stacks-sheaves-remark-flat

In the situation of Lemma \ref{lemma-functoriality-structure-sheaf} the morphism of ringed topoi $f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$ is flat as is clear from the equality $f^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_\mathcal{Y}$. This is a bit counter intuitive, for example because a closed immersion of algebraic stacks is typically not flat (as a morphism of algebraic stacks). However, exactly the same thing happens when taking a closed immersion $i : X \to Y$ of schemes: in this case the associated morphism of big $\tau$-sites $i : (\Sch/X)_\tau \to (\Sch/Y)_\tau$ also is flat.


stacks-sheaves-remark-cech-complex-presheaves

We can define the complex $\mathcal{K}^\bullet(f, \mathcal{F})$ also if $\mathcal{F}$ is a presheaf, only we cannot use the reference to Sites, Section \ref{sites-section-pullback} to define the pullback maps. To explain the pullback maps, suppose given a commutative diagram $$ \xymatrix{ \mathcal{V} \ar[rd]_g \ar[rr]_h & &  \mathcal{U} \ar[ld]^f \\ & \mathcal{X} } $$ of categories fibred in groupoids over $(\Sch/S)_{fppf}$ and a presheaf $\mathcal{G}$ on $\mathcal{U}$ we can define the pullback map $f_*\mathcal{G} \to g_*h^{-1}\mathcal{G}$ as the composition $$ f_*\mathcal{G} \longrightarrow f_*h_*h^{-1}\mathcal{G} = g_*h^{-1}\mathcal{G} $$ where the map comes from the adjunction map $\mathcal{G} \to h_*h^{-1}\mathcal{G}$. This works because in our situation the functors $h_*$ and $h^{-1}$ are adjoint in presheaves (and agree with their counter parts on sheaves). See Sections \ref{section-presheaves} and \ref{section-sheaves}.


stacks-sheaves-remark-cech-complex-sections

Let us ``compute'' the value of the relative {\v C}ech complex on an object $x$ of $\mathcal{X}$. Say $p(x) = U$. Consider the $2$-fibre product diagram (which serves to introduce the notation $g : \mathcal{V} \to \mathcal{Y}$) $$ \xymatrix{ \mathcal{V} \ar@{=}[r] \ar[d]_g & (\Sch/U)_{fppf} \times_{x, \mathcal{X}} \mathcal{U} \ar[r] \ar[d] & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar@{=}[r] & (\Sch/U)_{fppf} \ar[r]^-x & \mathcal{X} } $$ Note that the morphism $\mathcal{V}_n \to \mathcal{U}_n$ of the proof of Lemma \ref{lemma-generalities} induces an equivalence $\mathcal{V}_n = (\Sch/U)_{fppf} \times_{x, \mathcal{X}} \mathcal{U}_n$. Hence we see from (\ref{equation-pushforward}) that $$ \Gamma(x, \mathcal{K}^\bullet(f, \mathcal{F})) = \check{\mathcal{C}}^\bullet(\mathcal{V} \to \mathcal{Y}, x^{-1}\mathcal{F}) $$ In words: The value of the relative {\v C}ech complex on an object $x$ of $\mathcal{X}$ is the {\v C}ech complex of the base change of $f$ to $\mathcal{X}/x \cong (\Sch/U)_{fppf}$. This implies for example that Lemma \ref{lemma-homotopy} implies Lemma \ref{lemma-homotopy-sheafified} and more generally that results on the (usual) {\v C}ech complex imply results for the relative {\v C}ech complex.


resolve-remark-compare-Garel

Let $\mathbf{F}_p \subset \Lambda \subset R \subset S$ and $\text{Tr}$ be as in Lemma \ref{lemma-trace-higher}. By de Rham Cohomology, Proposition \ref{derham-proposition-Garel} there is a canonical map of complexes $$ \Theta_{S/R} : \Omega_{S/\Lambda}^\bullet \longrightarrow \Omega_{R/\Lambda}^\bullet $$ The computation in de Rham Cohomology, Example \ref{derham-example-Garel} shows that $\Theta_{S/R}(x^i \text{d}x) = \text{Tr}_x(x^i\text{d}x)$ for all $i$. Since $\text{Trace}_{S/R} = \Theta^0_{S/R}$ is identically zero and since $$ \Theta_{S/R}(a \wedge b) = a \wedge \Theta_{S/R}(b) $$ for $a \in \Omega^i_{R/\Lambda}$ and $b \in \Omega^j_{S/\Lambda}$ it follows that $\text{Tr} = \Theta_{S/R}$. The advantage of using $\text{Tr}$ is that it is a good deal more elementary to construct.


resolve-remark-dualizing-setup

Let $X$ be an integral Noetherian normal scheme of dimension $2$. In this case the following are equivalent \begin{enumerate} \item $X$ has a dualizing complex $\omega_X^\bullet$, \item there is a coherent $\mathcal{O}_X$-module $\omega_X$ such that $\omega_X[n]$ is a dualizing complex, where $n$ can be any integer. \end{enumerate} This follows from the fact that $X$ is Cohen-Macaulay (Properties, Lemma \ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}) and Duality for Schemes, Lemma \ref{duality-lemma-dualizing-module-CM-scheme}. In this situation we will say that $\omega_X$ is a {\it dualizing module} in accordance with Duality for Schemes, Section \ref{duality-section-dualizing-module}. In particular, when $A$ is a Noetherian normal local domain of dimension $2$, then we say {\it $A$ has a dualizing module $\omega_A$} if the above is true. In this case, if $X \to \Spec(A)$ is a normal modification, then $X$ has a dualizing module too, see Duality for Schemes, Example \ref{duality-example-proper-over-local}. In this situation we always denote $\omega_X$ the dualizing module normalized with respect to $\omega_A$, i.e., such that $\omega_X[2]$ is the dualizing complex normalized relative to $\omega_A[2]$. See Duality for Schemes, Section \ref{duality-section-glue}.


resolve-remark-pic-blowup

Let $b : X \to X'$ be the contraction of an exceptional curve of the first kind $E \subset X$. From Lemma \ref{lemma-pic-blowup} we obtain an identification $$ \Pic(X) = \Pic(X') \oplus \mathbf{Z} $$ where $\mathcal{L}$ corresponds to the pair $(\mathcal{L}', n)$ if and only if $\mathcal{L} = (b^*\mathcal{L}')(-nE)$, i.e., $\mathcal{L}(nE) = b^*\mathcal{L}'$. In fact the proof of Lemma \ref{lemma-pic-blowup} shows that $\mathcal{L}' = b_*\mathcal{L}(nE)$. Of course the assignment $\mathcal{L} \mapsto \mathcal{L}'$ is a group homomorphism.


exercises-remark-simple-geometric

Of course the idea of this exercise is to find a simple argument in each case rather than applying a ``big'' theorem. Nonetheless it is good to be guided by general principles.


exercises-remark-flat-not-projective

If $M$ is of finite presentation and flat over $A$, then $M$ is projective over $A$. Thus your example will have to involve a ring $A$ which is not Noetherian. I know of an example where $A$ is the ring of ${\mathcal C}^\infty$-functions on ${\mathbf R}$.


exercises-remark-flat-given-residue-field-extension-general

The same result holds for arbitrary field extensions $k \subset K$.


exercises-remark-not-hausdorff

Usually the word compact is reserved for quasi-compact and Hausdorff spaces.


exercises-remark-singularities

A singularity on a curve over a field $k$ is called an ordinary double point if the complete local ring of the curve at the point is of the form $k'[[x, y]]/(f)$, where (a) $k'$ is a finite separable extension of $k$, (b) the initial term of $f$ has degree two, i.e., it looks like $q = ax^2 + bxy + cy^2$ for some $a, b, c\in k'$ not all zero, and (c) $q$ is a nondegenerate quadratic form over $k'$ (in char 2 this means that $b$ is not zero). In general there is one isomorphism class of such rings for each isomorphism class of pairs $(k', q)$.


exercises-remark-HNSS

Let $k$ be a field. Then for every integer $n\in {\mathbf N}$ and every maximal ideal ${\mathfrak m} \subset k[x_1, \ldots, x_n]$ the quotient $k[x_1, \ldots, x_n]/{\mathfrak m}$ is a finite field extension of $k$. This will be shown later in the course. Of course (please check this) it implies a similar statement for maximal ideals of finitely generated $k$-algebras. The exercise above proves it in the case $k = {\mathbf C}$.


exercises-remark-Hilbert-Nullstellensatz

This is the Hilbert Nullstellensatz. Namely it says that the closed subsets of $\Spec(k[x_1, \ldots, x_n])$ (which correspond to radical ideals by a previous exercise) are determined by the closed points contained in them.


exercises-remark-cover

In algebraic geometric language this means that the property of ``being finitely generated'' or ``being flat'' is local for the Zariski topology (in a suitable sense). You can also show this for the property ``being of finite presentation''.


exercises-remark-elimination-theory

Finding the image as above usually is done by using elimination theory.


exercises-remark-continuous-proj-spec

There is a continuous map $ \text{Proj}(R) \longrightarrow \Spec(R_0) $.


exercises-remark-CM-dim-1-embedding-dim-2

This suggests that a local Noetherian Cohen-Macaulay ring of dimension 1 and embedding dimension 2 is of the form $B/FB$, where $B$ is a 2-dimensional regular local ring. This is more or less true (under suitable ``niceness'' properties of the ring).


exercises-remark-strange-fp

Let $h \in {\mathbf Z}[y]$ be a monic polynomial of degree $d$. Then: \begin{enumerate} \item The map $A = {\mathbf Z}[x] \to B ={\mathbf Z}[y]$, $x \mapsto h$ is finite locally free of rank $d$. \item For all primes $p$ the map $A_p = {\mathbf F}_p[x]\to B_p = {\mathbf F}_p[y]$, $y \mapsto h(y) \bmod p$ is finite locally free of rank $d$. \end{enumerate}


exercises-remark-direct-sum-stalk-abelian

Let $X$ be a topological space. In the category of abelian sheaves the direct sum of a family of sheaves $\{{\mathcal F}_i\}_{i\in I}$ is the sheaf associated to the presheaf $U \mapsto \oplus {\mathcal F}_i(U)$. Consequently the stalk of the direct sum at a point $x$ is the direct sum of the stalks of the ${\mathcal F}_i$ at $x$.


exercises-remark-open-immersion

When $(X, {\mathcal O}_X)$ is a ringed space and $U \subset X$ is an open subset then $(U, {\mathcal O}_X|_U)$ is a ringed space. Notation: ${\mathcal O}_U = {\mathcal O}_X|_U$. There is a canonical morphisms of ringed spaces $$ j : (U, {\mathcal O}_U) \longrightarrow (X, {\mathcal O}_X). $$ If $(X, {\mathcal O}_X)$ is a locally ringed space, so is $(U, {\mathcal O}_U)$ and $j$ is a morphism of locally ringed spaces. If $(X, {\mathcal O}_X)$ is a scheme so is $(U, {\mathcal O}_U)$ and $j$ is a morphism of schemes. We say that $(U, {\mathcal O}_U)$ is an {\it open subscheme} of $(X, {\mathcal O}_X)$ and that $j$ is an {\it open immersion}. More generally, any morphism $j' : (V, {\mathcal O}_V) \to (X, {\mathcal O}_X)$ that is {\it isomorphic} to a morphism $j : (U, {\mathcal O}_U) \to (X, {\mathcal O}_X)$ as above is called an open immersion.


exercises-remark-separated-base-fibre-product-affines-affine

It turns out this cannot happen with $S$ separated. Do you know why?


exercises-remark-affine-dimension

If your scheme is affine then dimension is the same as the Krull dimension of the underlying ring. So you can use last semesters results to compute dimension.


exercises-remark-tsen

Exercise \ref{exercise-has-rational-section} is a special case of ``Tsen's theorem''. Exercise \ref{exercise-no-section-curve} shows that the method is limited to low degree equations (conics when the base and fibre have dimension 1).


exercises-remark-interpretation-skolem-noether

The interpretation of the results of Exercise \ref{exercise-for-number-theorists} and \ref{exercise-quasi-section} is that given the morphism $X \to S$ all of whose fibres are nonempty, there exists a finite surjective morphism $S' \to S$ such that the base change $X_{S'} \to S'$ does have a section. This is not a general fact, but it holds if the base is the spectrum of a dedekind ring with finite residue fields at closed points, and the morphism $X \to S$ is flat with geometrically irreducible generic fibre. See Exercise \ref{exercise-no-quasi-section} below for an example where it doesn't work.


exercises-remark-tangent-space-relative

Exercise \ref{exercise-compute-TS} explains why it is necessary to consider the tangent space of $X$ over $S$ to get a good notion.


exercises-remark-extend-off-open

If $U \to X$ is a quasi-compact immersion then any quasi-coherent sheaf on $U$ is the restriction of a quasi-coherent sheaf on $X$. If $X$ is a Noetherian scheme, and $U \subset X$ is open, then any coherent sheaf on $U$ is the restriction of a coherent sheaf on $X$. Of course the exercise above is easier, and shouldn't use these general facts.


exercises-remark-fitting-omega-not-sings

The $k$th Fitting ideal of $\Omega_{X/S}$ is commonly used to define the singular scheme of the morphism $X \to S$ when $X$ has relative dimension $k$ over $S$. But as part (a) shows, you have to be careful doing this when your family does not have ``constant'' fibre dimension, e.g., when it is not flat. As part (b) shows, flatness doesn't guarantee it works either (and yes this is a flat family). In ``good cases'' -- such as in (c) -- for families of curves you expect the $0$-th Fitting ideal to be zero and the $1$st Fitting ideal to define (scheme-theoretically) the singular locus.


