algebra-remark-cohen-bound-cardinality

Let $R$ be a ring. Let $\kappa$ be an infinite cardinal. By applying Example \ref{example-oka-family-bound-cardinality} and Proposition \ref{proposition-oka} we see that any ideal maximal with respect to the property of not being generated by $\kappa$ elements is prime. This result is not so useful because there exists a ring for which every prime ideal of $R$ can be generated by $\aleph_0$ elements, but some ideal cannot. Namely, let $k$ be a field, let $T$ be a set whose cardinality is greater than $\aleph_0$ and let $$ R = k[\{x_n\}_{n \geq 1}, \{z_{t, n}\}_{t \in T, n \geq 0}]/ (x_n^2, z_{t, n}^2, x_n z_{t, n} - z_{t, n - 1}) $$ This is a local ring with unique prime ideal $\mathfrak m = (x_n)$. But the ideal $(z_{t, n})$ cannot be generated by countably many elements.


algebra-remark-intersection-powers-ideal

Lemma \ref{lemma-intersect-powers-ideal-module-zero} in particular implies that $\bigcap_n I^n = (0)$ when $I \subset R$ is a non-unit ideal in a Noetherian local ring $R$. More generally, let $R$ be a Noetherian ring and $I \subset R$ an ideal. Suppose that $f \in \bigcap_{n \in \mathbf{N}} I^n$. Then Lemma \ref{lemma-intersection-powers-ideal-module} says that for every prime ideal $I \subset \mathfrak p$ there exists a $g \in R$, $g \not \in \mathfrak p$ such that $f$ maps to zero in $R_g$. In algebraic geometry we express this by saying that ``$f$ is zero in an open neighbourhood of the closed set $V(I)$ of $\Spec(R)$''.


algebra-remark-period-polynomial

If $S$ is still Noetherian but $S$ is not generated in degree $1$, then the function associated to a graded $S$-module is a periodic polynomial (i.e., it is a numerical polynomial on the congruence classes of integers modulo $n$ for some $n$).


algebra-remark-ass-reverse-functorial

Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Then it is not always the case that $\Spec(\varphi)(\text{Ass}_S(M)) \supset \text{Ass}_R(M)$. For example, consider the ring map $R = k \to S = k[x_1, x_2, x_3, \ldots]/(x_i^2)$ and $M = S$. Then $\text{Ass}_R(M)$ is not empty, but $\text{Ass}_S(S)$ is empty.


algebra-remark-bourbaki

Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Let $\mathfrak p$ be a prime of $R$. Then $$ \text{Ass}_S(N \otimes_R \kappa(\mathfrak p)) = \text{Ass}_{S/\mathfrak pS}(N \otimes_R \kappa(\mathfrak p)) = \text{Ass}_{S \otimes_R \kappa(\mathfrak p)}(N \otimes_R \kappa(\mathfrak p)). $$ The first equality by Lemma \ref{lemma-ass-quotient-ring} and the second by Lemma \ref{lemma-localize-ass} part (1).


algebra-remark-weakly-ass-not-functorial

Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Then it is not always the case that $\Spec(\varphi)(\text{WeakAss}_S(M)) \subset \text{WeakAss}_R(M)$ contrary to the case of associated primes (see Lemma \ref{lemma-ass-functorial}). An example is to consider the ring map $$ R = k[x_1, x_2, x_3, \ldots] \to S = k[x_1, x_2, x_3, \ldots, y_1, y_2, y_3, \ldots]/ (x_1y_1, x_2y_2, x_3y_3, \ldots) $$ and $M = S$. In this case $\mathfrak q = \sum x_iS$ is a minimal prime of $S$, hence a weakly associated prime of $M = S$ (see Lemma \ref{lemma-weakly-ass-minimal-prime-support}). But on the other hand, for any nonzero element of $S$ the annihilator in $R$ is finitely generated, and hence does not have radical equal to $R \cap \mathfrak q = (x_1, x_2, x_3, \ldots)$ (details omitted).


algebra-remark-ass-functorial

Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Denote $f : \Spec(S) \to \Spec(R)$ the associated map on spectra. Then we have $$ f(\text{Ass}_S(M)) \subset \text{Ass}_R(M) \subset \text{WeakAss}_R(M) \subset f(\text{WeakAss}_S(M)) $$ see Lemmas \ref{lemma-ass-functorial}, \ref{lemma-weakly-ass-reverse-functorial}, and \ref{lemma-weakly-ass-support}. In general all of the inclusions may be strict, see Remarks \ref{remark-ass-reverse-functorial} and \ref{remark-weakly-ass-not-functorial}. If $S$ is Noetherian, then all the inclusions are equalities as the outer two are equal by Lemma \ref{lemma-ass-weakly-ass}.


algebra-remark-koszul-regular

In the paper \cite{Kabele} the author introduces two more regularity conditions for sequences $x_1, \ldots, x_r$ of elements of a ring $R$. Namely, we say the sequence is {\it Koszul-regular} if $H_i(K_{\bullet}(R, x_{\bullet})) = 0$ for $i \geq 1$ where $K_{\bullet}(R, x_{\bullet})$ is the Koszul complex. The sequence is called {\it $H_1$-regular} if $H_1(K_{\bullet}(R, x_{\bullet})) = 0$. If $R$ is a local ring (possibly non-Noetherian) and the sequence consists of elements of the maximal ideal, then one has the implications regular $\Rightarrow$ Koszul-regular $\Rightarrow$ $H_1$-regular $\Rightarrow$ quasi-regular. By examples the author shows that these implications cannot be reversed in general. We introduce these notions in more detail in More on Algebra, Section \ref{more-algebra-section-koszul-regular}.


algebra-remark-join-quasi-regular-sequences

Let $k$ be a field. Consider the ring $$ A = k[x, y, w, z_0, z_1, z_2, \ldots]/ (y^2z_0 - wx, z_0 - yz_1, z_1 - yz_2, \ldots) $$ In this ring $x$ is a nonzerodivisor and the image of $y$ in $A/xA$ gives a quasi-regular sequence. But it is not true that $x, y$ is a quasi-regular sequence in $A$ because $(x, y)/(x, y)^2$ isn't free of rank two over $A/(x, y)$ due to the fact that $wx = 0$ in $(x, y)/(x, y)^2$ but $w$ isn't zero in $A/(x, y)$. Hence the analogue of Lemma \ref{lemma-join-regular-sequences} does not hold for quasi-regular sequences.


algebra-remark-signs-double-complex

The isomorphism constructed above is the ``correct'' one only up to signs. A good part of homological algebra is concerned with choosing signs for various maps and showing commutativity of diagrams with intervention of suitable signs. For the moment we will simply use the isomorphism as given in the proof above, and worry about signs later.


algebra-remark-curiosity-signs-swap

An interesting case occurs when $M = N$ in the above. In this case we get a canonical map $\text{Tor}_i^R(M, M) \to \text{Tor}_i^R(M, M)$. Note that this map is not the identity, because even when $i = 0$ this map is not the identity! For example, if $V$ is a vector space of dimension $n$ over a field, then the switch map $V \otimes_k V \to V \otimes_k V$ has $(n^2 + n)/2$ eigenvalues $+1$ and $(n^2-n)/2$ eigenvalues $-1$. In characteristic $2$ it is not even diagonalizable. Note that even changing the sign of the map will not get rid of this.


algebra-remark-Tor-ring-mod-ideal

The proof of Lemma \ref{lemma-characterize-flat} actually shows that $$ \text{Tor}_1^R(M, R/I) = \Ker(I \otimes_R M \to M). $$


algebra-remark-warning

It is not true that a finite $R$-module which is $R$-flat is automatically projective. A counter example is where $R = \mathcal{C}^\infty(\mathbf{R})$ is the ring of infinitely differentiable functions on $\mathbf{R}$, and $M = R_{\mathfrak m} = R/I$ where $\mathfrak m = \{f \in R \mid f(0) = 0\}$ and $I = \{f \in R \mid \exists \epsilon, \epsilon > 0 : f(x) = 0\ \forall x, |x| < \epsilon\}$.


algebra-remark-flat-ML

Let $M$ be a flat $R$-module. By Lazard's theorem (Theorem \ref{theorem-lazard}) we can write $M = \colim M_i$ as the colimit of a directed system $(M_i, f_{ij})$ where the $M_i$ are free finite $R$-modules. For $M$ to be Mittag-Leffler, it is enough for the inverse system of duals $(\Hom_R(M_i, R), \Hom_R(f_{ij}, R))$ to be Mittag-Leffler. This follows from criterion (4) of Proposition \ref{proposition-ML-characterization} and the fact that for a free finite $R$-module $F$, there is a functorial isomorphism $\Hom_R(F, R) \otimes_R N \cong \Hom_R(F, N)$ for any $R$-module $N$.


algebra-remark-go-up-ML-modules

Let $R \to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module which is Mittag-Leffler as an $R$-module. Then it is in general not the case that $M$ is Mittag-Leffler as an $S$-module. For example suppose that $S$ is the ring of dual numbers over $R$, i.e., $S = R \oplus R\epsilon$ with $\epsilon^2 = 0$. Then an $S$-module consists of an $R$-module $M$ endowed with a square zero $R$-linear endomorphism $\epsilon : M \to M$. Now suppose that $M_0$ is an $R$-module which is not Mittag-Leffler. Choose a presentation $F_1 \xrightarrow{u} F_0 \to M_0 \to 0$ with $F_1$ and $F_0$ free $R$-modules. Set $M = F_1 \oplus F_0$ with $$ \epsilon = \left( \begin{matrix} 0 & 0 \\ u & 0 \end{matrix} \right) : M \longrightarrow M. $$ Then $M/\epsilon M \cong F_1 \oplus M_0$ is not Mittag-Leffler over $R = S/\epsilon S$, hence not Mittag-Leffler over $S$ (see Lemma \ref{lemma-mod-ideal-ML-modules}). On the other hand, $M/\epsilon M = M \otimes_S S/\epsilon S$ which would be Mittag-Leffler over $S$ if $M$ was, see Lemma \ref{lemma-tensor-ML-modules}.


algebra-remark-characterize-projective

Lemma \ref{lemma-countgen-projective} does not hold without the countable generation assumption.  For example, the $\mathbf Z$-module $M = \mathbf{Z}[[x]]$ is flat and Mittag-Leffler but not projective.  It is Mittag-Leffler by Lemma \ref{lemma-power-series-ML}.  Subgroups of free abelian groups are free, hence a projective $\mathbf Z$-module is in fact free and so are its submodules.  Thus to show $M$ is not projective it suffices to produce a non-free submodule.  Fix a prime $p$ and consider the submodule $N$ consisting of power series $f(x) = \sum a_i x^i$ such that for every integer $m \geq 1$, $p^m$ divides $a_i$ for all but finitely many $i$.  Then $\sum a_i p^i x^i$ is in $N$ for all $a_i \in \mathbf{Z}$, so $N$ is uncountable.  Thus if $N$ were free it would have uncountable rank and the dimension of $N/pN$ over $\mathbf{Z}/p$ would be uncountable.  This is not true as the elements $x^i \in N/pN$ for $i \geq 0$ span $N/pN$.


algebra-remark-matrices-associated-to-elements-epicenter

Let $R \to S$ be a ring map. Sometimes the set of elements $g \in S$ such that $g \otimes 1 = 1 \otimes g$ is called the {\it epicenter} of $S$. It is an $R$-algebra. By the construction of Lemma \ref{lemma-kernel-difference-projections} we get for each $g$ in the epicenter a matrix factorization $$ (g) = Y X Z $$ with $X \in \text{Mat}(n \times n, R)$, $Y \in \text{Mat}(1 \times n, S)$, and $Z \in \text{Mat}(n \times 1, S)$. Namely, let $x_{i, j}, y_i, z_j$ be as in part (2) of the lemma. Set $X = (x_{i, j})$, let $y$ be the row vector whose entries are the $y_i$ and let $z$ be the column vector whose entries are the $z_j$. With this notation conditions (b) and (c) of Lemma \ref{lemma-kernel-difference-projections} mean exactly that $Y X \in \text{Mat}(1 \times n, R)$, $X Z \in \text{Mat}(n \times 1, R)$. It turns out to be very convenient to consider the triple of matrices $(X, YX, XZ)$. Given $n \in \mathbf{N}$ and a triple $(P, U, V)$ we say that $(P, U, V)$ is a {\it $n$-triple associated to $g$} if there exists a matrix factorization as above such that $P = X$, $U = YX$ and $V = XZ$.


algebra-remark-resolution-dim-1

Suppose that $R$ is a $1$-dimensional semi-local Noetherian domain. If there is a maximal ideal $\mathfrak m \subset R$ such that $R_{\mathfrak m}$ is not regular, then we may apply Lemma \ref{lemma-nonregular-dimension-one} to $(R, \mathfrak m)$ to get a finite ring extension $R \subset R_1$. (For example one can do this so that $\Spec(R_1) \to \Spec(R)$ is the blowup of $\Spec(R)$ in the ideal $\mathfrak m$.) Of course $R_1$ is a $1$-dimensional semi-local Noetherian domain with the same fraction field as $R$. If $R_1$ is not a regular semi-local ring, then we may repeat the construction to get $R_1 \subset R_2$. Thus we get a sequence $$ R \subset R_1 \subset R_2 \subset R_3 \subset \ldots $$ of finite ring extensions which may stop if $R_n$ is regular for some $n$. Resolution of singularities would be the claim that eventually $R_n$ is indeed regular. In reality this is not the case. Namely, there exists a characteristic $0$ Noetherian local domain $A$ of dimension $1$ whose completion is nonreduced, see \cite[Proposition 3.1]{Ferrand-Raynaud} or our Examples, Section \ref{examples-section-local-completion-nonreduced}. For an example in characteristic $p > 0$ see Example \ref{example-bad-dvr-char-p}. Since the construction of blowing up commutes with completion it is easy to see the sequence never stabilizes. See \cite{Bennett} for a discussion (mostly in positive characteristic). On the other hand, if the completion of $R$ in all of its maximal ideals is reduced, then the procedure stops (insert future reference here).


