumerate}


Let $R$ be a local Artinian ring. There is a commutative diagram $$ \xymatrix{ K_0(R) \ar[rr] \ar[d]_{\text{rank}_R} & & K'_0(R) \ar[d]^{\text{length}_R} \\ \mathbf{Z} \ar[rr]^{\text{length}_R(R)} & & \mathbf{Z} } $$ where the vertical maps are isomorphisms by Lemmas \ref{lemma-K0prime-Artinian} and \ref{lemma-K0-local}.


algebra-lemma-integral-closure-graded

Let $R \to S$ be a homomorphism of graded rings. Let $S' \subset S$ be the integral closure of $R$ in $S$. Then $$ S' = \bigoplus\nolimits_{d \geq 0} S' \cap S_d, $$ i.e., $S'$ is a graded $R$-subalgebra of $S$.


algebra-lemma-minimal-over-r

Let $R$ be a Noetherian ring. Let $f_1, \ldots, f_r \in R$. \begin{enumerate} \item If $\mathfrak p$ is minimal over $(f_1, \ldots, f_r)$ then the height of $\mathfrak p$ is $\leq r$. \item If $\mathfrak p, \mathfrak q \in \Spec(R)$ and $\mathfrak q$ is minimal over $(\mathfrak p, f_1, \ldots, f_r)$, then every chain of primes between $\mathfrak p$ and $\mathfrak q$ has length at most $r$. \end{enumerate}


algebra-lemma-Noetherian-finite-nr-primes

A Noetherian ring with finitely many primes has dimension $\leq 1$.


algebra-lemma-ses-dimension

Let $R$ be a Noetherian ring. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of finite $R$-modules. Then $\max\{\dim(\text{Supp}(M')), \dim(\text{Supp}(M''))\} = \dim(\text{Supp}(M))$.


algebra-lemma-weakly-ass-quotient-ring

Let $R$ be a ring. Let $I$ be an ideal. Let $M$ be an $R/I$-module. Via the canonical injection $\Spec(R/I) \to \Spec(R)$ we have $\text{WeakAss}_{R/I}(M) = \text{WeakAss}_R(M)$.


algebra-lemma-tor-left-right

Let $R$ be a ring. For any $i \geq 0$ the functors $\text{Mod}_R \times \text{Mod}_R \to \text{Mod}_R$, $(M, N) \mapsto \text{Tor}_i^R(M, N)$ and $(M, N) \mapsto \text{Tor}_i^R(N, M)$ are canonically isomorphic.


algebra-lemma-quotient-module-ML

Let $M_1 \to M_2 \to M_3 \to 0$ be an exact sequence of $R$-modules. If $M_1$ is finitely generated and $M_2$ is Mittag-Leffler, then $M_3$ is Mittag-Leffler.


algebra-lemma-hathat

\begin{reference} Taken from an unpublished note of Lenstra and de Smit. \end{reference} Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. Denote $K_n = \Ker(M^\wedge \to M/I^nM)$. Then $M^\wedge$ is $I$-adically complete if and only if $K_n$ is equal to $I^nM^\wedge$ for all $n \geq 1$.


algebra-lemma-artinian-variant-local-criterion-flatness

Let $R$ be an Artinian local ring. Let $M$ be an $R$-module. Let $I \subset R$ be a proper ideal. The following are equivalent \begin{enumerate} \item $M$ is flat over $R$, and \item $M/IM$ is flat over $R/I$ and $\text{Tor}_1^R(R/I, M) = 0$. \end{enumerate}


algebra-lemma-exact-artinian-local

In Situation \ref{situation-complex}. Let $R$ be a Artinian local ring. Suppose that $0 \to R^{n_e} \to R^{n_{e-1}} \to \ldots \to R^{n_0}$ is exact at $R^{n_e}, \ldots, R^{n_1}$. Then the complex is isomorphic to a direct sum of trivial complexes.


algebra-lemma-epimorphism-cardinality

Let $R \to S$ be an epimorphism of rings. Then the cardinality of $S$ is at most the cardinality of $R$. In a formula: $|S| \leq |R|$.