exercises-remark-invertible-projective-space

Let $k$ be a field. Let $\mathbf{P}^2_k = \text{Proj}(k[X_0, X_1, X_2])$. Any invertible sheaf on $\mathbf{P}^2_k$ is isomorphic to $\mathcal{O}_{\mathbf{P}^2_k}(n)$ for some $n \in \mathbf{Z}$. Recall that $$ \Gamma(\mathbf{P}^2_k, \mathcal{O}_{\mathbf{P}^2_k}(n)) = k[X_0, X_1, X_2]_n $$ is the degree $n$ part of the polynomial ring. For a quasi-coherent sheaf $\mathcal{F}$ on $\mathbf{P}^2_k$ set $\mathcal{F}(n) = \mathcal{F} \otimes_{\mathcal{O}_{\mathbf{P}^2_k}} \mathcal{O}_{\mathbf{P}^2_k}(n)$ as usual.


exercises-remark-recall-dimension-theory

Freely use the following facts on dimension theory (and add more if you need more). \begin{enumerate} \item The dimension of a scheme is the supremum of the length of chains of irreducible closed subsets. \item The dimension of a finite type scheme over a field is the maximum of the dimensions of its affine opens. \item The dimension of a Noetherian scheme is the maximum of the dimensions of its irreducible components. \item The dimension of an affine scheme coincides with the dimension of the corresponding ring. \item Let $k$ be a field and let $A$ be a finite type $k$-algebra. If $A$ is a domain, and $x \not = 0$, then $\dim(A) = \dim(A/xA) + 1$. \end{enumerate}


exercises-remark-chi

Given a projective scheme $X$ over a field $k$ and a coherent sheaf $\mathcal{F}$ on $X$ we set $$ \chi(X, \mathcal{F}) = \sum\nolimits_{i \geq 0} (-1)^i\dim_k H^i(X, \mathcal{F}). $$


exercises-remarks-divisors

Here are some trivial remarks. \begin{enumerate} \item On a Noetherian integral scheme $X$ the sheaf ${\mathcal K}_X$ is constant with value the function field $K(X)$. \item To make sense out of the definitions above one needs to show that $$ \text{length}_{\mathcal O}({\mathcal O}/(ab)) = \text{length}_{\mathcal O}({\mathcal O}/(a)) + \text{length}_{\mathcal O}({\mathcal O}/(b)) $$ for any pair $(a, b)$ of nonzero elements of a Noetherian 1-dimensional local domain ${\mathcal O}$. This will be done in the lectures. \end{enumerate}


spaces-duality-remark-iso-on-RSheafHom

In the situation of Lemma \ref{lemma-iso-on-RSheafHom} we have $$ DQ_Y(Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K))) = Rf_* DQ_X(R\SheafHom_{\mathcal{O}_X}(L, a(K))) $$ by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-pushforward-better-coherator}. Thus if $R\SheafHom_{\mathcal{O}_X}(L, a(K)) \in D_\QCoh(\mathcal{O}_X)$, then we can ``erase'' the $DQ_Y$ on the left hand side of the arrow. On the other hand, if we know that $R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K) \in D_\QCoh(\mathcal{O}_Y)$, then we can ``erase'' the $DQ_Y$ from the right hand side of the arrow. If both are true then we see that (\ref{equation-sheafy-trace}) is an isomorphism. Combining this with Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-quasi-coherence-internal-hom} we see that $Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)$ is an isomorphism if \begin{enumerate} \item $L$ and $Rf_*L$ are perfect, or \item $K$ is bounded below and $L$ and $Rf_*L$ are pseudo-coherent. \end{enumerate} For (2) we use that $a(K)$ is bounded below if $K$ is bounded below, see Lemma \ref{lemma-twisted-inverse-image-bounded-below}.


spaces-duality-remark-going-around

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ X'' \ar[r]_{k'} \ar[d]_{f''} & X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{l'} \ar[d]_{g''} & Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z'' \ar[r]^{m'} & Z' \ar[r]^m & Z } $$ of quasi-compact and quasi-separated algebraic spaces over $S$ where all squares are cartesian and where $(f, l)$, $(g, m)$, $(f', l')$, $(g', m')$ are Tor independent pairs of maps. Let $a$, $a'$, $a''$, $b$, $b'$, $b''$ be the right adjoints of Lemma \ref{lemma-twisted-inverse-image} for $f$, $f'$, $f''$, $g$, $g'$, $g''$. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows $$ \begin{matrix} A & B \\ C & D \end{matrix} $$ Then the maps (\ref{equation-base-change-map}) for the squares are (where we use $k^* = Lk^*$, etc) $$ \begin{matrix} \gamma_A : (k')^* \circ a' \to a'' \circ (l')^* & \gamma_B : k^* \circ a \to a' \circ l^* \\ \gamma_C : (l')^* \circ b' \to b'' \circ (m')^* & \gamma_D : l^* \circ b \to b' \circ m^* \end{matrix} $$ For the $2 \times 1$ and $1 \times 2$ rectangles we have four further base change maps $$ \begin{matrix} \gamma_{A + B} : (k \circ k')^* \circ a \to a'' \circ (l \circ l')^* \\ \gamma_{C + D} : (l \circ l')^* \circ b \to b'' \circ (m \circ m')^* \\ \gamma_{A + C} : (k')^* \circ (a' \circ b') \to (a'' \circ b'') \circ (m')^* \\ \gamma_{A + C} : k^* \circ (a \circ b) \to (a' \circ b') \circ m^* \end{matrix} $$ By Lemma \ref{lemma-compose-base-change-maps-horizontal} we have $$ \gamma_{A + B} = \gamma_A \circ \gamma_B, \quad \gamma_{C + D} = \gamma_C \circ \gamma_D $$ and by Lemma \ref{lemma-compose-base-change-maps} we have $$ \gamma_{A + C} = \gamma_C \circ \gamma_A, \quad \gamma_{B + D} = \gamma_D \circ \gamma_B $$ Here it would be more correct to write $\gamma_{A + B} = (\gamma_A \star \text{id}_{l^*}) \circ (\text{id}_{(k')^*} \star \gamma_B)$ with notation as in Categories, Section \ref{categories-section-formal-cat-cat} and similarly for the others. However, we continue the abuse of notation used in the proofs of Lemmas \ref{lemma-compose-base-change-maps} and \ref{lemma-compose-base-change-maps-horizontal} of dropping $\star$ products with identities as one can figure out which ones to add as long as the source and target of the transformation is known. Having said all of this we find (a priori) two transformations $$ (k')^* \circ k^* \circ a \circ b \longrightarrow a'' \circ b'' \circ (m')^* \circ m^* $$ namely $$ \gamma_C \circ \gamma_A \circ \gamma_D \circ \gamma_B = \gamma_{A + C} \circ \gamma_{B + D} $$ and $$ \gamma_C \circ \gamma_D \circ \gamma_A \circ \gamma_B = \gamma_{C + D} \circ \gamma_{A + B} $$ The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that $$ \xymatrix{ (k')^* \circ a' \circ l^* \circ b \ar[r]_{\gamma_D} \ar[d]_{\gamma_A} & (k')^* \circ a' \circ b' \circ m^* \ar[d]^{\gamma_A} \\ a'' \circ (l')^* \circ l^* \circ b \ar[r]^{\gamma_D} & a'' \circ (l')^* \circ b' \circ m^* } $$ commutes. This is true by Categories, Lemma \ref{categories-lemma-properties-2-cat-cats} or more simply the discussion preceding Categories, Definition \ref{categories-definition-horizontal-composition}.


spaces-properties-remark-list-properties-local-etale-topology

Here is a list of properties which are local for the \'etale topology (keep in mind that the fpqc, fppf, syntomic, and smooth topologies are stronger than the \'etale topology): \begin{enumerate} \item locally Noetherian, see Descent, Lemma \ref{descent-lemma-Noetherian-local-fppf}, \item Jacobson, see Descent, Lemma \ref{descent-lemma-Jacobson-local-fppf}, \item locally Noetherian and $(S_k)$, see Descent, Lemma \ref{descent-lemma-Sk-local-syntomic}, \item Cohen-Macaulay, see Descent, Lemma \ref{descent-lemma-CM-local-syntomic}, \item Gorenstein, see Duality for Schemes, Lemma \ref{duality-lemma-gorenstein-local-syntomic}, \item reduced, see Descent, Lemma \ref{descent-lemma-reduced-local-smooth}, \item normal, see Descent, Lemma \ref{descent-lemma-normal-local-smooth}, \item locally Noetherian and $(R_k)$, see Descent, Lemma \ref{descent-lemma-Rk-local-smooth}, \item regular, see Descent, Lemma \ref{descent-lemma-regular-local-smooth}, \item Nagata, see Descent, Lemma \ref{descent-lemma-Nagata-local-smooth}. \end{enumerate}


spaces-properties-remark-list-properties-local-ring-local-etale-topology

Let $P$ be a property of local rings. Assume that for any \'etale ring map $A \to B$ and $\mathfrak q$ is a prime of $B$ lying over the prime $\mathfrak p$ of $A$, then $P(A_\mathfrak p) \Leftrightarrow P(B_\mathfrak q)$. Then we obtain an \'etale local property of germs $(U, u)$ of schemes by setting $\mathcal{P}(U, u) = P(\mathcal{O}_{U, u})$. In this situation we will use the terminology ``the local ring of $X$ at $x$ has $P$'' to mean $X$ has property $\mathcal{P}$ at $x$. Here is a list of such properties $P$: \begin{enumerate} \item Noetherian, see More on Algebra, Lemma \ref{more-algebra-lemma-Noetherian-etale-extension}, \item dimension $d$, see More on Algebra, Lemma \ref{more-algebra-lemma-dimension-etale-extension}, \item regular, see More on Algebra, Lemma \ref{more-algebra-lemma-regular-etale-extension}, \item discrete valuation ring, follows from (2), (3), and Algebra, Lemma \ref{algebra-lemma-characterize-dvr}, \item reduced, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-reduced}, \item normal, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-normal}, \item Noetherian and depth $k$, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-depth}, \item Noetherian and Cohen-Macaulay, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-CM}, \item Noetherian and Gorenstein, see Dualizing Complexes, Lemma \ref{dualizing-lemma-flat-under-gorenstein}. \end{enumerate} There are more properties for which this holds, for example G-ring and Nagata. If we every need these we will add them here as well as references to detailed proofs of the corresponding algebra facts.


spaces-properties-remark-cannot-decide-yet

Lemma \ref{lemma-point-like-spaces} holds for decent algebraic spaces, see Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-point-like-spaces}. In fact a decent algebraic space with one point is a scheme, see Decent Spaces, Lemma \ref{decent-spaces-lemma-when-field}. This also holds when $X$ is locally separated, because a locally separated algebraic space is decent, see Decent Spaces, Lemma \ref{decent-spaces-lemma-locally-separated-decent}.


spaces-properties-remark-explain-equivalence

Let us explain the meaning of Lemma \ref{lemma-compare-etale-sites}. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf on the small \'etale site $X_\etale$ of $X$. The lemma says that there exists a unique sheaf $\mathcal{F}'$ on $X_{spaces, \etale}$ which restricts back to $\mathcal{F}$ on the subcategory $X_\etale$. If $U \to X$ is an \'etale morphism of algebraic spaces, then how do we compute $\mathcal{F}'(U)$? Well, by definition of an algebraic space there exists a scheme $U'$ and a surjective \'etale morphism $U' \to U$. Then $\{U' \to U\}$ is a covering in $X_{spaces, \etale}$ and hence we get an equalizer diagram $$ \xymatrix{ \mathcal{F}'(U) \ar[r] & \mathcal{F}(U') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(U' \times_U U'). } $$ Note that $U' \times_U U'$ is a scheme, and hence we may write $\mathcal{F}$ and not $\mathcal{F}'$. Thus we see how to compute $\mathcal{F}'$ when given the sheaf $\mathcal{F}$.


spaces-properties-remark-stalk-pullback

This remark is the analogue of \'Etale Cohomology, Remark \ref{etale-cohomology-remark-stalk-pullback}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x} : \Spec(k) \to X$ be a geometric point of $X$. By \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-equivalence-sheaves-point} the category of sheaves on $\Spec(k)_\etale$ is equivalent to the category of sets (by taking a sheaf to its global sections). Hence it follows from Lemma \ref{lemma-stalk-pullback} part (4) applied to the morphism $\overline{x}$ that the functor $$ \Sh(X_\etale) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{x}} $$ is isomorphic to the functor $$ \Sh(X_\etale) \longrightarrow \Sh(\Spec(k)_\etale) = \textit{Sets}, \quad \mathcal{F} \longmapsto \overline{x}^*\mathcal{F} $$ Hence we may view the stalk functors as pullback functors along geometric morphisms (and not just some abstract morphisms of topoi as in the result of Lemma \ref{lemma-stalk-gives-point}).


spaces-properties-remark-map-stalks

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. We claim that for any pair of geometric points $\overline{x}$ and $\overline{x}'$ lying over $x$ the stalk functors are isomorphic. By definition of $|X|$ we can find a third geometric point $\overline{x}''$ so that there exists a commutative diagram $$ \xymatrix{ \overline{x}'' \ar[r] \ar[d] \ar[rd]^{\overline{x}''} & \overline{x}' \ar[d]^{\overline{x}'} \\ \overline{x} \ar[r]^{\overline{x}} & X. } $$ Since the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is given by pullback along the morphism $\overline{x}$ (and similarly for the others) we conclude by functoriality of pullbacks.