algebra-remark-suitable-systems-limits

Suppose that $R \to S$ is a local homomorphism of local rings, which is essentially of finite presentation. Take any system $(\Lambda, \leq)$, $R_\lambda \to S_\lambda$ with the properties listed in Lemma \ref{lemma-limit-essentially-finite-type}. What may happen is that this is the ``wrong'' system, namely, it may happen that property (4) of Lemma \ref{lemma-limit-essentially-finite-presentation} is not satisfied. Here is an example. Let $k$ be a field. Consider the ring $$ R = k[[z, y_1, y_2, \ldots]]/(y_i^2 - zy_{i + 1}). $$ Set $S = R/zR$. As system take $\Lambda = \mathbf{N}$ and $R_n = k[[z, y_1, \ldots, y_n]]/(\{y_i^2 - zy_{i + 1}\}_{i \leq n-1})$ and $S_n = R_n/(z, y_n^2)$. All the maps $S_n \otimes_{R_n} R_{n + 1} \to S_{n + 1}$ are not localizations (i.e., isomorphisms in this case) since $1 \otimes y_{n + 1}^2$ maps to zero. If we take instead $S_n' = R_n/zR_n$ then the maps $S'_n \otimes_{R_n} R_{n + 1} \to S'_{n + 1}$ are isomorphisms. The moral of this remark is that we do have to be a little careful in choosing the systems.


algebra-remark-functoriality-principal-parts

Suppose given a commutative diagram of rings $$ \xymatrix{ B \ar[r] & B' \\ A \ar[u] \ar[r] & A' \ar[u] } $$ a $B$-module $M$, a $B'$-module $M'$, and a $B$-linear map $M \to M'$. Then we get a compatible system of module maps $$ \xymatrix{ \ldots \ar[r] & P^2_{B'/A'}(M') \ar[r] & P^1_{B'/A'}(M') \ar[r] & P^0_{B'/A'}(M') \\ \ldots \ar[r] & P^2_{B/A}(M) \ar[r] \ar[u] & P^1_{B/A}(M) \ar[r] \ar[u] & P^0_{B/A}(M) \ar[u] } $$ These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules $P^k_{B/A}(M)$ in terms of generators and relations in the proof of Lemma \ref{lemma-module-principal-parts} but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma \ref{lemma-sequence-of-principal-parts}.


algebra-remark-composition-homotopy-equivalent-to-zero

Let $A \to B$ and $\phi : B \to C$ be ring maps. Then the composition $\NL_{B/A} \to \NL_{C/A} \to \NL_{C/B}$ is homotopy equivalent to zero. Namely, this composition is the functoriality of the naive cotangent complex for the square $$ \xymatrix{ B \ar[r]_\phi & C \\ A \ar[r] \ar[u] & B \ar[u] } $$ Write $J = \Ker(B[C] \to C)$. An explicit homotopy is given by the map $\Omega_{A[B]/A} \otimes_A B \to J/J^2$ which maps the basis element $\text{d}[b]$ to the class of $[\phi(b)] - b$ in $J/J^2$.


algebra-remark-lemma-characterize-formally-smooth

Lemma \ref{lemma-characterize-formally-smooth} holds more generally whenever $P$ is formally smooth over $R$.


algebra-remark-construct-sh-from-h

We can also construct $R^{sh}$ from $R^h$. Namely, for any finite separable subextension $\kappa \subset \kappa' \subset \kappa^{sep}$ there exists a unique (up to unique isomorphism) finite \'etale local ring extension $R^h \subset R^h(\kappa')$ whose residue field extension reproduces the given extension, see Lemma \ref{lemma-henselian-cat-finite-etale}. Hence we can set $$ R^{sh} = \bigcup\nolimits_{\kappa \subset \kappa' \subset \kappa^{sep}} R^h(\kappa') $$ The arrows in this system, compatible with the arrows on the level of residue fields, exist by Lemma \ref{lemma-henselian-cat-finite-etale}. This will produce a henselian local ring by Lemma \ref{lemma-colimit-henselian} since each of the rings $R^h(\kappa')$ is henselian by Lemma \ref{lemma-finite-over-henselian}. By construction the residue field extension induced by $R^h \to R^{sh}$ is the field extension $\kappa \subset \kappa^{sep}$. Hence $R^{sh}$ so constructed is strictly henselian. By Lemma \ref{lemma-composition-colimit-etale} the $R$-algebra $R^{sh}$ is a colimit of \'etale $R$-algebras. Hence the uniqueness of Lemma \ref{lemma-uniqueness-henselian} shows that $R^{sh}$ is the strict henselization.


algebra-remark-Noetherian-complete-local-ring-universally-catenary

If $k$ is a field then the power series ring $k[[X_1, \ldots, X_d]]$ is a Noetherian complete local regular ring of dimension $d$. If $\Lambda$ is a Cohen ring then $\Lambda[[X_1, \ldots, X_d]]$ is a complete local Noetherian regular ring of dimension $d + 1$. Hence the Cohen structure theorem implies that any Noetherian complete local ring is a quotient of a regular local ring. In particular we see that a Noetherian complete local ring is universally catenary, see Lemma \ref{lemma-CM-ring-catenary} and Lemma \ref{lemma-regular-ring-CM}.


algebra-remark-universally-catenary-does-not-descend

The property of being ``universally catenary'' does not descend; not even along \'etale ring maps. In Examples, Section \ref{examples-section-non-catenary-Noetherian-local} there is a construction of a finite ring map $A \to B$ with $A$ local Noetherian and not universally catenary, $B$ semi-local with two maximal ideals $\mathfrak m$, $\mathfrak n$ with $B_{\mathfrak m}$ and $B_{\mathfrak n}$ regular of dimension $2$ and $1$ respectively, and the same residue fields as that of $A$. Moreover, $\mathfrak m_A$ generates the maximal ideal in both $B_{\mathfrak m}$ and $B_{\mathfrak n}$ (so $A \to B$ is unramified as well as finite). By Lemma \ref{lemma-etale-makes-unramified-closed} there exists a local \'etale ring map $A \to A'$ such that $B \otimes_A A' = B_1 \times B_2$ decomposes with $A' \to B_i$ surjective. This shows that $A'$ has two minimal primes $\mathfrak q_i$ with $A'/\mathfrak q_i \cong B_i$. Since $B_i$ is regular local (since it is \'etale over either $B_{\mathfrak m}$ or $B_{\mathfrak n}$) we conclude that $A'$ is universally catenary.


moduli-curves-remark-boundedness-aut-does-not-work-surfaces

The boundedness argument in the proof of Lemma \ref{lemma-curves-diagonal-separated-fp} does not work for moduli of surfaces and in fact, the result is wrong, for example because K3 surfaces over fields can have infinite discrete automorphism groups. The ``reason'' the argument does not work is that on a projective surface $S$ over a field, given ample invertible sheaves $\mathcal{N}$ and $\mathcal{L}$ with Hilbert polynomials $Q$ and $P$, there is no a priori bound on the Hilbert polynomial of $\mathcal{N} \otimes_{\mathcal{O}_S} \mathcal{L}$. In terms of intersection theory, if $H_1$, $H_2$ are ample effective Cartier divisors on $S$, then there is no (upper) bound on the intersection number $H_1 \cdot H_2$ in terms of $H_1 \cdot H_1$ and $H_2 \cdot H_2$.


dpa-remark-forgetful

The forgetful functor $(A, I, \gamma) \mapsto A$ does not commute with colimits. For example, let $$ \xymatrix{ (B, J, \delta) \ar[r] & (B'', J'', \delta'') \\ (A, I, \gamma) \ar[r] \ar[u] & (B', J', \delta') \ar[u] } $$ be a pushout in the category of divided power rings. Then in general the map $B \otimes_A B' \to B''$ isn't an isomorphism. (It is always surjective.) An explicit example is given by $(A, I, \gamma) = (\mathbf{Z}, (0), \emptyset)$, $(B, J, \delta) = (\mathbf{Z}/4\mathbf{Z}, 2\mathbf{Z}/4\mathbf{Z}, \delta)$, and $(B', J', \delta') = (\mathbf{Z}/4\mathbf{Z}, 2\mathbf{Z}/4\mathbf{Z}, \delta')$ where $\delta_2(2) = 2$ and $\delta'_2(2) = 0$ and all higher divided powers equal to zero. Then $(B'', J'', \delta'') = (\mathbf{F}_2, (0), \emptyset)$ which doesn't agree with the tensor product. However, note that it is always true that $$ B''/J'' = B/J \otimes_{A/I} B'/J' $$ as can be seen from the universal property of the pushout by considering maps into divided power algebras of the form $(C, (0), \emptyset)$.


dpa-remark-divided-power-polynomial-algebra

Let $(A, I, \gamma)$ be a divided power ring. There is a variant of Lemma \ref{lemma-divided-power-polynomial-algebra} for infinitely many variables. First note that if $s < t$ then there is a canonical map $$ A\langle x_1, \ldots, x_s \rangle \to A\langle x_1, \ldots, x_t\rangle $$ Hence if $W$ is any set, then we set $$ A\langle x_w: w \in W\rangle = \colim_{E \subset W} A\langle x_e:e \in E\rangle $$ (colimit over $E$ finite subset of $W$) with transition maps as above. By the definition of a colimit we see that the universal mapping property of $A\langle x_w: w \in W\rangle$ is completely analogous to the mapping property stated in Lemma \ref{lemma-divided-power-polynomial-algebra}.


dpa-remark-adjoining-set-of-variables

We can also adjoin a set (possibly infinite) of exterior or divided power generators in a given degree $d > 0$, rather than just one as in Examples \ref{example-adjoining-odd} and \ref{example-adjoining-even}. Namely,  following Remark \ref{remark-divided-power-polynomial-algebra}: for $(A,\gamma)$ as above and a set $J$, let $A\langle T_j:j\in J\rangle$ be the directed colimit of the algebras $A\langle T_j:j\in S\rangle$ over all finite subsets $S$ of $J$. It is immediate that this algebra has a unique divided power structure, compatible with the given structure on $A$ and on each generator $T_j$.


dpa-remark-no-good-ci-map

It appears difficult to define an good notion of ``local complete intersection homomorphisms'' for maps between general Noetherian rings. The reason is that, for a local Noetherian ring $A$, the fibres of $A \to A^\wedge$ are not local complete intersection rings. Thus, if $A \to B$ is a local homomorphism of local Noetherian rings, and the map of completions $A^\wedge \to B^\wedge$ is a complete intersection homomorphism in the sense defined above, then $(A_\mathfrak p)^\wedge \to (B_\mathfrak q)^\wedge$ is in general {\bf not} a complete intersection homomorphism in the sense defined above. A solution can be had by working exclusively with excellent Noetherian rings. More generally, one could work with those Noetherian rings whose formal fibres are complete intersections, see \cite{Rodicio-ci}. We will develop this theory in Dualizing Complexes, Section \ref{dualizing-section-formal-fibres}.


moduli-remark-quot-numerical

Let $f : X \to B$ and $\mathcal{F}$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_i \in D(\mathcal{O}_X)$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that we have a morphism $$ \Quotfunctor_{\mathcal{F}/X/B} \longrightarrow \Cohstack_{X/B} $$ which sends the element $\mathcal{F}_T \to \mathcal{Q}$ of $\Quotfunctor_{\mathcal{F}/X/B}(T)$ to the object $\mathcal{Q}$ of $\Cohstack_{X/B}$ over $T$, see proof of Quot, Proposition \ref{quot-proposition-quot}. Hence we can form the fibre product diagram $$ \xymatrix{ \Quotfunctor^P_{\mathcal{F}/X/B} \ar[r] \ar[d] & \Cohstack^P_{X/B} \ar[d] \\ \Quotfunctor_{\mathcal{F}/X/B} \ar[r] & \Cohstack_{X/B} } $$ This is the defining diagram for the algebraic space in the upper left corner. The left vertical arrow is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \mathbf{Z}$ and $E_i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_X$-module $\mathcal{L}$ (in the last case we sometimes use the notation $\Quotfunctor^{P, \mathcal{L}}_{\mathcal{F}/X/B}$). See Situation \ref{situation-numerical} and Lemmas \ref{lemma-open-P} and \ref{lemma-finite-list-perfect-objects} and Example \ref{example-hilbert-polynomial}.