algebra-lemma-dimension-graded

Let $k$ be a field. Let $S$ be a finitely generated graded algebra over $k$. Assume $S_0 = k$. Let $P(T) \in \mathbf{Q}[T]$ be the polynomial such that $\dim(S_d) = P(d)$ for all $d \gg 0$. See Proposition \ref{proposition-graded-hilbert-polynomial}. Then \begin{enumerate} \item The irrelevant ideal $S_{+}$ is a maximal ideal $\mathfrak m$. \item Any minimal prime of $S$ is a homogeneous ideal and is contained in $S_{+} = \mathfrak m$. \item We have $\dim(S) = \deg(P) + 1 = \dim_x\Spec(S)$ (with the convention that $\deg(0) = -1$) where $x$ is the point corresponding to the maximal ideal $S_{+} = \mathfrak m$. \item The Hilbert function of the local ring $R = S_{\mathfrak m}$ is equal to the Hilbert function of $S$. \end{enumerate}


algebra-lemma-limit-finite-type

Suppose $R \to S$ is a ring map. Assume that $S$ is of finite type over $R$. Then there exists a directed set $(\Lambda, \leq)$, and a system of ring maps $R_\lambda \to S_\lambda$ such that \begin{enumerate} \item The colimit of the system $R_\lambda \to S_\lambda$ is equal to $R \to S$. \item Each $R_\lambda$ is of finite type over $\mathbf{Z}$. \item Each $S_\lambda$ is of finite type over $R_\lambda$. \item For each $\lambda \leq \mu$ the map $S_\lambda \otimes_{R_\lambda} R_\mu \to S_\mu$ presents $S_\mu$ as a quotient of $S_\lambda \otimes_{R_\lambda} R_\mu$. \end{enumerate}


algebra-lemma-variant-local-criterion-flatness-general

Let $R \to S$ be a local homomorphism of local rings. Let $I \not = R$ be an ideal in $R$. Let $M$ be an $S$-module. Assume \begin{enumerate} \item $S$ is essentially of finite presentation over $R$, \item $M$ is of finite presentation over $S$, \item $\text{Tor}_1^R(M, R/I) = 0$, and \item $M/IM$ is flat over $R/I$. \end{enumerate} Then $M$ is flat over $R$.


algebra-lemma-NL-of-localization

If $S \subset A$ is a multiplicative subset of $A$, then $\NL_{S^{-1}A/A}$ is homotopy equivalent to the zero complex.


algebra-lemma-syntomic-descends

\begin{slogan} Being syntomic is fpqc local on the base. \end{slogan} Let $R \to S$ be a ring map. Let $R \to R'$ be a faithfully flat ring map. Set $S' = R'\otimes_R S$. Then $R \to S$ is syntomic if and only if $R' \to S'$ is syntomic.


algebra-lemma-localize-smooth

Let $R \to S$ be a smooth ring map. Any localization $S_g$ is smooth over $R$. If $f \in R$ maps to an invertible element of $S$, then $R_f \to S$ is smooth.


algebra-lemma-section-smooth

\begin{slogan} If $R$ is a summand of $S$ and $S$ is smooth over $R$, then the $I$-adic completion of $S$ is often a power series over $R$ where $I$ is the kernel of the projection map from $S$ to $R$. \end{slogan} Let $\varphi : R \to S$ be a smooth ring map. Let $\sigma : S \to R$ be a left inverse to $\varphi$. Set $I = \Ker(\sigma)$. Then \begin{enumerate} \item $I/I^2$ is a finite locally free $R$-module, and \item if $I/I^2$ is free, then $S^\wedge \cong R[[t_1, \ldots, t_d]]$ as $R$-algebras, where $S^\wedge$ is the $I$-adic completion of $S$. \end{enumerate}


algebra-lemma-henselian-functorial-prepare

Let $R \to S$ be a local map of local rings. Let $S \to S^h$ be the henselization. Let $R \to A$ be an \'etale ring map and let $\mathfrak q$ be a prime of $A$ lying over $\mathfrak m_R$ such that $R/\mathfrak m_R \cong \kappa(\mathfrak q)$. Then there exists a unique morphism of rings $f : A \to S^h$ fitting into the commutative diagram $$ \xymatrix{ A \ar[r]_f & S^h \\ R \ar[u] \ar[r] & S \ar[u] } $$ such that $f^{-1}(\mathfrak m_{S^h}) = \mathfrak q$.


moduli-curves-lemma-pre-genus-in-CM-1

We have $\Curvesstack^{h0, 1} \subset \Curvesstack^{CM, 1}$ as open substacks of $\Curvesstack$.


moduli-curves-lemma-smooth-curves-smooth

The morphism $\Curvesstack^{smooth} \to \Spec(\mathbf{Z})$ is smooth.


moduli-curves-lemma-gorenstein-dualizing

Let $X \to S$ be a family of curves with Gorenstein fibres equidimensional of dimension $1$ (Lemma \ref{lemma-gorenstein-1-curves}). Then the relative dualizing sheaf $\omega_{X/S}$ is an invertible $\mathcal{O}_X$-module whose formation commutes with arbitrary base change.