intersection-remark-trivial-generalization

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $M$ be a finite $A$-module. Let $I \subset A$ be an ideal. The following are equivalent \begin{enumerate} \item $I' = I + \text{Ann}(M)$ is an ideal of definition (Algebra, Definition \ref{algebra-definition-ideal-definition}), \item the image $\overline{I}$ of $I$ in $\overline{A} = A/\text{Ann}(M)$ is an ideal of definition, \item $\text{Supp}(M/IM) \subset \{\mathfrak m\}$, \item $\dim(\text{Supp}(M/IM)) \leq 0$, and \item $\text{length}_A(M/IM) < \infty$. \end{enumerate} This follows from Algebra, Lemma \ref{algebra-lemma-support-point} (details omitted). If this is the case we have $M/I^nM = M/(I')^nM$ for all $n$ and $M/I^nM = M/\overline{I}^nM$ for all $n$ if $M$ is viewed as an $\overline{A}$-module. Thus we can define $$ \chi_{I, M}(n) = \text{length}_A(M/I^nM) = \sum\nolimits_{p = 0, \ldots, n - 1} \text{length}_A(I^pM/I^{p + 1}M) $$ and we get $$ \chi_{I, M}(n) = \chi_{I', M}(n) = \chi_{\overline{I}, M}(n) $$ for all $n$ by the equalities above. All the results of Algebra, Section \ref{algebra-section-Noetherian-local} and all the results in this section, have analogues in this setting. In particular we can define multiplicities $e_I(M, d)$ for $d \geq \dim(\text{Supp}(M))$ and we have $$ \chi_{I, M}(n) \sim e_I(M, d) \frac{n^d}{d!} + \text{lower order terms} $$ as in the case where $I$ is an ideal of definition.


intersection-remark-Serre-conjectures

Let $(A, \mathfrak m, \kappa)$ be a regular local ring. Let $M$ and $N$ be nonzero finite $A$-modules such that $M \otimes_A N$ is supported in $\{\mathfrak m\}$. Then $$ \chi(M, N) = \sum (-1)^i \text{length}_A \text{Tor}_i^A(M, N) $$ is finite. Let $r = \dim(\text{Supp}(M))$ and $s = \dim(\text{Supp}(N))$. In \cite{Serre_algebre_locale} it is shown that $r + s \leq \dim(A)$ and the following conjectures are made: \begin{enumerate} \item if $r + s < \dim(A)$, then $\chi(M, N) = 0$, and \item if $r + s = \dim(A)$, then $\chi(M, N) > 0$. \end{enumerate} The arguments that prove Lemma \ref{lemma-tor-sheaf} and Proposition \ref{proposition-positivity} can be leveraged (as is done in Serre's text) to show that (1) and (2) are true if $A$ contains a field. Currently, conjecture (1) is known in general and it is known that $\chi(M, N) \geq 0$ in general (Gabber). Positivity is, as far as we know, still an open problem.


intersection-remark-quasi-projective

Lemma \ref{lemma-moving-move} and Theorem \ref{theorem-well-defined} also hold for nonsingular quasi-projective varieties with the same proof. The only change is that one needs to prove the following version of the moving Lemma \ref{lemma-moving}: Let  $X \subset \mathbf{P}^N$ be a closed subvariety. Let $n = \dim(X)$ and $0 \leq d, d' < n$. Let $X^{reg} \subset X$ be the open subset of nonsingular points. Let $Z \subset X^{reg}$ be a closed subvariety of dimension $d$ and $T_i \subset X^{reg}$, $i \in I$ be a finite collection of closed subvarieties of dimension $d'$. Then there exists a subvariety $C \subset \mathbf{P}^N$ such that $C$ intersects $X$ properly and such that $$ (C \cdot X)|_{X^{reg}} = Z + \sum\nolimits_{j \in J} m_j Z_j $$ where $Z_j \subset X^{reg}$ are irreducible of dimension $d$, distinct from $Z$, and $$ \dim(Z_j \cap T_i) \leq \dim(Z \cap T_i) $$ with strict inequality if $Z$ does not intersect $T_i$ properly in $X^{reg}$.


homology-remark-direct-sum

Note that the proof of Lemma \ref{lemma-preadditive-direct-sum} shows that given $p$ and $q$ the morphisms $i$, $j$ are uniquely determined by the rules $p \circ i = \text{id}_x$, $q \circ j = \text{id}_y$, $p \circ j = 0$, $q \circ i = 0$. Moreover, we automatically have $i \circ p + j \circ q = \text{id}_{x \oplus y}$. Similarly, given $i$, $j$ the morphisms $p$ and $q$ are uniquely determined. Finally, given objects $x, y, z$ and morphisms $i : x \to z$, $j : y \to z$, $p : z \to x$ and $q : z \to y$ such that $p \circ i = \text{id}_x$, $q \circ j = \text{id}_y$, $p \circ j = 0$, $q \circ i = 0$ and $i \circ p + j \circ q = \text{id}_z$, then $z$ is the direct sum of $x$ and $y$ with the four morphisms equal to $i, j, p, q$.


homology-remark-direct-sums-not-exact

There are abelian categories $\mathcal{A}$ having countable direct sums but where countable direct sums are not exact. An example is the opposite of the category of abelian sheaves on $\mathbf{R}$. Namely, the category of abelian sheaves on $\mathbf{R}$ has countable products, but countable products are not exact. For such a category the functor $\text{Gr}(\mathcal{A}) \to \mathcal{A}$, $(A^i) \mapsto \bigoplus A^i$ described above is not exact. It is still true that $\text{Gr}(\mathcal{A})$ is equivalent to the category of graded objects $(A, k)$ of $\mathcal{A}$, but the kernel in the category of graded objects of a map $\varphi : (A, k) \to (B, k)$ is not equal to $\Ker(\varphi)$ endowed with a direct sum decomposition, but rather it is the direct sum of the kernels of the maps $k^iA \to k^iB$.


homology-remark-triple-complex

Let $\mathcal{A}$ be an additive category. Let $A^{\bullet, \bullet, \bullet}$ be a triple complex. The associated total complex is the complex with terms $$ \text{Tot}^n(A^{\bullet, \bullet, \bullet}) = \bigoplus\nolimits_{p + q + r = n} A^{p, q, r} $$ and differential $$ d^n_{\text{Tot}(A^{\bullet, \bullet, \bullet})} = \sum\nolimits_{p + q + r = n} d_1^{p, q, r} + (-1)^pd_2^{p, q, r} + (-1)^{p + q}d_3^{p, q, r} $$ With this definition a simple calculation shows that the associated total complex is equal to $$ \text{Tot}(A^{\bullet, \bullet, \bullet}) = \text{Tot}(\text{Tot}_{12}(A^{\bullet, \bullet, \bullet})) = \text{Tot}(\text{Tot}_{23}(A^{\bullet, \bullet, \bullet})) $$ In other words, we can either first combine the first two of the variables and then combine sum of those with the last, or we can first combine the last two variables and then combine the first with the sum of the last two.


homology-remark-shift-double-complex

Let $\mathcal{A}$ be an additive category. Let $A^{\bullet, \bullet}$ be a double complex with differentials $d_1^{p, q}$ and $d_2^{p, q}$. Denote $A^{\bullet, \bullet}[a, b]$ the double complex with $$ (A^{\bullet, \bullet}[a, b])^{p, q} = A^{p + a, q + b} $$ and differentials $$ d_{A^{\bullet, \bullet}[a, b], 1}^{p, q} = (-1)^a d_1^{p + a, q + b} \quad\text{and}\quad d_{A^{\bullet, \bullet}[a, b], 2}^{p, q} = (-1)^b d_2^{p + a, q + b} $$ In this situation there is a well defined isomorphism $$ \gamma : \text{Tot}(A^{\bullet, \bullet})[a + b] \longrightarrow \text{Tot}(A^{\bullet, \bullet}[a, b]) $$ which in degree $n$ is given by the map $$ \xymatrix{ (\text{Tot}(A^{\bullet, \bullet})[a + b])^n = \bigoplus_{p + q = n + a + b} A^{p, q} \ar[d]^{\epsilon(p, q, a, b)\text{id}_{A^{p, q}}} \\ \text{Tot}(A^{\bullet, \bullet}[a, b])^n = \bigoplus_{p' + q' = n} A^{p' + a, q' + b} } $$ for some sign $\epsilon(p, q, a, b)$. Of course the summand $A^{p, q}$ maps to the summand $A^{p' + a, q' + b}$ when $p = p' + a$ and $q = q' + b$. To figure out the conditions on these signs observe that on the source we have $$ d|_{A^{p, q}} = (-1)^{a + b}\left(d_1^{p, q} + (-1)^pd_2^{p, q}\right) $$ whereas on the target we have $$ d|_{A^{p' + a, q' + b}} = (-1)^ad_1^{p' + a, q' + b} + (-1)^{p'}(-1)^bd_2^{p' + a, q' + b} $$ Thus our constraints are that $$ (-1)^a \epsilon(p, q, a, b) = \epsilon(p + 1, q, a, b)(-1)^{a + b} \Leftrightarrow \epsilon(p + 1, q, a, b) = (-1)^b \epsilon(p, q, a, b) $$ and $$ (-1)^{p' + b}\epsilon(p, q, a, b) = \epsilon(p, q + 1, a, b) (-1)^{a + b + p} \Leftrightarrow \epsilon(p, q, a, b) = \epsilon(p, q + 1, a, b) $$ Thus we choose $\epsilon(p, q, a, b) = (-1)^{pb}$.


homology-remark-double-complex-complex-of-complexes-first

Let $\mathcal{A}$ be an additive category with countable direct sums. Let $\text{DoubleComp}(\mathcal{A})$ denote the category of double complexes. We can consider an object $A^{\bullet, \bullet}$ of $\text{DoubleComp}(\mathcal{A})$ as a complex of complexes as follows $$ \ldots \to A^{\bullet, -1} \to A^{\bullet, 0} \to A^{\bullet, 1} \to \ldots $$ For the variant where we switch the role of the indices, see Remark \ref{remark-double-complex-complex-of-complexes-second}. In this remark we show that taking the associated total complex is compatible with all the structures on complexes we have studied in the chapter so far. \medskip\noindent First, observe that the shift functor on double complexes viewed as complexes of complexes in the manner given above is the functor $[0, 1]$ defined in Remark \ref{remark-shift-double-complex}. By Remark \ref{remark-shift-double-complex} the functor $$ \text{Tot} : \text{DoubleComp}(\mathcal{A}) \to \text{Comp}(\mathcal{A}) $$ is compatible with shift functors, in the sense that we have a functorial isomorphism $\gamma : \text{Tot}(A^{\bullet, \bullet})[1] \to \text{Tot}(A^{\bullet, \bullet}[0, 1])$. \medskip\noindent Second, if $$ f, g : A^{\bullet, \bullet} \to B^{\bullet, \bullet} $$ are homotopic when $f$ and $g$ are viewed as morphisms of complexes of complexes in the manner given above, then $$ \text{Tot}(f), \text{Tot}(g) : \text{Tot}(A^{\bullet, \bullet}) \to \text{Tot}(B^{\bullet, \bullet}) $$ are homotopic maps of complexes. Indeed, let $h = (h^q)$ be a homotopy between $f$ and $g$. If we denote $h^{p, q} : A^{p, q} \to B^{p, q - 1}$ the component in degree $p$ of $h^q$, then this means that $$ f^{p, q} - g^{p, q} = d_2^{p, q - 1} \circ h^{p, q} + h^{p, q + 1} \circ d_2^{p, q} $$ The fact that $h^q : A^{\bullet, q} \to B^{\bullet, q - 1}$ is a map of complexes means that $$ d_1^{p, q - 1} \circ h^{p, q} = h^{p + 1, q} \circ d_1^{p, q} $$ Let us define $h' = ((h')^n)$ the homotopy given by the maps $(h')^n : \text{Tot}^n(A^{\bullet, \bullet}) \to \text{Tot}^{n - 1}(B^{\bullet, \bullet})$ using $(-1)^ph^{p, q}$ on the summand $A^{p, q}$ for $p + q = n$. Then we see that $$ d_{\text{Tot}(B^{\bullet, \bullet})} \circ h' + h' \circ d_{\text{Tot}(A^{\bullet, \bullet})} $$ restricted to the summand $A^{p, q}$ is equal to $$ d_1^{p, q - 1} \circ (-1)^p h^{p, q} + (-1)^p d_2^{p, q - 1} \circ (-1)^p h^{p, q} + (-1)^{p + 1} h^{p + 1, q} \circ d_1^{p, q} + (-1)^p h^{p, q + 1} \circ (-1)^p d_2^{p, q} $$ which evaluates to $f^{p, q} - g^{p, q}$ by the equations given above. This proves the second compatibility. \medskip\noindent Third, suppose that in the paragraph above we have $f = g$. Then the assignment $h \leadsto h'$ above is compatible with the identification of Lemma \ref{lemma-homotopy-shift-cochain}. More precisely, if we view $h$ as a morphism of complexes of complexes $A^{\bullet, \bullet} \to B^{\bullet, \bullet}[0, -1]$ via this lemma then $$ \text{Tot}(A^{\bullet, \bullet}) \xrightarrow{\text{Tot}(h)} \text{Tot}(B^{\bullet, \bullet}[0, -1]) \xrightarrow{\gamma^{-1}} \text{Tot}(B^{\bullet, \bullet})[-1] $$ is equal to $h'$ viewed as a morphism of complexes via the lemma. Here $\gamma$ is the identification of Remark \ref{remark-shift-double-complex}. The verification of this third point is immediate. \medskip\noindent Fourth, let $$ 0 \to A^{\bullet, \bullet} \to B^{\bullet, \bullet} \to C^{\bullet, \bullet} \to 0 $$ be a complex of double complexes and suppose we are given splittings $s^q : C^{\bullet, q} \to B^{\bullet, q}$ and $\pi^q : B^{\bullet, q} \to A^{\bullet, q}$ of this as in Lemma \ref{lemma-ses-termwise-split-cochain} when we view double complexes as complexes of complexes in the manner given above. This on the one hand produces a map $$ \delta : C^{\bullet, \bullet} \longrightarrow A^{\bullet, \bullet}[0, 1] $$ by the procedure in Lemma \ref{lemma-ses-termwise-split-cochain}. On the other hand taking $\text{Tot}$ we obtain a complex $$ 0 \to \text{Tot}(A^{\bullet, \bullet}) \to \text{Tot}(B^{\bullet, \bullet}) \to \text{Tot}(C^{\bullet, \bullet}) \to 0 $$ which is termwise split (see below) and hence comes with a morphism $$ \delta' : \text{Tot}(C^{\bullet, \bullet}) \longrightarrow \text{Tot}(A^{\bullet, \bullet})[1] $$ well defined up to homotopy by Lemmas \ref{lemma-ses-termwise-split-cochain} and \ref{lemma-ses-termwise-split-homotopy-cochain}. Claim: these maps agree in the sense that $$ \text{Tot}(C^{\bullet, \bullet}) \xrightarrow{\text{Tot}(\delta)} \text{Tot}(A^{\bullet, \bullet}[0, 1]) \xrightarrow{\gamma^{-1}} \text{Tot}(A^{\bullet, \bullet})[1] $$ is equal to $\delta'$ where $\gamma$ is as in Remark \ref{remark-shift-double-complex}. To see this denote $s^{p, q} : C^{p, q} \to B^{\bullet, q}$ and $\pi^{p, q} : B^{p, q} \to A^{p, q}$ the components of $s^q$ and $\pi^q$. As splittings $(s')^n : \text{Tot}^n(C^{\bullet, \bullet}) \to \text{Tot}^n(B^{\bullet, \bullet})$ and $(\pi')^n : \text{Tot}^n(B^{\bullet, \bullet}) \to \text{Tot}^n(A^{\bullet, \bullet})$ we use the maps whose components are $s^{p, q}$ and $\pi^{p, q}$ for $p + q = n$. We recall that $$ (\delta')^n = (\pi')^{n + 1} \circ d_{\text{Tot}(B^{\bullet, \bullet})}^n \circ (s')^n : \text{Tot}^n(C^{\bullet, \bullet}) \to \text{Tot}^{n + 1}(A^{\bullet, \bullet}) $$ The restriction of this to the summand $C^{p, q}$ is equal to $$ \pi^{p + 1, q} \circ d_1^{p, q} \circ s^{p, q} + \pi^{p, q + 1} \circ (-1)^p d_2^{p, q} \circ s^{p, q} = \pi^{p, q + 1} \circ (-1)^p d_2^{p, q} \circ s^{p, q} $$ The equality holds because $s^q$ is a morphism of complexes (with $d_1$ as differential) and because $\pi^{p + 1, q} \circ s^{p + 1, q} = 0$ as $s$ and $\pi$ correspond to a direct sum decomposition of $B$ in every bidegree. On the other hand, for $\delta$ we have $$ \delta^q =  \pi^q \circ d_2 \circ s^q : C^{\bullet, q} \to A^{\bullet, q + 1} $$ whose restriction to the summand $C^{p, q}$ is equal to $\pi^{p, q + 1} \circ d_2^{p, q} \circ s^{p, q}$. The difference in signs is exactly canceled out by the sign of $(-1)^p$ in the isomorphism $\gamma$ and the fourth claim is proven.