moduli-remark-hilb-numerical

Let $f : X \to B$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_i \in D(\mathcal{O}_X)$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that $\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}$, see Quot, Lemma \ref{quot-lemma-hilb-is-quot}. Thus we can define $$ \Hilbfunctor^P_{X/B} = \Quotfunctor^P_{\mathcal{O}_X/X/B} $$ where $\Quotfunctor^P_{\mathcal{O}_X/X/B}$ is as in Remark \ref{remark-quot-numerical}. The morphism $$ \Hilbfunctor^P_{X/B} \longrightarrow \Hilbfunctor_{X/B} $$ is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \mathbf{Z}$ and $E_i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_X$-module $\mathcal{L}$. In the last case we sometimes use the notation $\Hilbfunctor^{P, \mathcal{L}}_{X/B}$.


moduli-remark-Mor-numerical

Let $B, X, Y$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_i \in D(\mathcal{O}_{Y \times_B X})$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that $$ \mathit{Mor}_B(Y, X) \subset \Hilbfunctor_{Y \times_B X/B} $$ is an open subspace, see Quot, Lemma \ref{quot-lemma-Mor-into-Hilb-open}. Thus we can define $$ \mathit{Mor}^P_B(Y, X) = \mathit{Mor}_B(Y, X) \cap \Hilbfunctor^P_{Y \times_B X/B} $$ where $\Hilbfunctor^P_{Y \times_B X/B}$ is as in Remark \ref{remark-hilb-numerical}. The morphism $$ \mathit{Mor}^P_B(Y, X) \longrightarrow \mathit{Mor}_B(Y, X) $$ is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \mathbf{Z}$, $E_i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_{Y \times_B X}$-module $\mathcal{L}$. In the last case we sometimes use the notation $\mathit{Mor}^{P, \mathcal{L}}_B(Y, X)$.


derived-remark-special-triangles

Let $\mathcal{D}$ be an additive category with translation functors $[n]$ as in Definition \ref{definition-triangle}. Let us call a triangle $(X, Y, Z, f, g, h)$ {\it special}\footnote{This is nonstandard notation.} if for every object $W$ of $\mathcal{D}$ the long sequence of abelian groups $$ \ldots \to \Hom_\mathcal{D}(W, X) \to \Hom_\mathcal{D}(W, Y) \to \Hom_\mathcal{D}(W, Z) \to \Hom_\mathcal{D}(W, X[1]) \to \ldots $$ is exact.  The proof of Lemma \ref{lemma-third-isomorphism-triangle} shows that if $$ (a, b, c) : (X, Y, Z, f, g, h) \to (X', Y', Z', f', g', h') $$ is a morphism of special triangles and if two among $a, b, c$ are isomorphisms so is the third. There is a dual statement for {\it co-special} triangles, i.e., triangles which turn into long exact sequences on applying the functor $\Hom_\mathcal{D}(-, W)$. Thus distinguished triangles are special and co-special, but in general there are many more (co-)special triangles, than there are distinguished triangles.


derived-remark-compute-modules

To see the last displayed equality in the proof above we can argue with elements as follows. We have $s\pi(l, k, k^{+}) = (l, 0, 0)$. Hence the morphism of the left hand side maps $(l, k, k^{+})$ to $(0, k, k^{+})$. On the other hand $h(l, k, k^{+}) = (0, 0, k)$ and $d(l, k, k^{+}) = (dl, dk + k^{+}, -dk^{+})$. Hence $(dh + hd)(l, k, k^{+}) = d(0, 0, k) + h(dl, dk + k^{+}, -dk^{+}) = (0, k, -dk) + (0, 0, dk + k^{+}) = (0, k, k^{+})$ as desired.


derived-remark-make-commute

Let $\mathcal{A}$ be an additive category. Let $0 \to A_i^\bullet \to B_i^\bullet \to C_i^\bullet \to 0$, $i = 1, 2$ be termwise split exact sequences. Suppose that $a : A_1^\bullet \to A_2^\bullet$, $b : B_1^\bullet \to B_2^\bullet$, and $c : C_1^\bullet \to C_2^\bullet$ are morphisms of complexes such that $$ \xymatrix{ A_1^\bullet \ar[d]_a \ar[r] & B_1^\bullet \ar[r] \ar[d]_b & C_1^\bullet \ar[d]_c \\ A_2^\bullet \ar[r] & B_2^\bullet \ar[r] & C_2^\bullet } $$ commutes in $K(\mathcal{A})$. In general, there does {\bf not} exist a morphism $b' : B_1^\bullet \to B_2^\bullet$ which is homotopic to $b$ such that the diagram above commutes in the category of complexes. Namely, consider Examples, Equation (\ref{examples-equation-commutes-up-to-homotopy}). If we could replace the middle map there by a homotopic one such that the diagram commutes, then we would have additivity of traces which we do not.


derived-remark-boundedness-conditions-triangulated

Let $\mathcal{A}$ be an additive category. Exactly the same proof as the proof of Proposition \ref{proposition-homotopy-category-triangulated} shows that the categories $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, and $K^b(\mathcal{A})$ are triangulated categories. Namely, the cone of a morphism between bounded (above, below) is bounded (above, below). But we prove below that these are triangulated subcategories of $K(\mathcal{A})$ which gives another proof.


derived-remark-homotopy-double

Let $\mathcal{A}$ be an additive category with countable direct sums. Let $\text{DoubleComp}(\mathcal{A})$ denote the category of double complexes in $\mathcal{A}$, see Homology, Section \ref{homology-section-double-complexes}. We can use this category to construct two triangulated categories. \begin{enumerate} \item We can consider an object $A^{\bullet, \bullet}$ of $\text{DoubleComp}(\mathcal{A})$ as a complex of complexes as follows $$ \ldots \to A^{\bullet, -1} \to A^{\bullet, 0} \to A^{\bullet, 1} \to \ldots $$ and take the homotopy category $K_{first}(\text{DoubleComp}(\mathcal{A}))$ with the corresponding triangulated structure given by Proposition \ref{proposition-homotopy-category-triangulated}. By Homology, Remark \ref{homology-remark-double-complex-complex-of-complexes-first} the functor $$ \text{Tot} : K_{first}(\text{DoubleComp}(\mathcal{A})) \longrightarrow K(\mathcal{A}) $$ is an exact functor of triangulated categories. \item We can consider an object $A^{\bullet, \bullet}$ of $\text{DoubleComp}(\mathcal{A})$ as a complex of complexes as follows $$ \ldots \to A^{-1, \bullet} \to A^{0, \bullet} \to A^{1, \bullet} \to \ldots $$ and take the homotopy category $K_{second}(\text{DoubleComp}(\mathcal{A}))$ with the corresponding triangulated structure given by Proposition \ref{proposition-homotopy-category-triangulated}. By Homology, Remark \ref{homology-remark-double-complex-complex-of-complexes-second} the functor $$ \text{Tot} : K_{second}(\text{DoubleComp}(\mathcal{A})) \longrightarrow K(\mathcal{A}) $$ is an exact functor of triangulated categories. \end{enumerate}


derived-remark-double-complex-as-tensor-product-of

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be additive categories and assume $\mathcal{C}$ has countable direct sums. Suppose that $$ \otimes : \mathcal{A} \times \mathcal{B} \longrightarrow \mathcal{C}, \quad (X, Y) \longmapsto X \otimes Y $$ is a functor which is bilinear on morphisms. This determines a functor $$ \text{Comp}(\mathcal{A}) \times \text{Comp}(\mathcal{B}) \longrightarrow \text{DoubleComp}(\mathcal{C}), \quad (X^\bullet, Y^\bullet) \longmapsto X^\bullet \otimes Y^\bullet $$ See Homology, Example \ref{homology-example-double-complex-as-tensor-product-of}. \begin{enumerate} \item For a fixed object $X^\bullet$ of $\text{Comp}(\mathcal{A})$ the functor $$ K(\mathcal{B}) \longrightarrow K(\mathcal{C}), \quad Y^\bullet \longmapsto \text{Tot}(X^\bullet \otimes Y^\bullet) $$ is an exact functor of triangulated categories. \item For a fixed object $Y^\bullet$ of $\text{Comp}(\mathcal{B})$ the functor $$ K(\mathcal{A}) \longrightarrow K(\mathcal{C}), \quad X^\bullet \longmapsto \text{Tot}(X^\bullet \otimes Y^\bullet) $$ is an exact functor of triangulated categories. \end{enumerate} This follows from Remark \ref{remark-homotopy-double} since the functors $\text{Comp}(\mathcal{A}) \to \text{DoubleComp}(\mathcal{C})$, $Y^\bullet \mapsto X^\bullet \otimes Y^\bullet$ and $\text{Comp}(\mathcal{B}) \to \text{DoubleComp}(\mathcal{C})$, $X^\bullet \mapsto X^\bullet \otimes Y^\bullet$ are immediately seen to be compatible with homotopies and termwise split short exact sequences and hence induce exact functors of triangulated categories $$ K(\mathcal{B}) \to K_{first}(\text{DoubleComp}(\mathcal{C})) \quad\text{and}\quad K(\mathcal{A}) \to K_{second}(\text{DoubleComp}(\mathcal{C})) $$ Observe that for the first of the two the isomorphism $$ \text{Tot}(X^\bullet \otimes Y^\bullet[1]) \cong \text{Tot}(X^\bullet \otimes Y^\bullet)[1] $$ involves signs (this goes back to the signs chosen in Homology, Remark \ref{homology-remark-shift-double-complex}).


derived-remark-existence-derived

In this chapter, we consistently work with ``small'' abelian categories (as is the convention in the Stacks project). For a ``big'' abelian category $\mathcal{A}$, it isn't clear that the derived category $D(\mathcal{A})$ exists, because it isn't clear that morphisms in the derived category are sets. In fact, in general they aren't, see Examples, Lemma \ref{examples-lemma-big-abelian-category}. However, if $\mathcal{A}$ is a Grothendieck abelian category, and given $K^\bullet, L^\bullet$ in $K(\mathcal{A})$, then by Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck} there exists a quasi-isomorphism $L^\bullet \to I^\bullet$ to a K-injective complex $I^\bullet$ and Lemma \ref{lemma-K-injective} shows that $$ \Hom_{D(\mathcal{A})}(K^\bullet, L^\bullet) = \Hom_{K(\mathcal{A})}(K^\bullet, I^\bullet) $$ which is a set. Some examples of Grothendieck abelian categories are the category of modules over a ring, or more generally the category of sheaves of modules on a ringed site.


derived-remark-truncation-distinguished-triangle

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex of $\mathcal{A}$. Let $a \in \mathbf{Z}$. We claim there is a canonical distinguished triangle $$ \tau_{\leq a}K^\bullet \to K^\bullet \to \tau_{\geq a + 1}K^\bullet \to (\tau_{\leq a}K^\bullet)[1] $$ in $D(\mathcal{A})$. Here we have used the canonical truncation functors $\tau$ from Homology, Section \ref{homology-section-truncations}. Namely, we first take the distinguished triangle associated by our $\delta$-functor (Lemma \ref{lemma-derived-canonical-delta-functor}) to the short exact sequence of complexes $$ 0 \to \tau_{\leq a}K^\bullet \to K^\bullet \to K^\bullet/\tau_{\leq a}K^\bullet \to 0 $$ Next, we use that the map $K^\bullet \to \tau_{\geq a + 1}K^\bullet$ factors through a quasi-isomorphism $K^\bullet/\tau_{\leq a}K^\bullet \to \tau_{\geq a + 1}K^\bullet$ by the description of cohomology groups in Homology, Section \ref{homology-section-truncations}. In a similar way we obtain canonical distinguished triangles $$ \tau_{\leq a}K^\bullet \to \tau_{\leq a + 1}K^\bullet \to H^{a + 1}(K^\bullet)[-a-1] \to (\tau_{\leq a}K^\bullet)[1] $$ and $$ H^a(K^\bullet)[-a] \to \tau_{\geq a}K^\bullet \to \tau_{\geq a + 1}K^\bullet \to H^a(K^\bullet)[-a + 1] $$


derived-remark-easier-proofs

Let $\mathcal{A}$ be an abelian category. Using the fact that $K(\mathcal{A})$ is a triangulated category we may use Lemma \ref{lemma-acyclic-is-zero} to obtain proofs of some of the lemmas below which are usually proved by chasing through diagrams. Namely, suppose that $\alpha : K^\bullet \to L^\bullet$ is a quasi-isomorphism of complexes. Then $$ (K^\bullet, L^\bullet, C(\alpha)^\bullet, \alpha, i, -p) $$ is a distinguished triangle in $K(\mathcal{A})$ (Lemma \ref{lemma-the-same-up-to-isomorphisms}) and $C(\alpha)^\bullet$ is an acyclic complex (Lemma \ref{lemma-acyclic}). Next, let $I^\bullet$ be a bounded below complex of injective objects. Then $$ \xymatrix{ \Hom_{K(\mathcal{A})}(C(\alpha)^\bullet, I^\bullet) \ar[r] & \Hom_{K(\mathcal{A})}(L^\bullet, I^\bullet) \ar[r] & \Hom_{K(\mathcal{A})}(K^\bullet, I^\bullet) \ar[lld] \\ \Hom_{K(\mathcal{A})}(C(\alpha)^\bullet[-1], I^\bullet) } $$ is an exact sequence of abelian groups, see Lemma \ref{lemma-representable-homological}. At this point Lemma \ref{lemma-acyclic-is-zero} guarantees that the outer two groups are zero and hence $\Hom_{K(\mathcal{A})}(L^\bullet, I^\bullet) = \Hom_{K(\mathcal{A})}(K^\bullet, I^\bullet)$.