moduli-curves-lemma-semistable-one-piece-per-genus

There is a decomposition into open and closed substacks $$ \Curvesstack^{semistable} = \coprod\nolimits_{g \geq 1} \Curvesstack^{semistable}_g $$ where each $\Curvesstack^{semistable}_g$ is characterized as follows: \begin{enumerate} \item given a family of curves $f : X \to S$ the following are equivalent \begin{enumerate} \item the classifying morphism $S \to \Curvesstack$ factors through $\Curvesstack^{semistable}_g$, \item $X \to S$ is a semistable family of curves and $R^1f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_S$-module of rank $g$, \end{enumerate} \item given $X$ a scheme proper over a field $k$ with $\dim(X) \leq 1$ the following are equivalent \begin{enumerate} \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors through $\Curvesstack^{semistable}_g$, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, the genus of $X$ is $g$, and $X$ has no rational tail, \item the singularities of $X$ are at-worst-nodal, $\dim(X) = 1$, $k = H^0(X, \mathcal{O}_X)$, the genus of $X$ is $g$, and $\omega_{X_s}^{\otimes m}$ is globally generated for $m \geq 2$. \end{enumerate} \end{enumerate}


moduli-curves-lemma-semistable-curves-smooth

The morphisms $\Curvesstack^{semistable} \to \Spec(\mathbf{Z})$ and $\Curvesstack^{semistable}_g \to \Spec(\mathbf{Z})$ are smooth.


dpa-lemma-need-only-gamma-p

Let $p$ be a prime number. Let $A$ be a ring such that every integer $n$ not divisible by $p$ is invertible, i.e., $A$ is a $\mathbf{Z}_{(p)}$-algebra. Let $I \subset A$ be an ideal. Two divided power structures $\gamma, \gamma'$ on $I$ are equal if and only if $\gamma_p = \gamma'_p$. Moreover, given a map $\delta : I \to I$ such that \begin{enumerate} \item $p!\delta(x) = x^p$ for all $x \in I$, \item $\delta(ax) = a^p\delta(x)$ for all $a \in A$, $x \in I$, and \item $\delta(x + y) = \delta(x) + \sum\nolimits_{i + j = p, i,j \geq 1} \frac{1}{i!j!} x^i y^j + \delta(y)$ for all $x, y \in I$, \end{enumerate} then there exists a unique divided power structure $\gamma$ on $I$ such that $\gamma_p = \delta$.


dpa-lemma-check-lci-at-maximal-ideals

Let $A$ be a Noetherian ring. Then $A$ is a local complete intersection if and only if $A_\mathfrak m$ is a complete intersection for every maximal ideal $\mathfrak m$ of $A$.


dpa-lemma-avramov-map-finite-type

Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map such that the image of $\Spec(B) \to \Spec(A)$ contains all closed points of $\Spec(A)$. Then the following are equivalent \begin{enumerate} \item $B$ is a complete intersection and $A \to B$ has finite tor dimension, \item $A$ is a complete intersection and $A \to B$ is a local complete intersection in the sense of More on Algebra, Definition \ref{more-algebra-definition-local-complete-intersection}. \end{enumerate}


moduli-lemma-coherent-existence-part

Assume $X \to B$ is proper as well as of finite presentation. Then $\Cohstack_{X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition \ref{stacks-morphisms-definition-existence}).


moduli-lemma-coherent-closed

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be a closed immersion of algebraic spaces which are separated and of finite presentation over $B$. Then the morphism $\Cohstack_{X/B} \to \Cohstack_{Y/B}$ of Lemma \ref{lemma-coherent-functorial} is a closed immersion.


moduli-lemma-hilb-s-lfp

The morphism $\Hilbfunctor_{X/B} \to B$ is separated and locally of finite presentation.


moduli-lemma-hilb-existence-part

Assume $X \to B$ is proper as well as of finite presentation. Then $\Hilbfunctor_{X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-valuative-criterion}).


moduli-lemma-hilb-closed

Let $B$ be an algebraic space. Let $\pi : X \to Y$ be a closed immersion of algebraic spaces which are separated and of finite presentation over $B$. Then $\pi$ induces a closed immersion $\Hilbfunctor_{X/B} \to \Hilbfunctor_{Y/B}$.


derived-lemma-third-map-idempotent

Let $\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle. If $$ \xymatrix{ Z \ar[r]_h \ar[d]_c & X[1] \ar[d]^{a[1]} \\ Z \ar[r]^h & X[1] } $$ is commutative and $a^2 = a$, $c^2 = c$, then there exists a morphism $b : Y \to Y$ with $b^2 = b$ such that $(a, b, c)$ is an endomorphism of the triangle $(X, Y, Z, f, g, h)$.