homology-remark-double-complex-complex-of-complexes-second

Let $\mathcal{A}$ be an additive category with countable direct sums. Let $\text{DoubleComp}(\mathcal{A})$ denote the category of double complexes. We can consider an object $A^{\bullet, \bullet}$ of $\text{DoubleComp}(\mathcal{A})$ as a complex of complexes as follows $$ \ldots \to A^{-1, \bullet} \to A^{0, \bullet} \to A^{1, \bullet} \to \ldots $$ For the variant where we switch the role of the indices, see Remark \ref{remark-double-complex-complex-of-complexes-first}. In this remark we show that taking the associated total complex is compatible with all the structures on complexes we have studied in the chapter so far. \medskip\noindent First, observe that the shift functor on double complexes viewed as complexes of complexes in the manner given above is the functor $[1, 0]$ defined in Remark \ref{remark-shift-double-complex}. By Remark \ref{remark-shift-double-complex} the functor $$ \text{Tot} : \text{DoubleComp}(\mathcal{A}) \to \text{Comp}(\mathcal{A}) $$ is compatible with shift functors, in the sense that we have a functorial isomorphism $\gamma : \text{Tot}(A^{\bullet, \bullet})[1] \to \text{Tot}(A^{\bullet, \bullet}[1, 0])$. \medskip\noindent Second, if $$ f, g : A^{\bullet, \bullet} \to B^{\bullet, \bullet} $$ are homotopic when $f$ and $g$ are viewed as morphisms of complexes of complexes in the manner given above, then $$ \text{Tot}(f), \text{Tot}(g) : \text{Tot}(A^{\bullet, \bullet}) \to \text{Tot}(B^{\bullet, \bullet}) $$ are homotopic maps of complexes. Indeed, let $h = (h^p)$ be a homotopy between $f$ and $g$. If we denote $h^{p, q} : A^{p, q} \to B^{p - 1, q}$ the component in degree $p$ of $h^q$, then this means that $$ f^{p, q} - g^{p, q} = d_1^{p - 1, q} \circ h^{p, q} + h^{p + 1, q} \circ d_1^{p, q} $$ The fact that $h^p : A^{p, \bullet} \to B^{p - 1, \bullet}$ is a map of complexes means that $$ d_2^{p - 1, q} \circ h^{p, q} = h^{p, q + 1} \circ d_2^{p, q} $$ Let us define $h' = ((h')^n)$ the homotopy given by the maps $(h')^n : \text{Tot}^n(A^{\bullet, \bullet}) \to \text{Tot}^{n - 1}(B^{\bullet, \bullet})$ using $h^{p, q}$ on the summand $A^{p, q}$ for $p + q = n$. Then we see that $$ d_{\text{Tot}(B^{\bullet, \bullet})} \circ h' + h' \circ d_{\text{Tot}(A^{\bullet, \bullet})} $$ restricted to the summand $A^{p, q}$ is equal to $$ d_1^{p - 1, q} \circ h^{p, q} + (-1)^{p - 1} d_2^{p - 1, q} \circ h^{p, q} + h^{p + 1, q} \circ d_1^{p, q} + h^{p, q + 1} \circ (-1)^p d_2^{p, q} $$ which evaluates to $f^{p, q} - g^{p, q}$ by the equations given above. This proves the second compatibility. \medskip\noindent Third, suppose that in the paragraph above we have $f = g$. Then the assignment $h \leadsto h'$ above is compatible with the identification of Lemma \ref{lemma-homotopy-shift-cochain}. More precisely, if we view $h$ as a morphism of complexes of complexes $A^{\bullet, \bullet} \to B^{\bullet, \bullet}[-1, 0]$ via this lemma then $$ \text{Tot}(A^{\bullet, \bullet}) \xrightarrow{\text{Tot}(h)} \text{Tot}(B^{\bullet, \bullet}[-1, 0]) \xrightarrow{\gamma^{-1}} \text{Tot}(B^{\bullet, \bullet})[-1] $$ is equal to $h'$ viewed as a morphism of complexes via the lemma. Here $\gamma$ is the identification of Remark \ref{remark-shift-double-complex}. The verification of this third point is immediate. \medskip\noindent Fourth, let $$ 0 \to A^{\bullet, \bullet} \to B^{\bullet, \bullet} \to C^{\bullet, \bullet} \to 0 $$ be a complex of double complexes and suppose we are given splittings $s^p : C^{p, \bullet} \to B^{p, \bullet}$ and $\pi^p : B^{p, \bullet} \to A^{p, \bullet}$ of this as in Lemma \ref{lemma-ses-termwise-split-cochain} when we view double complexes as complexes of complexes in the manner given above. This on the one hand produces a map $$ \delta : C^{\bullet, \bullet} \longrightarrow A^{\bullet, \bullet}[0, 1] $$ by the procedure in Lemma \ref{lemma-ses-termwise-split-cochain}. On the other hand taking $\text{Tot}$ we obtain a complex $$ 0 \to \text{Tot}(A^{\bullet, \bullet}) \to \text{Tot}(B^{\bullet, \bullet}) \to \text{Tot}(C^{\bullet, \bullet}) \to 0 $$ which is termwise split (see below) and hence comes with a morphism $$ \delta' : \text{Tot}(C^{\bullet, \bullet}) \longrightarrow \text{Tot}(A^{\bullet, \bullet})[1] $$ well defined up to homotopy by Lemmas \ref{lemma-ses-termwise-split-cochain} and \ref{lemma-ses-termwise-split-homotopy-cochain}. Claim: these maps agree in the sense that $$ \text{Tot}(C^{\bullet, \bullet}) \xrightarrow{\text{Tot}(\delta)} \text{Tot}(A^{\bullet, \bullet}[1, 0]) \xrightarrow{\gamma^{-1}} \text{Tot}(A^{\bullet, \bullet})[1] $$ is equal to $\delta'$ where $\gamma$ is as in Remark \ref{remark-shift-double-complex}. To see this denote $s^{p, q} : C^{p, q} \to B^{\bullet, q}$ and $\pi^{p, q} : B^{p, q} \to A^{p, q}$ the components of $s^q$ and $\pi^q$. As splittings $(s')^n : \text{Tot}^n(C^{\bullet, \bullet}) \to \text{Tot}^n(B^{\bullet, \bullet})$ and $(\pi')^n : \text{Tot}^n(B^{\bullet, \bullet}) \to \text{Tot}^n(A^{\bullet, \bullet})$ we use the maps whose components are $s^{p, q}$ and $\pi^{p, q}$ for $p + q = n$. We recall that $$ (\delta')^n = (\pi')^{n + 1} \circ d_{\text{Tot}(B^{\bullet, \bullet})}^n \circ (s')^n : \text{Tot}^n(C^{\bullet, \bullet}) \to \text{Tot}^{n + 1}(A^{\bullet, \bullet}) $$ The restriction of this to the summand $C^{p, q}$ is equal to $$ \pi^{p + 1, q} \circ d_1^{p, q} \circ s^{p, q} + \pi^{p, q + 1} \circ (-1)^p d_2^{p, q} \circ s^{p, q} = \pi^{p + 1, q} \circ d_1^{p, q} \circ s^{p, q} $$ The equality holds because $s^p$ is a morphism of complexes (with $d_2$ as differential) and because $\pi^{p, q + 1} \circ s^{p, q + 1} = 0$ as $s$ and $\pi$ correspond to a direct sum decomposition of $B$ in every bidegree. On the other hand, for $\delta$ we have $$ \delta^p =  \pi^p \circ d_1 \circ s^p : C^{p, \bullet} \to A^{p + 1, \bullet} $$ whose restriction to the summand $C^{p, q}$ is equal to $\pi^{p + 1, q} \circ d_1^{p, q} \circ s^{p, q}$. Thus we get the same as before which matches with the fact that the isomorphism $\gamma : \text{Tot}(A^{\bullet, \bullet})[1] \to \text{Tot}(A^{\bullet, bullet}[1, 0])$ is defined without the intervetion of signs.


homology-remark-allow-translation-functors

It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. To accomodate this (and to get around certain technical issues) we introduce the following notion. Let $\mathcal{A}$ be an abelian category. Let $(T_r)_{r \geq 1}$ be a sequence of {\it translation} or {\it shift} functors, i.e., $T_r : \mathcal{A} \to \mathcal{A}$ is an isomorphism of categories. In this setting a {\it spectral sequence} is given by a system $(E_r, d_r)_{r \geq 1}$ where each $E_r$ is an object of $\mathcal{A}$, each $d_r : E_r \to T_rE_r$ is a morphism such that $T_rd_r \circ d_r = 0$ so that $$ \xymatrix{ \ldots \ar[r] & T_r^{-1}E_r \ar[r]^-{T_r^{-1}d_r} & E_r \ar[r]^-{d_r} & T_rE_r \ar[r]^{T_r d_r} & T_r^2E_r \ar[r] & \ldots } $$ is a complex and $E_{r + 1} = \Ker(d_r)/\Im(T_r^{-1}d_r)$ for $r \geq 1$. It is clear what a {\it morphism of spectral sequences} means in this setting. In this setting we can still define $$ 0 = B_1 \subset B_2 \subset \ldots \subset B_r \subset \ldots \subset Z_r \subset \ldots \subset Z_2 \subset Z_1 = E_1 $$ and $Z_\infty$ and $B_\infty$ (if they exist) as above.