derived-remark-easier-projective

Let $\mathcal{A}$ be an abelian category. Suppose that $\alpha : K^\bullet \to L^\bullet$ is a quasi-isomorphism of complexes. Let $P^\bullet$ be a bounded above complex of projectives. Then $$ \Hom_{K(\mathcal{A})}(P^\bullet, K^\bullet) \longrightarrow \Hom_{K(\mathcal{A})}(P^\bullet, L^\bullet) $$ is an isomorphism. This is dual to Remark \ref{remark-easier-proofs}.


derived-remark-functorial-ss

The spectral sequences of Lemma \ref{lemma-two-ss-complex-functor} are functorial in the complex $K^\bullet$. This follows from functoriality properties of Cartan-Eilenberg resolutions. On the other hand, they are both examples of a more general spectral sequence which may be associated to a filtered complex of $\mathcal{A}$. The functoriality will follow from its construction. We will return to this in the section on the filtered derived category, see Remark \ref{remark-final-functorial}.


derived-remark-big-localization

Suppose that $\mathcal{A}$ is a ``big'' abelian category with enough injectives such as the category of abelian groups. In this case we have to be slightly more careful in constructing our resolution functor since we cannot use the axiom of choice with a quantifier ranging over a class. But note that the proof of the lemma does show that any two localization functors are canonically isomorphic. Namely, given quasi-isomorphisms $i : K^\bullet \to I^\bullet$ and $i' : K^\bullet \to J^\bullet$ of a bounded below complex $K^\bullet$ into bounded below complexes of injectives there exists a unique(!) morphism $a : I^\bullet \to J^\bullet$ in $K^{+}(\mathcal{I})$ such that $i' = i \circ a$ as morphisms in $K^{+}(\mathcal{I})$. Hence the only issue is existence, and we will see how to deal with this in the next section.


derived-remark-match

Suppose $inj$ is a functor such that $s \circ inj = \text{id}$ as in part (2) of Lemma \ref{lemma-functorial-injective-resolutions}. Write $inj(K^\bullet) = (i_{K^\bullet} : K^\bullet \to j(K^\bullet))$ as in the proof of that lemma. Suppose $\alpha : K^\bullet \to L^\bullet$ is a map of bounded below complexes. Consider the map $inj(\alpha)$ in the category $\text{InjRes}(\mathcal{A})$. It induces a commutative diagram $$ \xymatrix{ K^\bullet \ar[rr]^-{\alpha} \ar[d]_{i_K} & & L^\bullet \ar[d]^{i_L} \\ j(K)^\bullet \ar[rr]^-{inj(\alpha)} & & j(L)^\bullet } $$ of morphisms of complexes. Hence, looking at the proof of Lemma \ref{lemma-resolution-functor} we see that the functor $j : K^{+}(\mathcal{A}) \to K^{+}(\mathcal{I})$ is given by the rule $$ j(\alpha\text{ up to homotopy}) = inj(\alpha)\text{ up to homotopy}\in \Hom_{K^{+}(\mathcal{I})}(j(K^\bullet), j(L^\bullet)) $$ Hence we see that $j$ matches $t \circ inj$ in this case, i.e., the diagram $$ \xymatrix{ \text{Comp}^{+}(\mathcal{A}) \ar[rr]_{t \circ inj} \ar[rd] & & K^{+}(\mathcal{I}) \\ & K^{+}(\mathcal{A}) \ar[ru]_j } $$ is commutative.


derived-remark-big-abelian-category

Let $\textit{Mod}(\mathcal{O}_X)$ be the category of $\mathcal{O}_X$-modules on a ringed space $(X, \mathcal{O}_X)$ (or more generally on a ringed site). We will see later that $\textit{Mod}(\mathcal{O}_X)$ has enough injectives and in fact functorial injective embeddings, see Injectives, Theorem \ref{injectives-theorem-sheaves-modules-injectives}. Note that the proof of Lemma \ref{lemma-into-derived-category} does not apply to $\textit{Mod}(\mathcal{O}_X)$. But the proof of Lemma \ref{lemma-functorial-injective-resolutions} does apply to $\textit{Mod}(\mathcal{O}_X)$. Thus we obtain $$ j : K^{+}(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K^{+}(\mathcal{I}) $$ which is a resolution functor where $\mathcal{I}$ is the additive category of injective $\mathcal{O}_X$-modules. This argument also works in the following cases: \begin{enumerate} \item The category $\text{Mod}_R$ of $R$-modules over a ring $R$. \item The category $\textit{PMod}(\mathcal{O})$ of presheaves of $\mathcal{O}$-modules on a site endowed with a presheaf of rings. \item The category $\textit{Mod}(\mathcal{O})$ of sheaves of $\mathcal{O}$-modules on a ringed site. \item Add more here as needed. \end{enumerate}


derived-remark-right-derived-functor

In the situation of Lemma \ref{lemma-right-derived-functor} we see that we have actually lifted the right derived functor to an exact functor $F \circ j' : D^{+}(\mathcal{A}) \to K^{+}(\mathcal{B})$. It is occasionally useful to use such a factorization.


derived-remark-filtered-localization-big

We can invert the arrow of the lemma only if $\mathcal{A}$ is a category in our sense, namely if it has a set of objects. However, suppose given a big abelian category $\mathcal{A}$ with enough injectives, such as $\textit{Mod}(\mathcal{O}_X)$ for example. Then for any given set of objects $\{A_i\}_{i\in I}$ there is an abelian subcategory $\mathcal{A}' \subset \mathcal{A}$ containing all of them and having enough injectives, see Sets, Lemma \ref{sets-lemma-abelian-injectives}. Thus we may use the lemma above for $\mathcal{A}'$. This essentially means that if we use a set worth of diagrams, etc then we will never run into trouble using the lemma.


derived-remark-final-functorial

As promised in Remark \ref{remark-functorial-ss} we discuss the connection of the lemma above with the constructions using Cartan-Eilenberg resolutions. Namely, let $T : \mathcal{A} \to \mathcal{B}$ be a left exact functor of abelian categories, assume $\mathcal{A}$ has enough injectives, and let $K^\bullet$ be a bounded below complex of $\mathcal{A}$. We give an alternative construction of the spectral sequences ${}'E$ and ${}''E$ of Lemma \ref{lemma-two-ss-complex-functor}. \medskip\noindent First spectral sequence. Consider the ``stupid'' filtration on $K^\bullet$ obtained by setting $F^p(K^\bullet) = \sigma_{\geq p}(K^\bullet)$, see Homology, Section \ref{homology-section-truncations}. Note that this stupid in the sense that $d(F^p(K^\bullet)) \subset F^{p + 1}(K^\bullet)$, compare Homology, Lemma \ref{homology-lemma-spectral-sequence-filtered-complex-d1}. Note that $\text{gr}^p(K^\bullet) = K^p[-p]$ with this filtration. According to Lemma \ref{lemma-ss-filtered-derived} there is a spectral sequence with $E_1$ term $$ E_1^{p, q} = R^{p + q}T(K^p[-p]) = R^qT(K^p) $$ as in the spectral sequence ${}'E_r$. Observe moreover that the differentials $E_1^{p, q} \to E_1^{p + 1, q}$ agree with the differentials in $'{}E_1$, see Homology, Lemma \ref{homology-lemma-spectral-sequence-filtered-complex-d1} part (2) and the description of ${}'d_1$ in the proof of Lemma \ref{lemma-two-ss-complex-functor}. \medskip\noindent Second spectral sequence. Consider the filtration on the complex $K^\bullet$ obtained by setting $F^p(K^\bullet) = \tau_{\leq -p}(K^\bullet)$, see Homology, Section \ref{homology-section-truncations}. The minus sign is necessary to get a decreasing filtration. Note that $\text{gr}^p(K^\bullet)$ is quasi-isomorphic to $H^{-p}(K^\bullet)[p]$ with this filtration. According to Lemma \ref{lemma-ss-filtered-derived} there is a spectral sequence with $E_1$ term $$ E_1^{p, q} = R^{p + q}T(H^{-p}(K^\bullet)[p]) = R^{2p + q}T(H^{-p}(K^\bullet)) = {}''E_2^{i, j} $$ with $i = 2p + q$ and $j = -p$. (This looks unnatural, but note that we could just have well developed the whole theory of filtered complexes using increasing filtrations, with the end result that this then looks natural, but the other one doesn't.) We leave it to the reader to see that the differentials match up. \medskip\noindent Actually, given a Cartan-Eilenberg resolution $K^\bullet \to I^{\bullet, \bullet}$ the induced morphism $K^\bullet \to \text{Tot}(I^{\bullet, \bullet})$ into the associated total complex will be a filtered injective resolution for either filtration using suitable filtrations on $\text{Tot}(I^{\bullet, \bullet})$. This can be used to match up the spectral sequences exactly.


derived-remark-uniqueness-derived-colimit

Let $\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be a system of objects of $\mathcal{D}$. We may think of a derived colimit as an object $K$ of $\mathcal{D}$ endowed with morphisms $i_n : K_n \to K$ such that $i_{n + 1} \circ f_n = i_n$ and such that there exists a morphism $c : K \to \bigoplus K_n$ with the property that $$ \bigoplus K_n \xrightarrow{1 - f_n} \bigoplus K_n \xrightarrow{i_n} K \xrightarrow{c} \bigoplus K_n[1] $$ is a distinguished triangle. If $(K', i'_n, c')$ is a second derived colimit, then there exists an isomorphism $\varphi : K \to K'$ such that $\varphi \circ i_n = i'_n$ and $c' \circ \varphi = c$. The existence of $\varphi$ is TR3 and the fact that $\varphi$ is an isomorphism is Lemma \ref{lemma-third-isomorphism-triangle}.


derived-remark-functoriality-derived-colimit

Let $\mathcal{D}$ be a triangulated category. Let $(a_n) : (K_n, f_n) \to (L_n, g_n)$ be a morphism of systems of objects of $\mathcal{D}$. Let $(K, i_n, c)$ be a derived colimit of the first system and let $(L, j_n, d)$ be a derived colimit of the second system with notation as in Remark \ref{remark-uniqueness-derived-colimit}. Then there exists a morphism $a : K \to L$ such that $a \circ i_n = j_n$ and $d \circ a = (a_n[1]) \circ c$. This follows from TR3 applied to the defining distinguished triangles.


derived-remark-operations-functor

Let $F : \mathcal{T} \to \mathcal{T}'$ be an exact functor of triangulated categories. Given a full subcategory $\mathcal{A}$ of $\mathcal{T}$ we denote $F(\mathcal{A})$ the full subcategory of $\mathcal{T}'$ whose objects consists of all objects $F(A)$ with $A \in \Ob(\mathcal{A})$. We have $$ F(\mathcal{A}[a, b]) = F(\mathcal{A})[a, b] $$ $$ F(smd(\mathcal{A})) \subset smd(F(\mathcal{A})), $$ $$ F(add(\mathcal{A})) \subset add(F(\mathcal{A})), $$ $$ F(\mathcal{A} \star \mathcal{B}) \subset F(\mathcal{A}) \star F(\mathcal{B}), $$ $$ F(\mathcal{A}^{\star n}) \subset F(\mathcal{A})^{\star n}. $$ We omit the trivial verifications.


derived-remark-operations-unions

Let $\mathcal{T}$ be a triangulated category. Given full subcategories $\mathcal{A}_1 \subset \mathcal{A}_2 \subset \mathcal{A}_3 \subset \ldots$ and $\mathcal{B}$ of $\mathcal{T}$ we have $$ \left(\bigcup \mathcal{A}_i\right)[a, b] = \bigcup \mathcal{A}_i[a, b] $$ $$ smd\left(\bigcup \mathcal{A}_i\right) = \bigcup smd(\mathcal{A}_i), $$ $$ add\left(\bigcup \mathcal{A}_i\right) = \bigcup add(\mathcal{A}_i), $$ $$ \left(\bigcup \mathcal{A}_i\right) \star \mathcal{B} = \bigcup \mathcal{A}_i \star \mathcal{B}, $$ $$ \mathcal{B} \star \left(\bigcup \mathcal{A}_i\right) = \bigcup \mathcal{B} \star \mathcal{A}_i, $$ $$ \left(\bigcup \mathcal{A}_i\right)^{\star n} = \bigcup \mathcal{A}_i^{\star n}. $$ We omit the trivial verifications.


derived-remark-check-on-generator

Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. Let $T$ be a property of objects of $\mathcal{D}$. Suppose that \begin{enumerate} \item if $K_i \in D(A)$, $i = 1, \ldots, r$ with $T(K_i)$ for $i = 1, \ldots, r$, then $T(\bigoplus K_i)$, \item if $K \to L \to M \to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object, \item if $T(K \oplus L)$ then $T(K)$ and $T(L)$, and \item $T(E[n])$ holds for all $n$. \end{enumerate} Then $T$ holds for all objects of $\langle E \rangle$.