derived-lemma-direct-sum-computes

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $X, Y$ be objects of $\mathcal{D}$. If $X \oplus Y$ computes $RF$, then $X$ and $Y$ compute $RF$. Similarly for $LF$.


derived-lemma-irrelevant

In Situation \ref{situation-classical}. \begin{enumerate} \item Let $X$ be an object of $K^{+}(\mathcal{A})$. The right derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ is defined at $X$ if and only if the right derived functor of $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is defined at $X$. Moreover, the values are canonically isomorphic. \item Let $X$ be an object of $K^{+}(\mathcal{A})$. Then $X$ computes the right derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ if and only if $X$ computes the right derived functor of $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$. \item Let $X$ be an object of $K^{-}(\mathcal{A})$. The left derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ is defined at $X$ if and only if the left derived functor of $K^{-}(\mathcal{A}) \to D^{-}(\mathcal{B})$ is defined at $X$. Moreover, the values are canonically isomorphic. \item Let $X$ be an object of $K^{-}(\mathcal{A})$. Then $X$ computes the left derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ if and only if $X$ computes the left derived functor of $K^{-}(\mathcal{A}) \to D^{-}(\mathcal{B})$. \end{enumerate}


derived-lemma-cohomology-bounded-above

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex of $\mathcal{A}$. \begin{enumerate} \item If $K^\bullet$ has a projective resolution then $H^n(K^\bullet) = 0$ for $n \gg 0$. \item If $H^n(K^\bullet) = 0$ for $n \gg 0$ then there exists a quasi-isomorphism $L^\bullet \to K^\bullet$ with $L^\bullet$ bounded above. \end{enumerate}


derived-lemma-filtered-localization-functor

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $\mathcal{I}^f \subset \text{Fil}^f(\mathcal{A})$ denote the strictly full additive subcategory whose objects are the filtered injective objects. The canonical functor $$ K^{+}(\mathcal{I}^f) \longrightarrow DF^{+}(\mathcal{A}) $$ is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories. Furthermore the diagrams $$ \xymatrix{ K^{+}(\mathcal{I}^f) \ar[d]_{\text{gr}^p} \ar[r] & DF^{+}(\mathcal{A}) \ar[d]_{\text{gr}^p} \\ K^{+}(\mathcal{I}) \ar[r] & D^{+}(\mathcal{A}) } \quad \xymatrix{ K^{+}(\mathcal{I}^f) \ar[d]^{\text{forget }F} \ar[r] & DF^{+}(\mathcal{A}) \ar[d]^{\text{forget }F} \\ K^{+}(\mathcal{I}) \ar[r] & D^{+}(\mathcal{A}) } $$ are commutative, where $\mathcal{I} \subset \mathcal{A}$ is the strictly full additive subcategory whose objects are the injective objects.


derived-lemma-enough-K-injectives-implies

Let $\mathcal{A}$ be an abelian category. Assume every complex has a quasi-isomorphism towards a K-injective complex. Then any exact functor $F : K(\mathcal{A}) \to \mathcal{D}'$ of triangulated categories has a right derived functor $$ RF : D(\mathcal{A}) \longrightarrow \mathcal{D}' $$ and $RF(I^\bullet) = F(I^\bullet)$ for K-injective complexes $I^\bullet$.


derived-lemma-classical-generator-generator

Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. If $E$ is a classical generator of $\mathcal{D}$, then $E$ is a generator.


derived-proposition-derived-category

Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives. Denote $\mathcal{I} \subset \mathcal{A}$ the strictly full additive subcategory whose objects are the injective objects of $\mathcal{A}$. The functor $$ K^{+}(\mathcal{I}) \longrightarrow D^{+}(\mathcal{A}) $$ is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories.


cohomology-lemma-vanishing-generated-one-section

\begin{reference} This is a special case of \cite[Proposition 3.6.1]{Tohoku}. \end{reference} Let $X$ be a topological space. Let $d \geq 0$ be an integer. Assume \begin{enumerate} \item $X$ is quasi-compact, \item the quasi-compact opens form a basis for $X$, and \item the intersection of two quasi-compact opens is quasi-compact. \item $H^p(X, j_!\underline{\mathbf{Z}}_U) = 0$ for all $p > d$ and any quasi-compact open $j : U \to X$. \end{enumerate} Then $H^p(X, \mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\mathcal{F}$ on $X$.


cohomology-lemma-proper-base-change-spectral

Let $f : X \to Y$ be a spectral map of spectral spaces. Let $y \in Y$. Let $E \subset Y$ be the set of points specializing to $y$. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then $(R^pf_*\mathcal{F})_y = H^p(f^{-1}(E), \mathcal{F}|_{f^{-1}(E)})$.