homology-remark-shifted-exact-couple

Let $\mathcal{A}$ be an abelian category. Let $S, T : \mathcal{A} \to \mathcal{A}$ be shift functors, i.e., isomorphisms of categories. We will indicate the $n$-fold compositions by $S^nA$ and $T^nA$ for $A \in \Ob(\mathcal{A})$ and $n \in \mathbf{Z}$. In this situation an {\it exact couple} is a datum $(A, E, \alpha, f, g)$ where $A$, $E$ are objects of $\mathcal{A}$ and $\alpha : A \to T^{-1}A$, $f : E \to A$, $g : A \to SE$ are morphisms such that $$ \xymatrix{ TE \ar[r]^-{Tf} & TA \ar[r]^-{T\alpha} & A \ar[r]^-{g} & SE \ar[r]^{Sf} & SA } $$ is an exact complex. Let's visualize this as follows $$ \xymatrix{ TA \ar[rrr]_{T\alpha} & & & A \ar[ld]^g \ar[rrr]_\alpha & & & T^{-1}A \ar[ld]^{T^{-1}g} \\ & TE \ar[lu]^{Tf} \ar@{..}[r] & SE & & E \ar[lu]^f \ar@{..}[r] & T^{-1}SE } $$ We set $d = g \circ f : E \to SE$. Then $d \circ S^{-1}d = g \circ f \circ S^{-1}g \circ S^{-1}f = 0$ because $f \circ S^{-1}g = 0$. Set $E' = \Ker(d)/\Im(S^{-1}d)$. Set $A' = \Im(T\alpha)$. Let $\alpha' : A' \to T^{-1}A'$ induced by $\alpha$. Let $f' : E' \to A'$ be induced by $f$ which works because $f(\Ker(d)) \subset \Ker(g) = \Im(T\alpha)$. Finally, let $g' : A' \to TSE'$ induced by ``$Tg \circ (T\alpha)^{-1}$''\footnote{This works because $TSE' = \Ker(TSd)/\Im(Td)$ and $Tg(\Ker(T\alpha)) = Tg(\Im(Tf)) = \Im(T(d))$ and $TS(d)(\Im(Tg)) = \Im(TSg \circ TSf \circ Tg) = 0$.}. \medskip\noindent In exactly the same way as above we find \begin{enumerate} \item $\Ker(d) = f^{-1}(\Ker(g)) = f^{-1}(\Im(T\alpha))$, \item $\Im(d) = g(\Im(f)) = g(\Ker(\alpha))$, \item $(A', E', \alpha', f', g')$ is an exact couple for the shift functors $TS$ and $T$. \end{enumerate} We obtain a spectral sequence (as in Remark \ref{remark-allow-translation-functors}) with $E_1 = E$, $E_2 = E'$, etc, with $d_r : E_r \to T^{r - 1}SE_r$ for all $r \geq 1$. Lemma \ref{lemma-spectral-sequence-associated-exact-couple} tells us that $$ SB_{r + 1} = g(\Ker(T^{-r + 1}\alpha \circ \ldots \circ T^{-1}\alpha \circ \alpha)) $$ and $$ Z_{r + 1} = f^{-1}(\Im(T\alpha \circ T^2\alpha \circ \ldots \circ T^r\alpha)) $$ in this situation. The description of the map $d_{r + 1}$ is similar to that given in the lemma. (It may be easier to use these explicit descriptions to prove one gets a spectral sequence from such an exact couple.)


homology-remark-differential-object-selfmap

Let $\mathcal{A}$ be an abelian category and let $S, T : \mathcal{A} \to \mathcal{A}$ be shift functors, i.e., isomorphisms of categories. Assume that $TS = ST$ as functors. Consider pairs $(A, d)$ consisting of an object $A$ of $\mathcal{A}$ and a morphism $d : A \to SA$ such that $d \circ S^{-1}d = 0$. The category of these objects is abelian. We define $H(A, d) = \Ker(d)/\Im(S^{-1}d)$ and we observe that $H(SA, Sd) = SH(A, d)$ (canonical isomorphism). Given a short exact sequence $$ 0 \to (A, d) \to (B, d) \to (C, d) \to 0 $$ we obtain a long exact homology sequence $$ \ldots \to S^{-1}H(C, d) \to H(A, d) \to H(B, d) \to H(C, d) \to SH(A, d) \to \ldots $$ (note the shifts in the boundary maps). Since $ST = TS$ the functor $T$ defines a shift functor on pairs by setting $T(A, d) = (TA, Td)$. Next, let $\alpha : (A, d) \to T^{-1}(A, d)$ be injective with cokernel $(Q, d)$. Then we get an exact couple as in Remark \ref{remark-shifted-exact-couple} with shift functors $TS$ and $T$ given by $$ (H(A, d), S^{-1}H(Q, d), \overline{\alpha}, f, g) $$ where $\overline{\alpha} : H(A, d) \to T^{-1}H(A, d)$ is induced by $\alpha$, the map $f : S^{-1}H(Q, d) \to H(A, d)$ is the boundary map and $g : H(A, d) \to TH(Q, d) = TS(S^{-1}H(Q, d))$ is induced by the quotient map $A \to TQ$. Thus we get a spectral sequence as above with $E_1 = S^{-1}H(Q, d)$ and differentials $d_r : E_r \to T^rSE_r$. As above we set $E_0 = S^{-1}Q$ and $d_0 : E_0 \to SE_0$ given by $S^{-1}d : S^{-1}Q \to Q$. If according to our conventions we define $B_r \subset Z_r \subset E_0$, then we have for $r \geq 1$ that \begin{enumerate} \item $SB_r$ is the image of $$ (T^{-r + 1}\alpha \circ \ldots \circ T^{-1}\alpha)^{-1} \Im(T^{-r}S^{-1}d) $$ under the natural map $T^{-1}A \to Q$, \item $Z_r$ is the image of $$ (S^{-1}T^{-1}d)^{-1} \Im(\alpha \circ \ldots \circ T^{r - 1}\alpha) $$ under the natural map $S^{-1}T^{-1}A \to S^{-1}Q$. \end{enumerate} The differentials can be described as follows: if $x \in Z_r$, then pick $x' \in S^{-1}T^{-1}A$ mapping to $x$. Then $S^{-1}T^{-1}d(x')$ is $(\alpha \circ \ldots \circ T^{r - 1}\alpha)(y)$ for some $y \in T^{r - 1}A$. Then $d_r(x) \in T^rSE_r$ is represented by the class of the image of $y$ in $T^rSE_0 = T^rQ$ modulo $T^rSB_r$.


homology-remark-need-left-exactness

Let $\mathcal{A}$, $\mathcal{B}$, $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be as in Lemma \ref{lemma-adjoint-preserve-injectives}. In the presence of assumption (1) assumption (2) is equivalent to requiring that $v$ is exact. Moreover, condition (2) is necessary. Here is an example. Let $A \to B$ be a ring map. Let $u : \text{Mod}_B \to \text{Mod}_A$ be $u(N) = N_A$ and let $v : \text{Mod}_A \to \text{Mod}_B$ be $v(M) = M \otimes_A B$. Then $u$ is right adjoint to $v$, and $u$ is exact and $v$ is right exact, but $v$ does not transform injective maps into injective maps in general (i.e., $v$ is not left exact). Moreover, it is {\bf not} the case that $u$ transforms injective $B$-modules into injective $A$-modules. For example, if $A = \mathbf{Z}$ and $B = \mathbf{Z}/p\mathbf{Z}$, then the injective $B$-module $\mathbf{Z}/p\mathbf{Z}$ is not an injective $\mathbf{Z}$-module. In fact, the lemma applies to this example if and only if the ring map $A \to B$ is flat.


homology-remark-faithfulness-needed

Let $\mathcal{A}$, $\mathcal{B}$, $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be as In Lemma \ref{lemma-adjoint-enough-injectives}. In the presence of conditions (1) and (2) condition (4) is equivalent to $v$ being faithful. Moreover, condition (4) is needed. An example is to consider the case where the functors $u$ and $v$ are both the zero functor.


categories-remark-big-categories

Big categories. In some texts a category is allowed to have a proper class of objects. We will allow this as well in these notes but only in the following list of cases (to be updated as we go along). In particular, when we say: ``Let $\mathcal{C}$ be a category'' then it is understood that $\Ob(\mathcal{C})$ is a set. \begin{enumerate} \item The category $\textit{Sets}$ of sets. \item The category $\textit{Ab}$ of abelian groups. \item The category $\textit{Groups}$ of groups. \item Given a group $G$ the category $G\textit{-Sets}$ of sets with a left $G$-action. \item Given a ring $R$ the category $\text{Mod}_R$ of $R$-modules. \item Given a field $k$ the category of vector spaces over $k$. \item The category of rings. \item The category of schemes. \item The category $\textit{Top}$ of topological spaces. \item Given a topological space $X$ the category $\textit{PSh}(X)$ of presheaves of sets over $X$. \item Given a topological space $X$ the category $\Sh(X)$ of sheaves of sets over $X$. \item Given a topological space $X$ the category $\textit{PAb}(X)$ of presheaves of abelian groups over $X$. \item Given a topological space $X$ the category $\textit{Ab}(X)$ of sheaves of abelian groups over $X$. \item Given a small category $\mathcal{C}$ the category of functors from $\mathcal{C}$ to $\textit{Sets}$. \item Given a category $\mathcal{C}$ the category of presheaves of sets over $\mathcal{C}$. \item Given a site $\mathcal{C}$ the category of sheaves of sets over $\mathcal{C}$. \end{enumerate} One of the reason to enumerate these here is to try and avoid working with something like the ``collection'' of ``big'' categories which would be like working with the collection of all classes which I think definitively is a meta-mathematical object.


categories-remark-unique-identity

It follows directly from the definition that any two identity morphisms of an object $x$ of $\mathcal{A}$ are the same. Thus we may and will speak of {\it the} identity morphism $\text{id}_x$ of $x$.


categories-remark-functor-into-sets

Suppose that $\mathcal{A}$ is a category. A functor $F$ from $\mathcal{A}$ to $\textit{Sets}$ is a mathematical object (i.e., it is a set not a class or a formula of set theory, see Sets, Section \ref{sets-section-sets-everything}) even though the category of sets is ``big''. Namely, the range of $F$ on objects will be a set $F(\Ob(\mathcal{A}))$ and then we may think of $F$ as a functor between $\mathcal{A}$ and the full subcategory of the category of sets whose objects are elements of $F(\Ob(\mathcal{A}))$.


categories-remark-functors-sets-sets

This is one instance where the same thing does not hold if $\mathcal{A}$ is a ``big'' category. For example consider functors $\textit{Sets} \to \textit{Sets}$. As we have currently defined it such a functor is a class and not a set. In other words, it is given by a formula in set theory (with some variables equal to specified sets)! It is not a good idea to try to consider all possible formulae of set theory as part of the definition of a mathematical object. The same problem presents itself when considering sheaves on the category of schemes for example. We will come back to this point later.


categories-remark-diagram-small

The index category of a (co)limit will never be allowed to have a proper class of objects. In this project it means that it cannot be one of the categories listed in Remark \ref{remark-big-categories}


categories-remark-limit-colim

We often write $\lim_i M_i$, $\colim_i M_i$, $\lim_{i\in \mathcal{I}} M_i$, or $\colim_{i\in \mathcal{I}} M_i$ instead of the versions indexed by $\mathcal{I}$. Using this notation, and using the description of limits and colimits of sets in Section \ref{section-limit-sets} below, we can say the following. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. \begin{enumerate} \item The object $\lim_i M_i$ if it exists satisfies the following property $$ \Mor_\mathcal{C}(W, \lim_i M_i) = \lim_i \Mor_\mathcal{C}(W, M_i) $$ where the limit on the right takes place in the category of sets. \item The object $\colim_i M_i$ if it exists satisfies the following property $$ \Mor_\mathcal{C}(\colim_i M_i, W) = \lim_{i\in \mathcal{I}^\text{opp}} \Mor_\mathcal{C}(M_i, W) $$ where on the right we have the limit over the opposite category with value in the category of sets. \end{enumerate} By the Yoneda lemma (and its dual) this formula completely determines the limit, respectively the colimit.


categories-remark-cones-and-cocones

Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. In this setting a {\it cone} for $M$ is given by an object $W$ and a family of morphisms $q_i : W \to M_i$, $i \in \Ob(\mathcal{I})$ such that for all morphisms $\phi : i \to i'$ of $\mathcal{I}$ the diagram $$ \xymatrix{ & W \ar[dl]_{q_i} \ar[dr]^{q_{i'}} \\ M_i \ar[rr]^{M(\phi)} & & M_{i'} } $$ is commutative. The collection of cones forms a category with an obvious notion of morphisms. Clearly, the limit of $M$, if it exists, is a final object in the category of cones. Dually, a {\it cocone} for $M$ is given by an object $W$ and a family of morphisms $t_i : M_i \to W$ such that for all morphisms $\phi : i \to i'$ in $\mathcal{I}$ the diagram $$ \xymatrix{ M_i \ar[rr]^{M(\phi)} \ar[dr]_{t_i} & & M_{i'} \ar[dl]^{t_{i'}} \\ & W } $$ commutes. The collection of cocones forms a category with an obvious notion of morphisms. Similarly to the above the colimit of $M$ exists if and only if the category of cocones has an initial object.


categories-remark-preorder-versus-partial-order

Let $I$ be a preordered set. From $I$ we can construct a canonical partially ordered set $\overline{I}$ and an order preserving map $\pi : I \to \overline{I}$. Namely, we can define an equivalence relation $\sim$ on $I$ by the rule $$ i \sim j \Leftrightarrow (i \leq j\text{ and }j \leq i). $$ We set $\overline{I} = I/\sim$ and we let $\pi : I \to \overline{I}$ be the quotient map. Finally, $\overline{I}$ comes with a unique partial ordering such that $\pi(i) \leq \pi(j) \Leftrightarrow i \leq j$. Observe that if $I$ is a directed set, then $\overline{I}$ is a directed partially ordered set. Given an (inverse) system $N$ over $\overline{I}$ we obtain an (inverse) system $M$ over $I$ by setting $M_i = N_{\pi(i)}$. This construction defines a functor between the category of inverse systems over $I$ and $\overline{I}$. In fact, this is an equivalence. The reason is that if $i \sim j$, then for any system $M$ over $I$ the maps $M_i \to M_j$ and $M_j \to M_i$ are mutually inverse isomorphisms. More precisely, choosing a section $s : \overline{I} \to I$ of $\pi$ a quasi-inverse of the functor above sends $M$ to $N$ with $N_{\overline{i}} = M_{s(\overline{i})}$. Finally, this correspondence is compatible with colimits of systems: if $M$ and $N$ are related as above and if either $\colim_{\overline{I}} N$ or $\colim_I M$ exists then so does the other and $\colim_{\overline{I}} N = \colim_I M$. Similar results hold for inverse systems and limits of inverse systems.


categories-remark-trick-needed

Note that a finite directed set $(I, \geq)$ always has a greatest object $i_\infty$. Hence any colimit of a system $(M_i, f_{ii'})$ over such a set is trivial in the sense that the colimit equals $M_{i_\infty}$. In contrast, a colimit indexed by a finite filtered category need not be trivial. For instance, let $\mathcal{I}$ be the category with a single object $i$ and a single non-trivial morphism $e$ satisfying $e = e \circ e$. The colimit of a diagram $M : \mathcal{I} \to Sets$ is the image of the idempotent $M(e)$. This illustrates that something like the trick of passing to $\mathcal{I}\times \omega$ in the proof of Lemma \ref{lemma-directed-category-system} is essential.