cohomology-remark-daniel

Here is a different approach to the proofs of Lemmas \ref{lemma-kill-cohomology-class-on-covering} and \ref{lemma-describe-higher-direct-images} above. Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i_X : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$ be the inclusion functor and let $\#$ be the sheafification functor. Recall that $i_X$ is left exact and $\#$ is exact. \begin{enumerate} \item First prove Lemma \ref{lemma-include} below which says that the right derived functors of $i_X$ are given by $R^pi_X\mathcal{F} = \underline{H}^p(\mathcal{F})$. Here is another proof: The equality is clear for $p = 0$. Both $(R^pi_X)_{p \geq 0}$ and $(\underline{H}^p)_{p \geq 0}$ are delta functors vanishing on injectives, hence both are universal, hence they are isomorphic. See Homology, Section \ref{homology-section-cohomological-delta-functor}. \item A restatement of Lemma \ref{lemma-kill-cohomology-class-on-covering} is that $(\underline{H}^p(\mathcal{F}))^\# = 0$, $p > 0$ for any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$. To see this is true, use that ${}^\#$ is exact so $$ (\underline{H}^p(\mathcal{F}))^\# = (R^pi_X\mathcal{F})^\# = R^p(\# \circ i_X)(\mathcal{F}) = 0 $$ because $\# \circ i_X$ is the identity functor. \item Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The presheaf $V \mapsto H^p(f^{-1}V, \mathcal{F})$ is equal to $R^p (i_Y \circ f_*)\mathcal{F}$. You can prove this by noticing that both give universal delta functors as in the argument of (1) above. Hence Lemma \ref{lemma-describe-higher-direct-images} says that $R^p f_* \mathcal{F}= (R^p (i_Y \circ f_*)\mathcal{F})^\#$. Again using that $\#$ is exact a that $\# \circ i_Y$ is the identity functor we see that $$ R^p f_* \mathcal{F} = R^p(\# \circ i_Y \circ f_*)\mathcal{F} = (R^p (i_Y \circ f_*)\mathcal{F})^\# $$ as desired. \end{enumerate}


cohomology-remark-elucidate-lemma

Here is a down-to-earth explanation of the meaning of Lemma \ref{lemma-before-Leray}. It says that given $f : X \to Y$ and $\mathcal{F} \in \textit{Mod}(\mathcal{O}_X)$ and given an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ we have $$ \begin{matrix} R\Gamma(X, \mathcal{F}) & \text{is represented by} & \Gamma(X, \mathcal{I}^\bullet) \\ Rf_*\mathcal{F} & \text{is represented by} & f_*\mathcal{I}^\bullet \\ R\Gamma(Y, Rf_*\mathcal{F}) & \text{is represented by} & \Gamma(Y, f_*\mathcal{I}^\bullet) \end{matrix} $$ the last fact coming from Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) and Lemma \ref{lemma-pushforward-injective}. Finally, it combines this with the trivial observation that $$ \Gamma(X, \mathcal{I}^\bullet) = \Gamma(Y, f_*\mathcal{I}^\bullet). $$ to arrive at the commutativity of the diagram of the lemma.


cohomology-remark-Leray-ss-more-structure

The Leray spectral sequence, the way we proved it in Lemma \ref{lemma-Leray} is a spectral sequence of $\Gamma(Y, \mathcal{O}_Y)$-modules. However, it is quite easy to see that it is in fact a spectral sequence of $\Gamma(X, \mathcal{O}_X)$-modules. For example $f$ gives rise to a morphism of ringed spaces $f' :  (X, \mathcal{O}_X) \to (Y, f_*\mathcal{O}_X)$. By Lemma \ref{lemma-modules-abelian} the terms $E_r^{p, q}$ of the Leray spectral sequence for an $\mathcal{O}_X$-module $\mathcal{F}$ and $f$ are identical with those for $\mathcal{F}$ and $f'$ at least for $r \geq 2$. Namely, they both agree with the terms of the Leray spectral sequence for $\mathcal{F}$ as an abelian sheaf. And since $(f_*\mathcal{O}_X)(Y) = \mathcal{O}_X(X)$ we see the result. It is often the case that the Leray spectral sequence carries additional structure.


cohomology-remark-explain-arrow

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}$ be an $\mathcal{O}_Y$-module. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $\varphi$ be an $f$-map from $\mathcal{G}$ to $\mathcal{F}$. Choose a resolution $\mathcal{F} \to \mathcal{I}^\bullet$ by a complex of injective $\mathcal{O}_X$-modules. Choose resolutions $\mathcal{G} \to \mathcal{J}^\bullet$ and $f_*\mathcal{I}^\bullet \to (\mathcal{J}')^\bullet$ by complexes of injective $\mathcal{O}_Y$-modules. By Derived Categories, Lemma \ref{derived-lemma-morphisms-lift} there exists a map of complexes $\beta$ such that the diagram \begin{equation} \label{equation-choice} \xymatrix{ \mathcal{G} \ar[d] \ar[r] & f_*\mathcal{F} \ar[r] & f_*\mathcal{I}^\bullet \ar[d] \\ \mathcal{J}^\bullet \ar[rr]^\beta & & (\mathcal{J}')^\bullet } \end{equation} commutes. Applying global section functors we see that we get a diagram $$ \xymatrix{  & & \Gamma(Y, f_*\mathcal{I}^\bullet) \ar[d]_{qis} \ar@{=}[r] & \Gamma(X, \mathcal{I}^\bullet) \\ \Gamma(Y, \mathcal{J}^\bullet) \ar[rr]^\beta & & \Gamma(Y, (\mathcal{J}')^\bullet) & } $$ The complex on the bottom left represents $R\Gamma(Y, \mathcal{G})$ and the complex on the top right represents $R\Gamma(X, \mathcal{F})$. The vertical arrow is a quasi-isomorphism by Lemma \ref{lemma-before-Leray} which becomes invertible after applying the localization functor $K^{+}(\mathcal{O}_Y(Y)) \to D^{+}(\mathcal{O}_Y(Y))$. The arrow (\ref{equation-functorial-derived}) is given by the composition of the horizontal map by the inverse of the vertical map.


cohomology-remark-correct-version-base-change-map

The ``correct'' version of the base change map is the map $$ Lg^* Rf_* \mathcal{F}^\bullet \longrightarrow R(f')_* L(g')^*\mathcal{F}^\bullet. $$ The construction of this map involves unbounded complexes, see Remark \ref{remark-base-change}.


cohomology-remark-compared-ordered-complexes

This means that if we have two total orderings $<_1$ and $<_2$ on the index set $I$, then we get an isomorphism of complexes $\tau = \pi_2 \circ c_1 : \check{\mathcal{C}}_{ord\text{-}1}(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}_{ord\text{-}2}(\mathcal{U}, \mathcal{F})$. It is clear that $$ \tau(s)_{i_0 \ldots i_p} = \text{sign}(\sigma) s_{i_{\sigma(0)} \ldots i_{\sigma(p)}} $$ where $i_0 <_1 i_1 <_1 \ldots <_1 i_p$ and $i_{\sigma(0)} <_2 i_{\sigma(1)} <_2 \ldots <_2 i_{\sigma(p)}$. This is the sense in which the ordered {\v C}ech complex is independent of the chosen total ordering.


cohomology-remark-locally-finite-sections

Let $X = \bigcup_{i \in I} U_i$ be a locally finite open covering. Denote $j_i : U_i \to X$ the inclusion map. Suppose that for each $i$ we are given an abelian sheaf $\mathcal{F}_i$ on $U_i$. Consider the abelian sheaf $\mathcal{G} = \bigoplus_{i \in I} (j_i)_*\mathcal{F}_i$. Then for $V \subset X$ open we actually have $$ \Gamma(V, \mathcal{G}) = \prod\nolimits_{i \in I} \mathcal{F}_i(V \cap U_i). $$ In other words we have $$ \bigoplus\nolimits_{i \in I} (j_i)_*\mathcal{F}_i = \prod\nolimits_{i \in I} (j_i)_*\mathcal{F}_i $$ This seems strange until you realize that the direct sum of a collection of sheaves is the sheafification of what you think it should be. See discussion in Modules, Section \ref{modules-section-kernels}. Thus we conclude that in this case the complex of Lemma \ref{lemma-covering-resolution} has terms $$ {\mathfrak C}^p(\mathcal{U}, \mathcal{F}) = \bigoplus\nolimits_{i_0 \ldots i_p} (j_{i_0 \ldots i_p})_* \mathcal{F}_{i_0 \ldots i_p} $$ which is sometimes useful.


cohomology-remark-shift-complex-cech-complex

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_X$-modules. Let $b$ be an integer. We claim there is a commutative diagram $$ \xymatrix{ \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))[b] \ar[r] \ar[d]_\gamma & R\Gamma(X, \mathcal{F}^\bullet)[b] \ar[d] \\ \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet[b])) \ar[r] & R\Gamma(X, \mathcal{F}^\bullet[b]) } $$ in the derived category where the map $\gamma$ is the map on complexes constructed in Homology, Remark \ref{homology-remark-shift-double-complex}. This makes sense because the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet[b])$ is clearly the same as the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)[0, b]$ introduced in Homology, Remark \ref{homology-remark-shift-double-complex}. To check that the diagram commutes, we may choose an injective resolution $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$ as in the proof of Lemma \ref{lemma-cech-complex-complex}. Chasing diagrams, we see that it suffices to check the diagram commutes when we replace $\mathcal{F}^\bullet$ by $\mathcal{I}^\bullet$. Then we consider the extended diagram $$ \xymatrix{ \Gamma(X, \mathcal{I}^\bullet)[b] \ar[r] \ar[d] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))[b] \ar[r] \ar[d]_\gamma & R\Gamma(X, \mathcal{I}^\bullet)[b] \ar[d] \\ \Gamma(X, \mathcal{I}^\bullet[b]) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet[b])) \ar[r] & R\Gamma(X, \mathcal{I}^\bullet[b]) } $$ where the left horizontal arrows are (\ref{equation-global-sections-to-cech}). Since in this case the horizonal arrows are isomorphisms in the derived category (see proof of Lemma \ref{lemma-cech-complex-complex}) it suffices to show that the left square commutes. This is true because the map $\gamma$ uses the sign $1$ on the summands $\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^{q + b})$, see formula in Homology, Remark \ref{homology-remark-shift-double-complex}.


cohomology-remark-base-change

The construction of unbounded derived functor $Lf^*$ and $Rf_*$ allows one to construct the base change map in full generality. Namely, suppose that $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ is a commutative diagram of ringed spaces. Let $K$ be an object of $D(\mathcal{O}_X)$. Then there exists a canonical base change map $$ Lg^*Rf_*K \longrightarrow R(f')_*L(g')^*K $$ in $D(\mathcal{O}_{S'})$. Namely, this map is adjoint to a map $L(f')^*Lg^*Rf_*K \to L(g')^*K$ Since $L(f')^*Lg^* = L(g')^*Lf^*$ we see this is the same as a map $L(g')^*Lf^*Rf_*K \to L(g')^*K$ which we can take to be $L(g')^*$ of the adjunction map $Lf^*Rf_*K \to K$.


cohomology-remark-compose-base-change

Consider a commutative diagram $$ \xymatrix{ X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z' \ar[r]^m & Z } $$ of ringed spaces. Then the base change maps of Remark \ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \begin{align*} Lm^* \circ R(g \circ f)_* & = Lm^* \circ Rg_* \circ Rf_* \\ & \to Rg'_* \circ Ll^* \circ Rf_* \\ & \to Rg'_* \circ Rf'_* \circ Lk^* \\ & = R(g' \circ f')_* \circ Lk^* \end{align*} is the base change map for the rectangle. We omit the verification.


cohomology-remark-compose-base-change-horizontal

Consider a commutative diagram $$ \xymatrix{ X'' \ar[r]_{g'} \ar[d]_{f''} & X' \ar[r]_g \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{h'} & Y' \ar[r]^h & Y } $$ of ringed spaces. Then the base change maps of Remark \ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \begin{align*} L(h \circ h')^* \circ Rf_* & = L(h')^* \circ Lh_* \circ Rf_* \\ & \to L(h')^* \circ Rf'_* \circ Lg^* \\ & \to Rf''_* \circ L(g')^* \circ Lg^* \\ & = Rf''_* \circ L(g \circ g')^* \end{align*} is the base change map for the rectangle. We omit the verification.


cohomology-remark-cup-product

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product $$ Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L \longrightarrow Rf_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} L) $$ in $D(\mathcal{O}_Y)$ for all $K, L$ in $D(\mathcal{O}_X)$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L) \to K \otimes_{\mathcal{O}_X}^\mathbf{L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L) = Lf^*Rf_*K \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*Rf_*L$ (Lemma \ref{lemma-pullback-tensor-product}) with the map $Lf^*Rf_*K \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*Rf_*L \to K \otimes_{\mathcal{O}_X}^\mathbf{L} L$ coming from the counit $Lf^* \circ Rf_* \to \text{id}$.


cohomology-remark-spectral-sequence-filtered-object

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a filtered complex of $\mathcal{O}_X$-modules. If $\mathcal{F}^\bullet$ is bounded from below and for each $n$ the filtration on $\mathcal{F}^n$ is finite, then there is a construction of the spectral sequence in Lemma \ref{lemma-spectral-sequence-filtered-object} avoiding Injectives, Lemma \ref{injectives-lemma-K-injective-embedding-filtration}. Namely, by Derived Categories, Lemma \ref{derived-lemma-right-resolution-by-filtered-injectives} there is a filtered quasi-isomorphism $i : \mathcal{F}^\bullet \to \mathcal{I}^\bullet$ of filtered complexes with $\mathcal{I}^\bullet$ bounded below, the filtration on $\mathcal{I}^n$ is finite for all $n$, and with each $\text{gr}^p\mathcal{I}^n$ an injective $\mathcal{O}_X$-module. Then we take the spectral sequence associated to $$ \Gamma(X, \mathcal{I}^\bullet) \quad\text{with}\quad F^p\Gamma(X, \mathcal{I}^\bullet) = \Gamma(X, F^p\mathcal{I}^\bullet) $$ Since cohomology can be computed by evaluating on bounded below complexes of injectives we see that the $E_1$ page is as stated in the lemma. The convergence and boundedness under the stated conditions follows from Homology, Lemma \ref{homology-lemma-biregular-ss-converges}. In fact, this is a special case of the spectral sequence in Derived Categories, Lemma \ref{derived-lemma-ss-filtered-derived}.