categories-remark-ind-category

Let $\mathcal{C}$ be a category. There exists a big category $\text{Ind-}\mathcal{C}$ of {\it ind-objects of} $\mathcal{C}$. Namely, if $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ are filtered diagrams in $\mathcal{C}$, then we can define $$ \Mor_{\text{Ind-}\mathcal{C}}(F, G) = \lim_i \colim_j \Mor_\mathcal{C}(F(i), G(j)). $$ There is a canonical functor $\mathcal{C} \to \text{Ind-}\mathcal{C}$ which maps $X$ to the {\it constant system} on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here.


categories-remark-pro-category

Let $\mathcal{C}$ be a category. There exists a big category $\text{Pro-}\mathcal{C}$ of {\it pro-objects} of $\mathcal{C}$. Namely, if $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ are cofiltered diagrams in $\mathcal{C}$, then we can define $$ \Mor_{\text{Pro-}\mathcal{C}}(F, G) = \lim_j \colim_i \Mor_\mathcal{C}(F(i), G(j)). $$ There is a canonical functor $\mathcal{C} \to \text{Pro-}\mathcal{C}$ which maps $X$ to the {\it constant system} on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here.


categories-remark-pro-category-copresheaves

Let $\mathcal{C}$ be a category. Let $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ be cofiltered diagrams in $\mathcal{C}$. Consider the functors $A, B : \mathcal{C} \to \textit{Sets}$ defined by $$ A(X) = \colim_i \Mor_\mathcal{C}(F(i), X) \quad\text{and}\quad B(X) = \colim_j \Mor_\mathcal{C}(G(j), X) $$ We claim that a morphism of pro-systems from $F$ to $G$ is the same thing as a transformation of functors $t : B \to A$. Namely, given $t$ we can apply $t$ to the class of $\text{id}_{G(j)}$ in $B(G(j))$ to get a compatible system of elements $\xi_j \in A(G(j)) = \colim_i \Mor_\mathcal{C}(F(i), G(j))$ which is exactly our definition of a morphism in $\text{Pro-}\mathcal{C}$ in Remark \ref{remark-pro-category}. We omit the construction of a transformation $B \to A$ given a morphism of pro-objects from $F$ to $G$.


categories-remark-how-to-use-it

The lemma above is often used to construct the free something on something. For example the free abelian group on a set, the free group on a set, etc. The idea, say in the case of the free group on a set $E$ is to consider the functor $$ F : \textit{Groups} \to \textit{Sets},\quad G \longmapsto \text{Map}(E, G) $$ This functor commutes with limits. As our family of objects we can take a family $E \to G_i$ consisting of groups $G_i$ of cardinality at most $\max(\aleph_0, |E|)$ and set maps $E \to G_i$ such that every isomorphism class of such a structure occurs at least once. Namely, if $E \to G$ is a map from $E$ to a group $G$, then the subgroup $G'$ generated by the image has cardinality at most $\max(\aleph_0, |E|)$. The lemma tells us the functor is representable, hence there exists a group $F_E$ such that $\Mor_{\textit{Groups}}(F_E, G) = \text{Map}(E, G)$. In particular, the identity morphism of $F_E$ corresponds to a map $E \to F_E$ and one can show that $F_E$ is generated by the image without imposing any relations. \medskip\noindent Another typical application is that we can use the lemma to construct colimits once it is known that limits exist. We illustrate it using the category of topological spaces which has limits by Topology, Lemma \ref{topology-lemma-limits}. Namely, suppose that $\mathcal{I} \to \textit{Top}$, $i \mapsto X_i$ is a functor. Then we can consider $$ F : \textit{Top} \longrightarrow \textit{Sets},\quad Y \longmapsto \lim_\mathcal{I} \Mor_{\textit{Top}}(X_i, Y) $$ This functor commutes with limits. Moreover, given any topological space $Y$ and an element $(\varphi_i : X_i \to Y)$ of $F(Y)$, there is a subspace $Y' \subset Y$ of cardinality at most $|\coprod X_i|$ such that the morphisms $\varphi_i$ map into $Y'$. Namely, we can take the induced topology on the union of the images of the $\varphi_i$. Thus it is clear that the hypotheses of the lemma are satisfied and we find a topological space $X$ representing the functor $F$, which precisely means that $X$ is the colimit of the diagram $i \mapsto X_i$.


categories-remark-motivation-localization

The motivation for the construction of $S^{-1} \mathcal{C}$ is to ``force'' the morphisms in $S$ to be invertible by artificially creating inverses to them (at the cost of some existing morphisms possibly becoming identified with each other). This is similar to the localization of a commutative ring at a multiplicative subset, and more generally to the localization of a noncommutative ring at a right denominator set (see \cite[Section 10A]{Lam}). This is more than just a similarity: The construction of $S^{-1} \mathcal{C}$ (or, more precisely, its version for additive categories $\mathcal{C}$) actually generalizes the latter type of localization. Namely, a noncommutative ring can be viewed as a pre-additive category with a single object (the morphisms being the elements of the ring); a multiplicative subset of this ring then becomes a set $S$ of morphisms satisfying LMS1 (aka RMS1). Then, the conditions RMS2 and RMS3 for this category and this subset $S$ translate into the two conditions (``right permutable'' and ``right reversible'') of a right denominator set (and similarly for LMS and left denominator sets), and $S^{-1} \mathcal{C}$ (with a properly defined additive structure) is the one-object category corresponding to the localization of the ring.


categories-remark-left-localization-morphisms-colimit

Let $\mathcal{C}$ be a category. Let $S$ be a left multiplicative system. Given an object $Y$ of $\mathcal{C}$ we denote $Y/S$ the category whose objects are $s : Y \to Y'$ with $s \in S$ and whose morphisms are commutative diagrams $$ \xymatrix{ & Y \ar[ld]_s \ar[rd]^t & \\ Y' \ar[rr]^a & & Y'' } $$ where $a : Y' \to Y''$ is arbitrary. We claim that the category $Y/S$ is filtered (see Definition \ref{definition-directed}). Namely, LMS1 implies that $\text{id}_Y : Y \to Y$ is in $Y/S$; hence $Y/S$ is nonempty. LMS2 implies that given $s_1 : Y \to Y_1$ and $s_2 : Y \to Y_2$ we can find a diagram $$ \xymatrix{ Y \ar[d]_{s_1} \ar[r]_{s_2} & Y_2 \ar[d]^t \\ Y_1 \ar[r]^a & Y_3 } $$ with $t \in S$. Hence $s_1 : Y \to Y_1$ and $s_2 : Y \to Y_2$ both have maps to $t \circ s_2 : Y \to Y_3$ in $Y/S$. Finally, given two morphisms $a, b$ from $s_1 : Y \to Y_1$ to $s_2 : Y \to Y_2$ in $Y/S$ we see that $a \circ s_1 = b \circ s_1$; hence by LMS3 there exists a $t : Y_2 \to Y_3$ in $S$ such that $t \circ a = t \circ b$. Now the combined results of Lemmas \ref{lemma-morphisms-left-localization} and \ref{lemma-equality-morphisms-left-localization} tell us that \begin{equation} \label{equation-left-localization-morphisms-colimit} \Mor_{S^{-1}\mathcal{C}}(X, Y) = \colim_{(s : Y \to Y') \in Y/S} \Mor_\mathcal{C}(X, Y') \end{equation} This formula expressing morphism sets in $S^{-1}\mathcal{C}$ as a filtered colimit of morphism sets in $\mathcal{C}$ is occasionally useful.


categories-remark-right-localization-morphisms-colimit

Let $\mathcal{C}$ be a category. Let $S$ be a right multiplicative system. Given an object $X$ of $\mathcal{C}$ we denote $S/X$ the category whose objects are $s : X' \to X$ with $s \in S$ and whose morphisms are commutative diagrams $$ \xymatrix{ X' \ar[rd]_s \ar[rr]_a & & X'' \ar[ld]^t \\ & X } $$ where $a : X' \to X''$ is arbitrary. The category $S/X$ is cofiltered (see Definition \ref{definition-codirected}). (This is dual to the corresponding statement in Remark \ref{remark-left-localization-morphisms-colimit}.) Now the combined results of Lemmas \ref{lemma-morphisms-right-localization} and \ref{lemma-equality-morphisms-right-localization} tell us that \begin{equation} \label{equation-right-localization-morphisms-colimit} \Mor_{S^{-1}\mathcal{C}}(X, Y) = \colim_{(s : X' \to X) \in (S/X)^{opp}} \Mor_\mathcal{C}(X', Y) \end{equation} This formula expressing morphisms in $S^{-1}\mathcal{C}$ as a filtered colimit of morphisms in $\mathcal{C}$ is occasionally useful.


categories-remark-big-2-categories

Big $2$-categories. In many texts a $2$-category is allowed to have a class of objects (but hopefully a ``class of classes'' is not allowed). We will allow these ``big'' $2$-categories as well, but only in the following list of cases (to be updated as we go along): \begin{enumerate} \item The $2$-category of categories $\textit{Cat}$. \item The $(2, 1)$-category of categories $\textit{Cat}$. \item The $2$-category of groupoids $\textit{Groupoids}$; this is a $(2, 1)$-category. \item The $2$-category of fibred categories over a fixed category. \item The $(2, 1)$-category of fibred categories over a fixed category. \end{enumerate} See Definition \ref{definition-2-1-category}. Note that in each case the class of objects of the $2$-category $\mathcal{C}$ is a proper class, but for all objects $x, y \in \Ob(C)$ the category $\Mor_\mathcal{C}(x, y)$ is ``small'' (according to our conventions).


categories-remark-other-2-categories

Thus there are variants of the construction of Example \ref{example-2-1-category-of-categories} above where we look at the $2$-category of groupoids, or categories fibred in groupoids over a fixed category, or stacks. And so on.


categories-remark-other-description-2-fibre-product

Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be categories. Let $F : \mathcal{A} \to \mathcal{C}$ and $G : \mathcal{B} \to \mathcal{C}$ be functors. Another, slightly more symmetrical, construction of a $2$-fibre product $\mathcal{A} \times_\mathcal{C} \mathcal{B}$ is as follows. An object is a quintuple $(A, B, C, a, b)$ where $A, B, C$ are objects of $\mathcal{A}, \mathcal{B}, \mathcal{C}$ and where $a : F(A) \to C$ and $b : G(B) \to C$ are isomorphisms. A morphism $(A, B, C, a, b) \to (A', B', C', a', b')$ is given by a triple of morphisms $A \to A', B \to B', C \to C'$ compatible with the morphisms $a, b, a', b'$. We can prove directly that this leads to a $2$-fibre product. However, it is easier to observe that the functor $(A, B, C, a, b) \mapsto (A, B, b^{-1} \circ a)$ gives an equivalence from the category of quintuples to the category constructed in Example \ref{example-2-fibre-product-categories}.


categories-remark-alternative-fibred-groupoids-strict

We can use the $2$-Yoneda lemma to give an alternative proof of Lemma \ref{lemma-fibred-groupoids-strict}. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. We define a contravariant functor $F$ from $\mathcal{C}$ to the category of groupoids as follows: for $U\in \Ob(\mathcal{C})$ let $$ F(U) = \Mor_{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}). $$ If $f : U \to V$ the induced functor $\mathcal{C}/U \to \mathcal{C}/V$ induces the morphism $F(f) : F(V) \to F(U)$. Clearly $F$ is a functor. Let $\mathcal{S}'$ be the associated category fibred in groupoids from Example \ref{example-functor-groupoids}. There is an obvious functor $G : \mathcal{S}' \to \mathcal{S}$ over $\mathcal{C}$ given by taking the pair $(U, x)$, where $U \in \Ob(\mathcal{C})$ and $x \in F(U)$, to $x(\text{id}_U) \in \mathcal{S}$.  Now Lemma \ref{lemma-yoneda-2category} implies that for each $U$, $$ G_U : \mathcal{S}'_U = F(U)= \Mor_{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \to \mathcal{S}_U $$ is an equivalence, and thus $G$ is an equivalence between $\mathcal{S}$ and $\mathcal{S}'$ by Lemma \ref{lemma-equivalence-fibred-categories}.


categories-remark-left-dual-adjoint

Lemma \ref{lemma-left-dual} says in particular that $Z \mapsto Z \otimes Y$ is the right adjoint of $Z' \mapsto Z' \otimes X$. In particular, uniqueness of adjoint functors guarantees that a left dual of $X$, if it exists, is unique up to unique isomorphism. Conversely, assume the functor $Z \mapsto Z \otimes Y$ is a right adjoint of the functor $Z' \mapsto Z' \otimes X$, i.e., we're given a bijection $$ \Mor(Z' \otimes X, Z) \longrightarrow \Mor(Z', Z \otimes Y) $$ functorial in both $Z$ and $Z'$. The unit of the adjunction produces maps $$ \eta_Z : Z \to Z \otimes X \otimes Y $$ functorial in $Z$ and the counit of the adjoint produces maps $$ \epsilon_{Z'} : Z' \otimes Y \otimes X \to Z' $$ functorial in $Z'$. In particular, we find $\eta = \eta_\mathbf{1} : \mathbf{1} \to X \otimes Y$ and $\epsilon = \epsilon_\mathbf{1} : Y \otimes X \to \mathbf{1}$. As an exercise in the relationship between units, counits, and the adjunction isomorphism, the reader can show that we have $$ (\epsilon \otimes \text{id}_Y) \circ \eta_Y = \text{id}_Y \quad\text{and}\quad \epsilon_X \circ (\eta \otimes \text{id}_X) = \text{id}_X $$ However, this isn't enough to show that $(\epsilon \otimes \text{id}_Y) \circ (\text{id}_Y \otimes \eta) = \text{id}_Y$ and $(\text{id}_X \otimes \epsilon) \circ (\eta \otimes \text{id}_X) = \text{id}_X$, because we don't know in general that $\eta_Y = \text{id}_Y \otimes \eta$ and we don't know that $\epsilon_X = \epsilon \otimes \text{id}_X$. For this it would suffice to know that our adjunction isomorphism has the following property: for every $W, Z, Z'$ the diagram $$ \xymatrix{ \Mor(Z' \otimes X, Z) \ar[r] \ar[d]_{\text{id}_W \otimes -} & \Mor(Z', Z \otimes Y) \ar[d]^{\text{id}_W \otimes -} \\ \Mor(W \otimes Z' \otimes X, W \otimes Z) \ar[r] & \Mor(W \otimes Z', W \otimes Z \otimes Y) } $$ If this holds, we will say {\it the adjunction is compatible with the given tensor structure}. Thus the requirement that $Z \mapsto Z \otimes Y$ be the right adjoint of $Z' \mapsto Z' \otimes X$ compatible with the given tensor structure is an equivalent formulation of the property of being a left dual.


topologies-remark-choice-sites

Take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set of schemes $\{X, Y, S\}$. Choose any set of coverings $\text{Cov}_{fppf}$ on $\Sch_\alpha$ as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of fppf coverings. Let $\Sch_{fppf}$ denote the big fppf site so obtained. Next, for $\tau \in \{Zariski, \etale, smooth, syntomic\}$ let $\Sch_\tau$ have the same underlying category as $\Sch_{fppf}$ with coverings $\text{Cov}_\tau \subset \text{Cov}_{fppf}$ simply the subset of $\tau$-coverings. It is straightforward to check that this gives rise to a big site $\Sch_\tau$.


pic-remark-when-proposition-applies

Let $f : X \to S$ be a morphism of schemes. The assumption of Proposition \ref{proposition-hilb-d-representable} and hence the conclusion holds in each of the following cases: \begin{enumerate} \item $X$ is quasi-affine, \item $f$ is quasi-affine, \item $f$ is quasi-projective, \item $f$ is locally projective, \item there exists an ample invertible sheaf on $X$, \item there exists an $f$-ample invertible sheaf on $X$, and \item there exists an $f$-very ample invertible sheaf on $X$. \end{enumerate} Namely, in each of these cases, every finite set of points of a fibre $X_s$ is contained in a quasi-compact open $U$ of $X$ which comes with an ample invertible sheaf, is isomorphic to an open of an affine scheme, or is isomorphic to an open of $\text{Proj}$ of a graded ring (in each case this follows by unwinding the definitions). Thus the existence of suitable affine opens by Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}.


pic-remark-universal-object-hilb-d

Let $X$ be a geometrically irreducible smooth proper curve over a field $k$ as in Proposition \ref{proposition-hilb-d}. Let $d \geq 0$. The universal closed object is a relatively effective divisor $$ D_{univ} \subset \underline{\Hilbfunctor}^{d + 1}_{X/k} \times_k X $$ over $\underline{\Hilbfunctor}^{d + 1}_{X/k}$ by Lemma \ref{lemma-divisors-on-curves}. In fact, $D_{univ}$ is isomorphic as a scheme to $\underline{\Hilbfunctor}^d_{X/k} \times_k X$, see proof of Lemma \ref{lemma-universal-object}. In particular, $D_{univ}$ is an effective Cartier divisor and we obtain an invertible module $\mathcal{O}(D_{univ})$. If $[D] \in \underline{\Hilbfunctor}^{d + 1}_{X/k}$ denotes the $k$-rational point corresponding to the effective Cartier divisor $D \subset X$ of degree $d + 1$, then the restriction of $\mathcal{O}(D_{univ})$ to the fibre $[D] \times X$ is $\mathcal{O}_X(D)$.


constructions-remark-relative-glueing-functorial

There is a functoriality property for the constructions explained in Lemmas \ref{lemma-relative-glueing} and \ref{lemma-relative-glueing-sheaves}. Namely, suppose given two collections of data $(f_U : X_U \to U, \rho^U_V)$ and $(g_U : Y_U \to U, \sigma^U_V)$ as in Lemma \ref{lemma-relative-glueing}. Suppose for every $U \in \mathcal{B}$ given a morphism $h_U : X_U \to Y_U$ over $U$ compatible with the restrictions $\rho^U_V$ and $\sigma^U_V$. Functoriality means that this gives rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_U$, where $f : X \to S$ is obtained from the datum $(f_U : X_U \to U, \rho^U_V)$ and $g : Y \to S$ is obtained from the datum $(g_U : Y_U \to U, \sigma^U_V)$. \medskip\noindent Similarly, suppose given two collections of data $(f_U : X_U \to U, \mathcal{F}_U, \rho^U_V, \theta^U_V)$ and $(g_U : Y_U \to U, \mathcal{G}_U, \sigma^U_V, \eta^U_V)$ as in Lemma \ref{lemma-relative-glueing-sheaves}. Suppose for every $U \in \mathcal{B}$ given a morphism $h_U : X_U \to Y_U$ over $U$ compatible with the restrictions $\rho^U_V$ and $\sigma^U_V$, and a morphism $\tau_U : h_U^*\mathcal{G}_U \to \mathcal{F}_U$ compatible with the maps $\theta^U_V$ and $\eta^U_V$. Functoriality means that these give rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_U$, and a morphism $h^*\mathcal{G} \to \mathcal{F}$ restricting back to the maps $h_U$ where $(f : X \to S, \mathcal{F})$ is obtained from the datum $(f_U : X_U \to U, \mathcal{F}_U, \rho^U_V, \theta^U_V)$ and where $(g : Y \to S, \mathcal{G})$ is obtained from the datum $(g_U : Y_U \to U, \mathcal{G}_U, \sigma^U_V, \eta^U_V)$. \medskip\noindent We omit the verifications and we omit a suitable formulation of ``equivalence of categories'' between relative glueing data and relative objects.


constructions-remark-global-sections-not-isomorphism

The map from $M_0$ to the global sections of $\widetilde M$ is generally far from being an isomorphism. A trivial example is to take $S = k[x, y, z]$ with $1 = \deg(x) = \deg(y) = \deg(z)$ (or any number of variables) and to take $M = S/(x^{100}, y^{100}, z^{100})$. It is easy to see that $\widetilde M = 0$, but $M_0 = k$.


constructions-remark-not-isomorphism

In general the map constructed in Lemma \ref{lemma-widetilde-tensor} above is not an isomorphism. Here is an example. Let $k$ be a field. Let $S = k[x, y, z]$ with $k$ in degree $0$ and $\deg(x) = 1$, $\deg(y) = 2$, $\deg(z) = 3$. Let $M = S(1)$ and $N = S(2)$, see Algebra, Section \ref{algebra-section-graded} for notation. Then $M \otimes_S N = S(3)$. Note that \begin{eqnarray*} S_z & = & k[x, y, z, 1/z] \\ S_{(z)} & = & k[x^3/z, xy/z, y^3/z^2] \cong k[u, v, w]/(uw - v^3) \\ M_{(z)} & = & S_{(z)} \cdot x + S_{(z)} \cdot y^2/z \subset S_z \\ N_{(z)} & = & S_{(z)} \cdot y + S_{(z)} \cdot x^2 \subset S_z \\ S(3)_{(z)} & = & S_{(z)} \cdot z \subset S_z \end{eqnarray*} Consider the maximal ideal $\mathfrak m = (u, v, w) \subset S_{(z)}$. It is not hard to see that both $M_{(z)}/\mathfrak mM_{(z)}$ and $N_{(z)}/\mathfrak mN_{(z)}$ have dimension $2$ over $\kappa(\mathfrak m)$. But $S(3)_{(z)}/\mathfrak mS(3)_{(z)}$ has dimension $1$. Thus the map $M_{(z)} \otimes N_{(z)} \to S(3)_{(z)}$ is not an isomorphism.


constructions-remark-missing-finite-type

What's missing in the list of properties above? Well to be sure the property of being of finite type. The reason we do not list this here is that we have not yet defined the notion of finite type at this point. (Another property which is missing is ``smoothness''. And I'm sure there are many more you can think of.)


constructions-remark-not-in-invertible-locus

Assumptions as in Lemma \ref{lemma-invertible-map-into-proj} above. The image of the morphism $r_{\mathcal{L}, \psi}$ need not be contained in the locus where the sheaf $\mathcal{O}_X(1)$ is invertible. Here is an example. Let $k$ be a field. Let $S = k[A, B, C]$ graded by $\deg(A) = 1$, $\deg(B) = 2$, $\deg(C) = 3$. Set $X = \text{Proj}(S)$. Let $T = \mathbf{P}^2_k = \text{Proj}(k[X_0, X_1, X_2])$. Recall that $\mathcal{L} = \mathcal{O}_T(1)$ is invertible and that $\mathcal{O}_T(n) = \mathcal{L}^{\otimes n}$. Consider the composition $\psi$ of the maps $$ S \to k[X_0, X_1, X_2] \to \Gamma_*(T, \mathcal{L}). $$ Here the first map is $A \mapsto X_0$, $B \mapsto X_1^2$, $C \mapsto X_2^3$ and the second map is (\ref{equation-global-sections}). By the lemma this corresponds to a morphism $r_{\mathcal{L}, \psi} : T \to X = \text{Proj}(S)$ which is easily seen to be surjective. On the other hand, in Remark \ref{remark-not-isomorphism} we showed that the sheaf $\mathcal{O}_X(1)$ is not invertible at all points of $X$.


algebraization-remark-compare-with-completion

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge$ be the derived completion functor of Proposition \ref{proposition-derived-completion}. For any $n \geq 1$ the object $K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n$ is derived complete as it is annihilated by powers of local sections of $\mathcal{I}$. Hence there is a canonical factorization $$ K \to K^\wedge \to K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n $$ of the canonical map $K \to K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n$. These maps are compatible for varying $n$ and we obtain a comparison map $$ K^\wedge \longrightarrow R\lim \left(K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n\right) $$ The right hand side is more recognizable as a kind of completion. In general this comparison map is not an isomorphism.


algebraization-remark-localization-and-completion

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge$ be the derived completion functor of Proposition \ref{proposition-derived-completion}. It follows from the construction in the proof of the proposition that $K^\wedge|_U$ is the derived completion of $K|_U$ for any $U \in \Ob(\mathcal{C})$. But we can also prove this as follows. From the definition of derived complete objects it follows that $K^\wedge|_U$ is derived complete. Thus we obtain a canonical map $a : (K|_U)^\wedge \to K^\wedge|_U$. On the other hand, if $E$ is a derived complete object of $D(\mathcal{O}_U)$, then $Rj_*E$ is a derived complete object of $D(\mathcal{O})$ by Lemma \ref{lemma-pushforward-derived-complete}. Here $j$ is the localization morphism (Modules on Sites, Section \ref{sites-modules-section-localize}). Hence we also obtain a canonical map $b : K^\wedge \to Rj_*((K|_U)^\wedge)$. We omit the (formal) verification that the adjoint of $b$ is the inverse of $a$.


algebraization-remark-completed-tensor-product

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals.  Denote $K \mapsto K^\wedge$ the adjoint of Proposition \ref{proposition-derived-completion}. Then we set $$ K \otimes^\wedge_\mathcal{O} L = (K \otimes_\mathcal{O}^\mathbf{L} L)^\wedge $$ This {\it completed tensor product} defines a functor $D_{comp}(\mathcal{O}) \times D_{comp}(\mathcal{O}) \to D_{comp}(\mathcal{O})$ such that we have $$ \Hom_{D_{comp}(\mathcal{O})}(K, R\SheafHom_\mathcal{O}(L, M)) = \Hom_{D_{comp}(\mathcal{O})}(K \otimes_\mathcal{O}^\wedge L, M) $$ for $K, L, M \in D_{comp}(\mathcal{O})$. Note that $R\SheafHom_\mathcal{O}(L, M) \in D_{comp}(\mathcal{O})$ by Lemma \ref{lemma-derived-complete-internal-hom}.


algebraization-remark-local-calculation-derived-completion

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge$ be the derived completion of Proposition \ref{proposition-derived-completion}. Let $U \in \Ob(\mathcal{C})$ be an object such that $\mathcal{I}$ is generated as an ideal sheaf by $f_1, \ldots, f_r \in \mathcal{I}(U)$. Set $A = \mathcal{O}(U)$ and $I = (f_1, \ldots, f_r) \subset A$. Warning: it may not be the case that $I = \mathcal{I}(U)$. Then we have $$ R\Gamma(U, K^\wedge) = R\Gamma(U, K)^\wedge $$ where the right hand side is the derived completion of the object $R\Gamma(U, K)$ of $D(A)$ with respect to $I$. This is true because derived completion commutes with localization (Remark \ref{remark-localization-and-completion}) and Lemma \ref{lemma-formal-functions-general}.