cohomology-remark-cup-with-element-map-total-cohomology

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K, M$ be objects of $D(\mathcal{O}_X)$. Set $A = \Gamma(X, \mathcal{O}_X)$. Given $\xi \in H^i(X, K)$ we get an associated map $$ \xi = ``\xi \cup -'' : R\Gamma(X, M)[-i] \to R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) $$ by representing $\xi$ as a map $\xi : A[-i] \to R\Gamma(X, K)$ as in the proof of Lemma \ref{lemma-second-cup-equals-first} and then using the composition $$ R\Gamma(X, M)[-i] = A[-i] \otimes_A^\mathbf{L} R\Gamma(X, M) \xrightarrow{\xi \otimes 1} R\Gamma(X, K) \otimes_A^\mathbf{L} R\Gamma(X, M) \to R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) $$ where the second arrow is the global cup product $\mu$ above. On cohomology this recovers the cup product by $\xi$ as is clear from Lemma \ref{lemma-second-cup-equals-first} and its proof.


cohomology-remark-support-cup-product

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. Given $K$ and $M$ in $D(\mathcal{O}_X)$ there is a canonical map $$ K|_Z \otimes_{\mathcal{O}_X|_Z}^\mathbf{L} R\mathcal{H}_Z(M) \longrightarrow R\mathcal{H}_Z(K \otimes_{\mathcal{O}_X}^\mathbf{L} M) $$ in $D(\mathcal{O}_X|_Z)$. Here $K|_Z = i^{-1}K$ is the restriction of $K$ to $Z$ viewed as an object of $D(\mathcal{O}_X|_Z)$. By adjointness of $i_*$ and $R\mathcal{H}_Z$ of Lemma \ref{lemma-cohomology-with-support-sheaf-on-support} to construct this map it suffices to produce a canonical map $$ i_*\left(K|_Z \otimes_{\mathcal{O}_X|_Z}^\mathbf{L} R\mathcal{H}_Z(M)\right) \longrightarrow K \otimes_{\mathcal{O}_X}^\mathbf{L} M $$ To construct this map, we choose a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_X$-modules representing $M$ and a K-flat complex $\mathcal{K}^\bullet$ of $\mathcal{O}_X$-modules representing $K$. Observe that $\mathcal{K}^\bullet|_Z$ is a K-flat complex of $\mathcal{O}_X|_Z$-modules representing $K|_Z$, see Lemma \ref{lemma-pullback-K-flat}. Hence we need to produce a map of complexes $$ i_*\text{Tot}\left( \mathcal{K}^\bullet|_Z \otimes_{\mathcal{O}_X|_Z} \mathcal{H}_Z(\mathcal{I}^\bullet)\right) \longrightarrow \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{I}^\bullet) $$ of $\mathcal{O}_X$-modules. For this it suffices to produce maps $$ i_*(\mathcal{K}^a|_Z \otimes_{\mathcal{O}_X|_Z} \mathcal{H}_Z(\mathcal{I}^b)) \longrightarrow \mathcal{K}^a \otimes_{\mathcal{O}_X} \mathcal{I}^b $$ Looking at stalks (for example), we see that the left hand side of this formula is equal to $\mathcal{K}^a \otimes_{\mathcal{O}_X} i_*\mathcal{H}_Z(\mathcal{I}^b)$ and we can use the inclusion $\mathcal{H}_Z(\mathcal{I}^b) \to \mathcal{I}^b$ to get our map.


cohomology-remark-support-cup-product-global

With notation as in Remark \ref{remark-support-cup-product} we obtain a canonical cup product \begin{align*} H^a(X, K) \times H^b_Z(X, M) & = H^a(X, K) \times H^b(Z, R\mathcal{H}_Z(M)) \\ & \to H^a(Z, K|_Z) \times H^b(Z, R\mathcal{H}_Z(M)) \\ & \to H^{a + b}(Z, K|_Z \otimes_{\mathcal{O}_X|_Z}^\mathbf{L} R\mathcal{H}_Z(M)) \\ & \to H^{a + b}(Z, R\mathcal{H}_Z(K \otimes_{\mathcal{O}_X}^\mathbf{L} M)) \\ & = H^{a + b}_Z(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) \end{align*} Here the equal signs are given by Lemma \ref{lemma-local-to-global-sections-with-support}, the first arrow is restriction to $Z$, the second arrow is the cup product (Section \ref{section-cup-product}), and the third arrow is the map from Remark \ref{remark-support-cup-product}.


cohomology-remark-support-functorial

Let $f : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_X)$ be a morphism of ringed spaces. Let $Z \subset X$ be a closed subset and $Z' = f^{-1}(Z)$. Denote $f|_{Z'} : (Z', \mathcal{O}_{X'}|_{Z'}) \to (Z, \mathcal{O}_X|Z)$ be the induced morphism of ringed spaces. For any $K$ in $D(\mathcal{O}_X)$ there is a canonical map $$ L(f|_{Z'})^*R\mathcal{H}_Z(K) \longrightarrow R\mathcal{H}_{Z'}(Lf^*K) $$ in $D(\mathcal{O}_{X'}|_{Z'})$. Denote $i : Z \to X$ and $i' : Z' \to X'$ the inclusion maps. By Lemma \ref{lemma-complexes-with-support-on-closed} part (2) applied to $i'$ it is the same thing to give a map $$ i'_* L(f|_{Z'})^* R\mathcal{H}_Z(K) \longrightarrow i'_*R\mathcal{H}_{Z'}(Lf^*K) $$ in $D_{Z'}(\mathcal{O}_{X'})$. The map of functors $Lf^* \circ i_* \to i'_* \circ L(f|_{Z'})^*$ of Remark \ref{remark-base-change} is an isomorphism in this case (follows by checking what happens on stalks using that $i_*$ and $i'_*$ are exact and that $\mathcal{O}_{Z, z} = \mathcal{O}_{X, z}$ and similarly for $Z'$). Hence it suffices to construct a the top horizonal arrow in the following diagram $$ \xymatrix{ Lf^* i_* R\mathcal{H}_Z(K) \ar[rr] \ar[rd] & & i'_* R\mathcal{H}_{Z'}(Lf^*K) \ar[ld] \\ & Lf^*K } $$ The complex $Lf^* i_* R\mathcal{H}_Z(K)$ is supported on $Z'$. The south-east arrow comes from the adjunction mapping $i_*R\mathcal{H}_Z(K) \to K$ (Lemma \ref{lemma-cohomology-with-support-sheaf-on-support}). Since the adjunction mapping $i'_* R\mathcal{H}_{Z'}(Lf^*K) \to Lf^*K$ is universal by Lemma \ref{lemma-complexes-with-support-on-closed} part (3), we find that the south-east arrow factors uniquely over the south-west arrow and we obtain the desired arrow.


cohomology-remark-discuss-derived-limit

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an inverse system in $D(\mathcal{O}_X)$. Set $K = R\lim K_n$. For each $n$ and $m$ let $\mathcal{H}^m_n = H^m(K_n)$ be the $m$th cohomology sheaf of $K_n$ and similarly set $\mathcal{H}^m = H^m(K)$. Let us denote $\underline{\mathcal{H}}^m_n$ the presheaf $$ U \longmapsto \underline{\mathcal{H}}^m_n(U) = H^m(U, K_n) $$ Similarly we set $\underline{\mathcal{H}}^m(U) = H^m(U, K)$. By Lemma \ref{lemma-sheafification-cohomology} we see that $\mathcal{H}^m_n$ is the sheafification of $\underline{\mathcal{H}}^m_n$ and $\mathcal{H}^m$ is the sheafification of $\underline{\mathcal{H}}^m$. Here is a diagram $$ \xymatrix{ K \ar@{=}[d] & \underline{\mathcal{H}}^m \ar[d] \ar[r] &  \mathcal{H}^m \ar[d] \\ R\lim K_n & \lim \underline{\mathcal{H}}^m_n \ar[r] &  \lim \mathcal{H}^m_n } $$ In general it may not be the case that $\lim \mathcal{H}^m_n$ is the sheafification of $\lim \underline{\mathcal{H}}^m_n$. If $U \subset X$ is an open, then we have short exact sequences \begin{equation} \label{equation-ses-Rlim-over-U} 0 \to R^1\lim \underline{\mathcal{H}}^{m - 1}_n(U) \to \underline{\mathcal{H}}^m(U) \to \lim \underline{\mathcal{H}}^m_n(U) \to 0 \end{equation} by Lemma \ref{lemma-RGamma-commutes-with-Rlim}.


cohomology-remark-tensor-internal-hom

Let $(X, \mathcal{O}_X)$ be a ringed space. For $K, K', M, M'$ in $D(\mathcal{O}_X)$ there is a canonical map $$ R\SheafHom(K, K') \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, M') \longrightarrow R\SheafHom(K \otimes_{\mathcal{O}_X}^\mathbf{L} M, K' \otimes_{\mathcal{O}_X}^\mathbf{L} M') $$ Namely, by (\ref{equation-internal-hom}) is the same thing as a map $$ R\SheafHom(K, K') \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, M') \otimes_{\mathcal{O}_X}^\mathbf{L} K \otimes_{\mathcal{O}_X}^\mathbf{L} M \longrightarrow K' \otimes_{\mathcal{O}_X}^\mathbf{L} M' $$ For this we can first flip the middle two factors (with sign rules as in More on Algebra, Section \ref{more-algebra-section-sign-rules}) and use the maps $$ R\SheafHom(K, K') \otimes_{\mathcal{O}_X}^\mathbf{L} K \to K' \quad\text{and}\quad R\SheafHom(M, M') \otimes_{\mathcal{O}_X}^\mathbf{L} M \to M' $$ from Lemma \ref{lemma-internal-hom-composition} when thinking of $K = R\SheafHom(\mathcal{O}_X, K)$ and similarly for $K'$, $M$, and $M'$.


cohomology-remark-projection-formula-for-internal-hom

Let $f : X \to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_X)$. We claim there is a canonical map $$ Rf_*R\SheafHom(L, K) \longrightarrow R\SheafHom(Rf_*L, Rf_*K) $$ Namely, by (\ref{equation-internal-hom}) this is the same thing as a map $Rf_*R\SheafHom(L, K) \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L \to Rf_*K$. For this we can use the composition $$ Rf_*R\SheafHom(L, K) \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L \to Rf_*(R\SheafHom(L, K) \otimes_{\mathcal{O}_X}^\mathbf{L} L) \to Rf_*K $$ where the first arrow is the relative cup product (Remark \ref{remark-cup-product}) and the second arrow is $Rf_*$ applied to the canonical map $R\SheafHom(L, K) \otimes_{\mathcal{O}_X}^\mathbf{L} L \to K$ coming from Lemma \ref{lemma-internal-hom-composition} (with $\mathcal{O}_X$ in one of the spots).


cohomology-remark-relative-cup-and-composition

Let $h : X \to Y$ be a morphism of ringed spaces. Let $K, L, M$ be objects of $D(\mathcal{O}_Y)$. The diagram $$ \xymatrix{ Rf_*R\SheafHom_{\mathcal{O}_X}(K, M) \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*M \ar[r] \ar[d] & Rf_*\left(R\SheafHom_{\mathcal{O}_X}(K, M) \otimes_{\mathcal{O}_X}^\mathbf{L} M\right) \ar[d] \\ R\SheafHom_{\mathcal{O}_Y}(Rf_*K, Rf_*M) \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*M \ar[r] & Rf_*M } $$ is commutative. Here the left vertical arrow comes from Remark \ref{remark-projection-formula-for-internal-hom}. The top horizontal arrow is Remark \ref{remark-cup-product}. The other two arrows are instances of the map in Lemma \ref{lemma-internal-hom-composition} (with one of the entries replaced with $\mathcal{O}_X$ or $\mathcal{O}_Y$).


cohomology-remark-prepare-fancy-base-change

Let $h : X \to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_Y)$. We claim there is a canonical map $$ Lh^*R\SheafHom(K, L) \longrightarrow R\SheafHom(Lh^*K, Lh^*L) $$ in $D(\mathcal{O}_X)$. Namely, by (\ref{equation-internal-hom}) proved in Lemma \ref{lemma-internal-hom} such a map is the same thing as a map $$ Lh^*R\SheafHom(K, L) \otimes^\mathbf{L} Lh^*K \longrightarrow Lh^*L $$ The source of this arrow is $Lh^*(\SheafHom(K, L) \otimes^\mathbf{L} K)$ by Lemma \ref{lemma-pullback-tensor-product} hence it suffices to construct a canonical map $$ R\SheafHom(K, L) \otimes^\mathbf{L} K \longrightarrow L. $$ For this we take the arrow corresponding to $$ \text{id} : R\SheafHom(K, L) \longrightarrow R\SheafHom(K, L) $$ via (\ref{equation-internal-hom}).