algebraization-remark-references

Here are some references to discussions of related material the literature. It seems that a ``derived formal functions theorem'' for proper maps goes back to \cite[Theorem 6.3.1]{lurie-thesis}. There is the discussion in \cite{dag12}, especially Chapter 4 which discusses the affine story, see More on Algebra, Section \ref{more-algebra-section-derived-completion}. In \cite[Section 2.9]{G-R} one finds a discussion of proper base change and derived completion using (ind) coherent modules. An analogue of (\ref{equation-formal-functions}) for complexes of quasi-coherent modules can be found as \cite[Theorem 6.5]{HL-P}


algebraization-remark-interesting-case-variant

In Lemma \ref{lemma-algebraization-principal-variant} if $A$ is universally catenary with Cohen-Macaulay formal fibres (for example if $A$ has a dualizing complex), then the condition that $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$ are finite $A$-modules, is equivalent with $$ \text{depth}((A/f)_\mathfrak q) + \dim((A/\mathfrak q)_\mathfrak p) > 2 $$ for all $\mathfrak q \in V(f) \setminus V(\mathfrak a)$ and $\mathfrak p \in V(\mathfrak q) \cap V(\mathfrak a)$ by Local Cohomology, Theorem \ref{local-cohomology-theorem-finiteness}. \medskip\noindent For example, if $A/fA$ is $(S_2)$ and if every irreducible component of $Z = V(\mathfrak a)$ has codimension $\geq 3$ in $Y = \Spec(A/fA)$, then we get the finiteness of $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$. This should be contrasted with the slightly weaker conditions found in Lemma \ref{lemma-algebraization-principal} (see also Remark \ref{remark-interesting-case}).


algebraization-remark-discussion

Let $Y$ be a Noetherian scheme and let $Z \subset Y$ be a closed subset. By Lemma \ref{lemma-discussion} we have $$ \delta_Z(y) \leq \min \left\{ k \middle| \begin{matrix} \text{ there exist specializations in }Y \\ y_0 \leftarrow y'_0 \rightarrow y_1 \leftarrow y'_1 \rightarrow \ldots \leftarrow y'_{k - 1} \rightarrow y_k = y \\ \text{ with }y_0 \in Z\text{ and }y_i' \leadsto y_i \text{ immediate} \end{matrix} \right\} $$ We claim that if $Y$ is of finite type over a field, then equality holds. If we ever need this result we will formulate a precise result and prove it here. However, in general if we define $\delta_Z$ by the right hand side of this inequality, then we don't know if Lemma \ref{lemma-change-distance-function} remains true.


algebraization-remark-interesting-case

Let $(A, \mathfrak m)$ be a complete Noetherian normal local domain of dimension $\geq 4$ and let $f \in \mathfrak m$ be nonzero. Then assumptions (1), (2), (3), (5), and (6) of Lemma \ref{lemma-algebraization-principal} are satisfied. Thus vectorbundles on the formal completion of $U$ along $U \cap V(f)$ can be algebraized. In Lemma \ref{lemma-algebraization-principal-bis} we will generalize this to more general coherent formal modules; please also compare with Remark \ref{remark-interesting-case-bis}.


algebraization-remark-interesting-case-bis

Let $(A, \mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to $f \in \mathfrak m$. Let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, f\mathcal{O}_U)$ where $U$ is the punctured spectrum of $A$. Set $Y = V(f) \subset X = \Spec(A)$. If for $y \in U \cap V(f)$ closed in $U$, i.e., with $\dim(\overline{\{y\}}) = 1$, we assume the $\mathcal{O}_{X, y}^\wedge$-module $\mathcal{F}_y^\wedge$ satisfies the following two conditions \begin{enumerate} \item $\mathcal{F}_y^\wedge[1/f]$ is $(S_2)$ as a $\mathcal{O}_{X, y}^\wedge[1/f]$-module, and \item for $\mathfrak p \in \text{Ass}(\mathcal{F}_y^\wedge[1/f])$ we have $\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) \geq 3$. \end{enumerate} Then $(\mathcal{F}_n)$ is the completion of a coherent module on $U$. This follows from Lemmas \ref{lemma-algebraization-principal-bis} and \ref{lemma-unwinding-conditions}.


algebraization-remark-question

We are unable to prove or disprove the analogue of Proposition \ref{proposition-d-generators} where the assumption that $I$ has $d$ generators is replaced with the assumption $\text{cd}(A, I) \leq d$. If you know a proof or have a counter example, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}. Another obvious question is to what extend the conditions in Proposition \ref{proposition-d-generators} are necessary.


algebraization-remark-interesting-case-ter

In the situation of Proposition \ref{proposition-algebraization-regular-sequence} if we assume $A$ has a dualizing complex, then the condition that $H^0(U, \mathcal{F}_1)$ and $H^1(U, \mathcal{F}_1)$ are finite is equivalent to $$ \text{depth}(\mathcal{F}_{1, y}) + \dim(\mathcal{O}_{\overline{\{y\}}, z}) > 2 $$ for all $y \in U \cap Y$ and $z \in Z \cap \overline{\{y\}}$. See Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-Rjstar}. This holds for example if $\mathcal{F}_1$ is a finite locally free $\mathcal{O}_{U \cap Y}$-module, $Y$ is $(S_2)$, and $\text{codim}(Z', Y') \geq 3$ for every pair of irreducible components $Y'$ of $Y$, $Z'$ of $Z$ with $Z' \subset Y'$.


algebraization-remark-interesting-case-quater

Proposition \ref{proposition-algebraization-flat} is a local version of \cite[Theorem 2.10 (i)]{Baranovsky}. It is straightforward to deduce the global results from the local one; we will sketch the argument. Namely, suppose $(R, \mathfrak m)$ is a complete Noetherian local ring and $X \to \Spec(R)$ is a proper morphism. For $n \geq 1$ set $X_n = X \times_{\Spec(R)} \Spec(R/\mathfrak m^n)$. Let $Z \subset X_1$ be a closed subset of the special fibre. Set $U = X \setminus Z$ and denote $j : U \to X$ the inclusion morphism. Suppose given an object $$ (\mathcal{F}_n) \text{ of } \textit{Coh}(U, \mathfrak m\mathcal{O}_U) $$ which is flat over $R$ in the sense that $\mathcal{F}_n$ is flat over $R/\mathfrak m^n$ for all $n$. Assume that $j_*\mathcal{F}_1$ and $R^1j_*\mathcal{F}_1$ are coherent modules. Then affine locally on $X$ we get a canonical extension of $(\mathcal{F}_n)$ by Proposition \ref{proposition-algebraization-flat} and formation of this extension commutes with localization (by Lemma \ref{lemma-algebraization-principal-variant}). Thus we get a canonical global object $(\mathcal{G}_n)$ of $\textit{Coh}(X, \mathfrak m\mathcal{O}_X)$ whose restriction of $U$ is $(\mathcal{F}_n)$. By Grothendieck's existence theorem (Cohomology of Schemes, Proposition \ref{coherent-proposition-existence-proper}) we see there exists a coherent $\mathcal{O}_X$-module $\mathcal{G}$ whose completion is $(\mathcal{G}_n)$. In this way we see that $(\mathcal{F}_n)$ is algebraizable, i.e., it is the completion of a coherent $\mathcal{O}_U$-module. \medskip\noindent We add that the coherence of $j_*\mathcal{F}_1$ and $R^1j_*\mathcal{F}_1$ is a condition on the special fibre. Namely, if we denote $j_1 : U_1 \to X_1$ the special fibre of $j : U \to X$, then we can think of $\mathcal{F}_1$ as a coherent sheaf on $U_1$ and we have $j_*\mathcal{F}_1 = j_{1, *}\mathcal{F}_1$ and $R^1j_*\mathcal{F}_1 = R^1j_{1, *}\mathcal{F}_1$. Hence for example if $X_1$ is $(S_2)$ and irreducible, we have $\dim(X_1) - \dim(Z) \geq 3$, and $\mathcal{F}_1$ is a locally free $\mathcal{O}_{U_1}$-module, then $j_{1, *}\mathcal{F}_1$ and $R^1j_{1, *}\mathcal{F}_1$ are coherent modules.


algebraization-remark-compare-SGA2

In SGA2 we find the following result. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume $A$ is a quotient of a regular ring, the element $f$ is a nonzerodivisor, and \begin{enumerate} \item[(a)] if $\mathfrak p \subset A$ is a prime ideal with $\dim(A/\mathfrak p) = 1$, then $\text{depth}(A_\mathfrak p) \geq 2$, and \item[(b)] $\text{depth}(A/fA) \geq 3$, or equivalently $\text{depth}(A) \geq 4$. \end{enumerate} Let $U$, resp.\ $U_0$ be the punctured spectrum of $A$, resp.\ $A/fA$. Then the map $$ \Pic(U) \to \Pic(U_0) $$ is injective. This is \cite[Exposee XI, Lemma 3.16]{SGA2}\footnote{Condition (a) follows from condition (b), see Algebra, Lemma \ref{algebra-lemma-depth-localization}.}. This result from SGA2 follows from Proposition \ref{proposition-injective-pic} because \begin{enumerate} \item a quotient of a regular ring has a dualizing complex (see Dualizing Complexes, Lemma \ref{dualizing-lemma-regular-gorenstein} and Proposition \ref{dualizing-proposition-dualizing-essentially-finite-type}), and \item if $\text{depth}(A) \geq 4$ then $\text{depth}(A_\mathfrak p) \geq 2$ for all primes $\mathfrak p$ with $\dim(A/\mathfrak p) = 2$, see Algebra, Lemma \ref{algebra-lemma-depth-localization}. \end{enumerate}


algebraization-remark-surjective-Pic-second

Let $(A, \mathfrak m)$ be a Noetherian local ring and $f \in \mathfrak m$. The conclusion of Lemma \ref{lemma-surjective-Pic-first} holds if we assume \begin{enumerate} \item $A$ has a dualizing complex, \item $A$ is $f$-adically complete, \item $f$ is a nonzerodivisor, \item one of the following is true \begin{enumerate} \item $A_f$ is $(S_2)$ and for $\mathfrak p \subset A$, $f \not \in \mathfrak p$ minimal we have $\dim(A/\mathfrak p) \geq 4$, or \item if $\mathfrak p \not \in V(f)$ and $V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}$, then $\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3$. \end{enumerate} \item $H^3_{\mathfrak m}(A/fA) = 0$. \end{enumerate} The proof is exactly the same as the proof of Lemma \ref{lemma-surjective-Pic-first} using Lemma \ref{lemma-equivalence-better} instead of Lemma \ref{lemma-equivalence}. Two points need to be made here: (a) it seems hard to find examples where one knows $H^3_{\mathfrak m}(A/fA) = 0$ without assuming $\text{depth}(A/fA) \geq 4$, and (b) the proof of Lemma \ref{lemma-equivalence-better} is a good deal harder than the proof of Lemma \ref{lemma-equivalence}.


spaces-over-fields-remark-alternate-proof-scheme-codim-1

Here is a sketch of a proof of Lemma \ref{lemma-codim-1-point-in-schematic-locus} which avoids using More on Groupoids, Lemma \ref{more-groupoids-lemma-find-affine-codimension-1}. \medskip\noindent Step 1. We may assume $X$ is a reduced Noetherian separated algebraic space (for example by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian} or by Limits of Spaces, Lemma \ref{spaces-limits-lemma-reduction-scheme}) and we may choose a finite surjective morphism $Y \to X$ where $Y$ is a Noetherian scheme (by Limits of Spaces, Proposition \ref{spaces-limits-proposition-there-is-a-scheme-finite-over}). \medskip\noindent Step 2. After replacing $X$ by an open neighbourhood of $x$, there exists a birational finite morphism $X' \to X$ and a closed subscheme $Y' \subset X' \times_X Y$ such that $Y' \to X'$ is surjective finite locally free. Namely, because $X$ is reduced there is a dense open subspace $U \subset X$ over which $Y$ is flat (Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-generic-flatness-reduced}). Then we can choose a $U$-admissible blowup $b : \tilde X \to X$ such that the strict transform $\tilde Y$ of $Y$ is flat over $\tilde X$, see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-flat-after-blowing-up}. (An alternative is to use Hilbert schemes if one wants to avoid using the result on blowups). Then we let $X' \subset \tilde X$ be the scheme theoretic closure of $b^{-1}(U)$ and $Y' = X' \times_{\tilde X} \tilde Y$. Since $x$ is a codimension $1$ point, we see that $X' \to X$ is finite over a neighbourhood of $x$ (Lemma \ref{lemma-finite-in-codim-1}). \medskip\noindent Step 3. After shrinking $X$ to a smaller neighbourhood of $x$ we get that $X'$ is a scheme. This holds because $Y'$ is a scheme and $Y' \to X'$ being finite locally free and because every finite set of codimension $1$ points of $Y'$ is contained in an affine open. Use Properties of Spaces, Proposition \ref{spaces-properties-proposition-finite-flat-equivalence-global} and Varieties, Proposition \ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}. \medskip\noindent Step 4. There exists an affine open $W' \subset X'$ containing all points lying over $x$ which is the inverse image of an open subspace of $X$. To prove this let $Z \subset X$ be the closure of the set of points where $X' \to X$ is not an isomorphism. We may assume $x \in Z$ otherwise we are already done. Then $x$ is a generic point of an irreducible component of $Z$ and after shrinking $X$ we may assume $Z$ is an affine scheme (Lemma \ref{lemma-generic-point-in-schematic-locus}). Then the inverse image $Z' \subset X'$ is an affine scheme as well. Say $x_1, \ldots, x_n \in Z'$ are the points mapping to $x$. Then we can find an affine open $W'$ in $X'$ whose intersection with $Z'$ is the inverse image of a principal open of $Z$ containing $x$. Namely, we first pick an affine open $W' \subset X'$ containing $x_1, \ldots, x_n$ using Varieties, Proposition \ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}. Then we pick a principal open $D(f) \subset Z$ containing $x$ whose inverse image $D(f|_{Z'})$ is contained in $W' \cap Z'$. Then we pick $f' \in \Gamma(W', \mathcal{O}_{W'})$ restricting to $f|_{Z'}$ and we replace $W'$ by $D(f') \subset W'$. Since $X' \to X$ is an isomorphism away from $Z' \to Z$ the choice of $W'$ guarantees that the image $W \subset X$ of $W'$ is open with inverse image $W'$ in $X'$. \medskip\noindent Step 5. Then $W' \to W$ is a finite surjective morphism and $W$ is a scheme by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian} and the proof is complete.