cohomology-remark-fancy-base-change

Suppose that $$ \xymatrix{ X' \ar[r]_h \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ is a commutative diagram of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_X)$. We claim there exists a canonical base change map $$ Lg^*Rf_*R\SheafHom(K, L) \longrightarrow R(f')_*R\SheafHom(Lh^*K, Lh^*L) $$ in $D(\mathcal{O}_{S'})$. Namely, we take the map adjoint to the composition \begin{align*} L(f')^*Lg^*Rf_*R\SheafHom(K, L) & = Lh^*Lf^*Rf_*R\SheafHom(K, L) \\ & \to Lh^*R\SheafHom(K, L) \\ & \to R\SheafHom(Lh^*K, Lh^*L) \end{align*} where the first arrow uses the adjunction mapping $Lf^*Rf_* \to \text{id}$ and the second arrow is the canonical map constructed in Remark \ref{remark-prepare-fancy-base-change}.


cohomology-remark-uniqueness

With notation and assumptions as in Lemma \ref{lemma-uniqueness}. Suppose that $U, V \in \mathcal{B}$. Let $\mathcal{B}'$ be the set of elements of $\mathcal{B}$ contained in $U \cap V$. Then $$ (\{K_{U'}\}_{U' \in \mathcal{B}'}, \{\rho_{V'}^{U'}\}_{V' \subset U'\text{ with }U', V' \in \mathcal{B}'}) $$ is a system on the ringed space $U \cap V$ satisfying the assumptions of Lemma \ref{lemma-uniqueness}. Moreover, both $(K_U|_{U \cap V}, \rho^U_{U'})$ and $(K_V|_{U \cap V}, \rho^V_{U'})$ are solutions to this system. By the lemma we find a unique isomorphism $$ \rho_{U, V} : K_U|_{U \cap V} \longrightarrow K_V|_{U \cap V} $$ such that for every $U' \subset U \cap V$, $U' \in \mathcal{B}$ the diagram $$ \xymatrix{ K_U|_{U'} \ar[rr]_{\rho_{U, V}|_{U'}} \ar[rd]_{\rho^U_{U'}} & & K_V|_{U'} \ar[ld]^{\rho^V_{U'}} \\ & K_{U'} } $$ commutes. Pick a third element $W \in \mathcal{B}$. We obtain isomorphisms $\rho_{U, W} : K_U|_{U \cap W} \to K_W|_{U \cap W}$ and $\rho_{V, W} : K_U|_{V \cap W} \to K_W|_{V \cap W}$ satisfying similar properties to those of $\rho_{U, V}$. Finally, we have $$ \rho_{U, W}|_{U \cap V \cap W} = \rho_{V, W}|_{U \cap V \cap W} \circ \rho_{U, V}|_{U \cap V \cap W} $$ This is true by the uniqueness in the lemma because both sides of the equality are the unique isomorphism compatible with the maps $\rho^U_{U''}$ and $\rho^W_{U''}$ for $U'' \subset U \cap V \cap W$, $U'' \in \mathcal{B}$. Some minor details omitted. The collection $(K_U, \rho_{U, V})$ is a descent datum in the derived category for the open covering $\mathcal{U} : X = \bigcup_{U \in \mathcal{B}} U$ of $X$. In this language we are looking for ``effectiveness of the descent datum'' when we look for the existence of a solution.


cohomology-remark-compatible-with-diagram

The map (\ref{equation-projection-formula-map}) is compatible with the base change map of Remark \ref{remark-base-change} in the following sense. Namely, suppose that $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ is a commutative diagram of ringed spaces.  Let $E \in D(\mathcal{O}_X)$ and $K \in D(\mathcal{O}_Y)$. Then the diagram $$ \xymatrix{ Lg^*(Rf_*E \otimes^\mathbf{L}_{\mathcal{O}_Y} K) \ar[r]_p \ar[d]_t & Lg^*Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*K) \ar[d]_b \\ Lg^*Rf_*E \otimes^\mathbf{L}_{\mathcal{O}_{Y'}} Lg^*K \ar[d]_b & Rf'_*L(g')^*(E \otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*K) \ar[d]_t \\ Rf'_*L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{Y'}} Lg^*K \ar[rd]_p & Rf'_*(L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{Y'}} L(g')^*Lf^*K) \ar[d]_c \\ & Rf'_*(L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{Y'}} L(f')^*Lg^*K) } $$ is commutative. Here arrows labeled $t$ are gotten by an application of Lemma \ref{lemma-pullback-tensor-product}, arrows labeled $b$ by an application of Remark \ref{remark-base-change}, arrows labeled $p$ by an application of (\ref{equation-projection-formula-map}), and $c$ comes from $L(g')^* \circ Lf^* = L(f')^* \circ Lg^*$. We omit the verification.


stacks-introduction-remark-diagonal

We have the formula $S \times_{\mathcal{M}_{1, 1}} S' = (S \times S') \times_{\mathcal{M}_{1, 1} \times \mathcal{M}_{1, 1}} \mathcal{M}_{1, 1}$. Hence the key fact is a property of the diagonal $\Delta_{\mathcal{M}_{1, 1}}$ of $\mathcal{M}_{1, 1}$.


stacks-introduction-remark-quotient-stack

The argument sketched above actually shows that $\mathcal{M}_{1, 1} = [W/H]$ is a global quotient stack. It is true about 50\% of the time that an argument proving a moduli stack is algebraic will show that it is a global quotient stack.


restricted-remark-base-change

Let $\varphi : A_1 \to A_2$ be a ring map and let $I_i \subset A_i$ be ideals such that $\varphi(I_1^c) \subset I_2$ for some $c \geq 1$. This induces ring maps $A_{1, cn} = A_1/I_1^{cn} \to A_2/I_2^n = A_{2, n}$ for all $n \geq 1$. Let $\mathcal{C}_i$ be the category (\ref{equation-C}) for $(A_i, I_i)$. There is a base change functor \begin{equation} \label{equation-base-change-systems} \mathcal{C}_1 \longrightarrow \mathcal{C}_2,\quad (B_n) \longmapsto (B_{cn} \otimes_{A_{1, cn}} A_{2, n}) \end{equation} Let $\mathcal{C}_i'$ be the category (\ref{equation-C-prime}) for $(A_i, I_i)$. If $I_2$ is finitely generated, then there is a base change functor \begin{equation} \label{equation-base-change-complete} \mathcal{C}_1' \longrightarrow \mathcal{C}_2',\quad B \longmapsto (B \otimes_{A_1} A_2)^\wedge \end{equation} because in this case the completion is complete (Algebra, Lemma \ref{algebra-lemma-hathat-finitely-generated}). If both $I_1$ and $I_2$ are finitely generated, then the two base change functors agree via the functors (\ref{equation-from-complete-to-systems}) which are equivalences by Lemma \ref{lemma-topologically-finite-type}.


restricted-remark-take-bar

Let $A$ be a Noetherian ring and $I \subset A$ an ideal. Let $\mathfrak a \subset A$ be an ideal. Denote $\bar A = A/\mathfrak a$. Let $\bar I \subset \bar A$ be an ideal such that $I^c \bar A \subset \bar I$ and $\bar I^d \subset I\bar A$ for some $c, d \geq 1$. In this case the base change functor (\ref{equation-base-change-complete}) for $(A, I)$ to $(\bar A, \bar I)$ is given by $B \mapsto \bar B = B/\mathfrak aB$. Namely, we have \begin{equation} \label{equation-base-change-to-closed} \bar B = (B \otimes_A \bar A)^\wedge = (B/\mathfrak a B)^\wedge = B/\mathfrak a B \end{equation} the last equality because any finite $B$-module is $I$-adically complete by Algebra, Lemma \ref{algebra-lemma-completion-tensor} and if annihilated by $\mathfrak a$ also $\bar I$-adically complete by Algebra, Lemma \ref{algebra-lemma-change-ideal-completion}.


restricted-remark-linear-approximation

Let $A$ be a ring and $I \subset A$ be a finitely generated ideal. Let $C$ be an $I$-adically complete $A$-algebra. Let $\psi : A[x_1, \ldots, x_r]^\wedge \to C$ be a continuous $A$-algebra map. Suppose given $\delta_i \in C$, $i = 1, \ldots, r$. Then we can consider $$ \psi' : A[x_1, \ldots, x_r]^\wedge \to C,\quad x_i \longmapsto \psi(x_i) + \delta_i $$ see Formal Spaces, Remark \ref{formal-spaces-remark-universal-property}. Then we have $$ \psi'(g) = \psi(g) + \sum \psi(\partial g/\partial x_i)\delta_i + \xi $$ with error term $\xi \in (\delta_i\delta_j)$. This follows by writing $g$ as a power series and working term by term. Convergence is automatic as the coefficients of $g$ tend to zero. Details omitted.


restricted-remark-improve-homomorphism

Let $A$ be a Noetherian ring and $I \subset A$ be an ideal. Let $B$ be an object of (\ref{equation-C-prime}). Let $C$ be an $I$-adically complete $A$-algebra. Let $\psi_n : B \to C/I^nC$ be an $A$-algebra homomorphism. The obstruction to lifting $\psi_n$ to an $A$-algebra homomorphism into $C/I^{2n}C$ is an element $$ o(\psi_n) \in \Ext^1_B(\NL_{B/A}^\wedge, I^nC/I^{2n}C) $$ as we will explain. Namely, choose a presentation $B = A[x_1, \ldots, x_r]^\wedge/J$. Choose a lift $\psi : A[x_1, \ldots, x_r]^\wedge \to C$ of $\psi_n$. Since $\psi(J) \subset I^nC$ we get $\psi(J^2) \subset I^{2n}C$ and hence we get a $B$-linear homomorphism $$ o(\psi) : J/J^2 \longrightarrow I^nC/I^{2n}C, \quad g \longmapsto \psi(g) $$ which of course extends to a $C$-linear map $J/J^2 \otimes_B C \to I^nC/I^{2n}C$. Since $\NL_{B/A}^\wedge = (J/J^2 \to \bigoplus B \text{d}x_i)$ we get $o(\psi_n)$ as the image of $o(\psi)$ by the identification \begin{align*} & \Ext^1_B(\NL_{B/A}^\wedge, I^nC/I^{2n}C) \\ & = \Coker\left(\Hom_B(\bigoplus B\text{d}x_i, I^nC/I^{2n}C) \to \Hom_B(J/J^2, I^nC/I^{2n}C)\right) \end{align*} See More on Algebra, Lemma \ref{more-algebra-lemma-map-out-of-almost-free} part (1) for the equality. \medskip\noindent Suppose that $o(\psi_n)$ maps to zero in $\Ext^1_B(\NL_{B/A}^\wedge, I^{n'}C/I^{2n'}C)$ for some integer $n'$ with $n > n' > n/2$. We claim that this means we can find an $A$-algebra homomorphism $\psi'_{2n'} : B \to C/I^{2n'}C$ which agrees with $\psi_n$ as maps into $C/I^{n'}C$. The extreme case $n' = n$ explains why we previously said $o(\psi_n)$ is the obstruction to lifting $\psi_n$ to $C/I^{2n}C$. Proof of the claim: the hypothesis that $o(\psi_n)$ maps to zero tells us we can find a $B$-module map $$ h : \bigoplus B\text{d}x_i \longrightarrow I^{n'}C/I^{2n'}C $$ such that $o(\psi)$ and $h \circ \text{d}$ agree as maps into $I^{n'}C/I^{2n'}C$. Say $h(\text{d}x_i) = \delta_i \bmod I^{2n'}C$ for some $\delta_i \in I^{n'}C$. Then we look at the map $$ \psi' : A[x_1, \ldots, x_r]^\wedge \to C,\quad x_i \longmapsto \psi(x_i) - \delta_i $$ A computation with power series shows that $\psi'(J) \subset I^{2n'}C$. Namely, for $g \in J$ we get $$ \psi'(g) \equiv \psi(g) - \sum \psi(\partial g/\partial x_i)\delta_i \equiv o(\psi)(g) - (h \circ \text{d})(g) \equiv 0 \bmod I^{2n'}C $$ See Remark \ref{remark-linear-approximation} for the first equality. Hence $\psi'$ induces an $A$-algebra homomorphism $\psi'_{2n'} : B \to C/I^{2n'}C$ as desired.


restricted-remark-discussion

Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object of (\ref{equation-C-prime}) which is rig-smooth over $(A, I)$. As far as we know, it is an open question as to whether $B$ is isomorphic to the $I$-adic completion of a finite type $A$-algebra. Here are some things we do know: \begin{enumerate} \item If $A$ is a G-ring, then the answer is yes by Proposition \ref{proposition-approximate}. \item If $B$ is rig-\'etale over $(A, I)$, then the answer is yes by Lemma \ref{lemma-approximate}. \item If $I$ is principal, then the answer is yes by \cite[III Theorem 7]{Elkik}. \item In general there exists an ideal $J = (b_1, \ldots, b_s) \subset B$ such that $V(J) \subset V(IB)$ and such that the $I$-adic completion of each of the affine blowup algebras $B[\frac{J}{b_i}]$ are isomorphic to the $I$-adic completion of a finite type $A$-algebra. \end{enumerate} To see the last statement, choose $b_1, \ldots, b_s$ as in Lemma \ref{lemma-equivalent-with-artin-smooth} part (4) and use the properties mentioned there to see that Lemma \ref{lemma-approximate-presentation-rig-smooth} applies to each completion $(B[\frac{J}{b_i}])^\wedge$. Part (4) tells us that ``rig-locally a rig-smooth formal algebraic space is the completion of a finite type scheme over $A$'' and it tells us that ``there is an admissible formal blowing up of $\text{Spf}(B)$ which is affine locally algebraizable''.


restricted-remark-NL-well-defined-topological

Let $A \to B$ be an arrow of $\text{WAdm}^{adic*}$ which is adic and topologically of finite type (see Lemma \ref{lemma-finite-type}). Write $B = A\{x_1, \ldots, x_r\}/J$. Then we can set\footnote{In fact, this construction works for arrows of $\text{WAdm}^{count}$ satisfying the equivalent conditions of Formal Spaces, Lemma \ref{formal-spaces-lemma-quotient-restricted-power-series}.} $$ \NL_{B/A}^\wedge = \left(J/J^2 \longrightarrow \bigoplus B\text{d}x_i\right) $$ Exactly as in the proof of Lemma \ref{lemma-NL-up-to-homotopy} the reader can show that this complex of $B$-modules is well defined up to (unique isomorphism) in the homotopy category $K(B)$. Now, if $A$ is Noetherian and $I \subset A$ is an ideal of definition, then this construction reproduces the naive cotangent complex of $B$ over $(A, I)$ defined by Equation (\ref{equation-NL}) in Section \ref{section-naive-cotangent-complex} simply because $A[x_1, \ldots, x_n]^\wedge$ agrees with $A\{x_1, \ldots, x_r\}$ by Formal Spaces, Remark \ref{formal-spaces-remark-I-adic-completion-and-restricted-power-series}. In particular, we find that, still when $A$ is an adic Noetherian topological ring, the object $\NL_{B/A}^\wedge$ is independent of the choice of the ideal of definition $I \subset A$.


restricted-remark-rig-surjective-more-general

The condition as formulated in Definition \ref{definition-rig-surjective} is not right even for morphisms of finite type of locally adic* formal algebraic spaces. For example, if $A = (\bigcup_{n \geq 1} k[t^{1/n}])^\wedge$ where the completion is the $t$-adic completion, then there are no adic morphisms $\text{Spf}(R) \to \text{Spf}(A)$ where $R$ is a complete discrete valuation ring. Thus any morphism $X \to \text{Spf}(A)$ would be rig-surjective, but since $A$ is a domain and $t \in A$ is not zero, we want to think of $A$ as having at least one ``rig-point'', and we do not want to allow $X = \emptyset$. To cover this particular case, one can consider adic morphisms $$ \text{Spf}(R) \longrightarrow Y $$ where $R$ is a valuation ring complete with respect to a principal ideal $J$ whose radical is $\mathfrak m_R = \sqrt{J}$. In this case the value group of $R$ can be embedded into $(\mathbf{R}, +)$ and one obtains the point of view used by Berkovich in defining an analytic space associated to $Y$, see \cite{Berkovich}. Another approach is championed by Huber. In his theory, one drops the hypothesis that $\Spec(R/J)$ is a singleton, see \cite{Huber-continuous-valuations}.


restricted-remark-diagonal-gives-diagonal

In the situation above consider the diagonal morphisms $\Delta_f : X' \to X' \times_X X'$ and $\Delta_{f_{/T}} : X'_{/T'} \to X'_{/T'} \times_{X_{/T}} X'_{/T'}$. It is easy to see that $$ X'_{/T'} \times_{X_{/T}} X'_{/T'} = (X' \times_X X')_{/T''} $$ as subfunctors of $X' \times_X X'$ where $T'' \subset |X' \times_X X'|$ is the inverse image of $T$. Hence we see that $\Delta_{f_{/T}} = (\Delta_f)_{/T''}$. We will use this below to show that properties of $\Delta_f$ are inherited by $\Delta_{f_{/T}}$.


restricted-remark-compare-formal-modification-artin

In \cite[Definition 1.7]{ArtinII} a formal modification is defined as a proper morphism $f : X \to Y$ of locally Noetherian formal algebraic spaces satisfying the following three conditions\footnote{We will not completely translate these conditions into the language developed in the Stacks project. We hope nonetheless the discussion here will be useful to the reader.} \begin{enumerate} \item[(\romannumeral1)] the Cramer and Jacobian ideal of $f$ each contain an ideal of definition of $X$, \item[(\romannumeral2)] the ideal defining the diagonal map $\Delta : X \to X \times_Y X$ is annihilated by an ideal of definition of $X \times_Y X$, and \item[(\romannumeral3)] any adic morphism $\text{Spf}(R) \to Y$ lifts to $\text{Spf}(R) \to X$ whenever $R$ is a complete discrete valuation ring. \end{enumerate} Let us compare these to our list of conditions above. \medskip\noindent Ad (\romannumeral1). Property (\romannumeral1) agrees with our condition that $f$ be a rig-\'etale morphism: this follows from Lemma \ref{lemma-equivalent-with-artin} part (\ref{item-condition-artin}). \medskip\noindent Ad (\romannumeral2). Assume $f$ is rig-\'etale. Then $\Delta_f : X \to X \times_Y X$ is rig-\'etale as a morphism of locally Noetherian formal algebraic spaces which are rig-\'etale over $X$ (via $\text{id}_X$ for the first one and via $\text{pr}_1$ for the second one). See Lemmas \ref{lemma-base-change-rig-etale} and \ref{lemma-rig-etale-permanence}. Hence property (\romannumeral2) agrees with our condition that $\Delta_f$ be rig-surjective by Lemma \ref{lemma-closed-immersion-rig-smooth-rig-surjective}. \medskip\noindent Ad (\romannumeral3). Property (\romannumeral3) does not quite agree with our notion of a rig-surjective morphism, as Artin requires all adic morphisms $\text{Spf}(R) \to Y$ to lift to morphisms into $X$ whereas our notion of rig-surjective only asserts the existence of a lift after replacing $R$ by an extension. However, since we already have that $\Delta_f$ is rig-\'etale and rig-surjective by (\romannumeral1) and (\romannumeral2), these conditions are equivalent by Lemma \ref{lemma-rig-monomorphism-rig-surjective}.


more-groupoids-remark-local-source-warning

Warning: Lemma \ref{lemma-local-source} should be used with care. For example, it applies to $\mathcal{P}=$``flat'', $\mathcal{Q}=$``empty'', and $\mathcal{R}=$``flat and locally of finite presentation''. But given a morphism of schemes $f : X \to Y$ the largest open $W \subset X$ such that $f|_W$ is flat is {\it not} the set of points where $f$ is flat!


more-groupoids-remark-local-source-apply

Notwithstanding the warning in Remark \ref{remark-local-source-warning} there are some cases where Lemma \ref{lemma-local-source} can be used without causing too much ambiguity. We give a list. In each case we omit the verification of assumptions (1) and (2) and we give references which imply (3) and (4). Here is the list: \begin{enumerate} \item $\mathcal{Q} = \mathcal{R} =$``locally of finite type'', and $\mathcal{P} =$``relative dimension $\leq d$''. See Morphisms, Definition \ref{morphisms-definition-relative-dimension-d} and Morphisms, Lemmas \ref{morphisms-lemma-openness-bounded-dimension-fibres} and \ref{morphisms-lemma-dimension-fibre-after-base-change}. \item $\mathcal{Q} = \mathcal{R} =$``locally of finite type'', and $\mathcal{P} =$``locally quasi-finite''. This is the case $d = 0$ of the previous item, see Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}. \item $\mathcal{Q} = \mathcal{R} =$``locally of finite type'', and $\mathcal{P} =$``unramified''. See Morphisms, Lemmas \ref{morphisms-lemma-unramified-characterize} and \ref{morphisms-lemma-set-points-where-fibres-unramified}. \end{enumerate} What is interesting about the cases listed above is that we do not need to assume that $s, t$ are flat to get a conclusion about the locus where the morphism $h$ has property $\mathcal{P}$. We continue the list: \begin{enumerate} \item[(4)] $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P} =$``flat''. See More on Morphisms, Theorem \ref{more-morphisms-theorem-openness-flatness} and Lemma \ref{more-morphisms-lemma-flat-locus-base-change}. \item[(5)] $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P}=$``Cohen-Macaulay''. See More on Morphisms, Definition \ref{more-morphisms-definition-CM} and More on Morphisms, Lemmas \ref{more-morphisms-lemma-base-change-CM} and \ref{more-morphisms-lemma-flat-finite-presentation-CM-open}. \item[(6)] $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P}=$``syntomic'' use Morphisms, Lemma \ref{morphisms-lemma-set-points-where-fibres-lci} (the locus is automatically open). \item[(7)] $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P}=$``smooth''. See Morphisms, Lemma \ref{morphisms-lemma-set-points-where-fibres-smooth} (the locus is automatically open). \item[(8)] $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P}=$``\'etale''. See Morphisms, Lemma \ref{morphisms-lemma-set-points-where-fibres-etale} (the locus is automatically open). \end{enumerate}


more-groupoids-remark-warn-dimension-groupoid-on-field

Warning: Lemma \ref{lemma-groupoid-on-field-dimension-equal-stabilizer} is wrong without the condition that $s$ and $t$ are locally of finite type. An easy example is to start with the action $$ \mathbf{G}_{m, \mathbf{Q}} \times_{\mathbf{Q}} \mathbf{A}^1_{\mathbf{Q}} \to \mathbf{A}^1_{\mathbf{Q}} $$ and restrict the corresponding groupoid scheme to the generic point of $\mathbf{A}^1_{\mathbf{Q}}$. In other words restrict via the morphism $\Spec(\mathbf{Q}(x)) \to \Spec(\mathbf{Q}[x]) = \mathbf{A}^1_{\mathbf{Q}}$. Then you get a groupoid scheme $(U, R, s, t, c)$ with $U = \Spec(\mathbf{Q}(x))$ and $$ R = \Spec\left( \mathbf{Q}(x)[y]\left[ \frac{1}{P(xy)}, P \in \mathbf{Q}[T], P \not = 0 \right] \right) $$ In this case $\dim(R) = 1$ and $\dim(G) = 0$.


examples-remark-reference-existence-regular-nonexcellent-rings

Non-excellent regular rings whose residue fields have a finite $p$-basis can be constructed even in the function field of $\mathbb{P}^2_k$, over a  characteristic $p$ field $k = \overline{k}$. See \cite[$\mathsection 4.1$]{DS18}.


examples-remark-specialization

Here are some remarks: \begin{enumerate} \item The presheaves $F$ and $F_n$ are separated presheaves. \item It turns out that $F$, $F_n$ are not sheaves. \item One can show that $G$, $G_n$ is actually a sheaf for the fppf topology. \end{enumerate} We will prove these results if we need them.


examples-remark-contradict-aoki

Proposition \ref{proposition-nonalghomstack} contradicts \cite[Theorem 1.1]{AokiHomStacks}. The problem is the non-effectivity of formal objects for $\underline{\Mor}_S(X, [S/A])$. The same problem is mentioned in the Erratum \cite{AokiHomStacksErr} to \cite{AokiHomStacks}. Unfortunately, the Erratum goes on to assert that $\underline{\Mor}_S(\mathcal{Y}, \mathcal{Z})$ is algebraic if $\mathcal{Z}$ is separated, which also contradicts Proposition \ref{proposition-nonalghomstack} as $[S/A]$ is separated.


spaces-perfect-remark-match-total-direct-images

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of representable algebraic spaces $X$ and $Y$ over $S$. Let $f_0 : X_0 \to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Then the diagram $$ \xymatrix{ D_\QCoh(\mathcal{O}_{X_0}) \ar@{=}[rrrrrr]_{\text{Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\QCoh(\mathcal{O}_X) \\ D_\QCoh(\mathcal{O}_{Y_0}) \ar[u]^{Lf^*_0} \ar@{=}[rrrrrr]^{\text{Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\QCoh(\mathcal{O}_Y) \ar[u]_{Lf^*} } $$ (Lemma \ref{lemma-quasi-coherence-pullback} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-pullback}) is commutative. This follows as the equivalences $D_\QCoh(\mathcal{O}_{X_0}) \to D_\QCoh(\mathcal{O}_X)$ and $D_\QCoh(\mathcal{O}_{Y_0}) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} come from pulling back by the (flat) morphisms of ringed sites $\epsilon : X_\etale \to X_{0, Zar}$ and $\epsilon : Y_\etale \to Y_{0, Zar}$ and the diagram of ringed sites $$ \xymatrix{ X_{0, Zar} \ar[d]_{f_0} & X_\etale \ar[l]^\epsilon \ar[d]^f \\ Y_{0, Zar} & Y_\etale \ar[l]_\epsilon } $$ is commutative (details omitted). If $f$ is quasi-compact and quasi-separated, equivalently if $f_0$ is quasi-compact and quasi-separated, then we claim $$ \xymatrix{ D_\QCoh(\mathcal{O}_{X_0}) \ar[d]_{Rf_{0, *}} \ar@{=}[rrrrrr]_{\text{Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D_\QCoh(\mathcal{O}_{Y_0}) \ar@{=}[rrrrrr]^{\text{Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\QCoh(\mathcal{O}_Y) } $$ (Lemma \ref{lemma-quasi-coherence-direct-image} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-direct-image}) is commutative as well. This also follows from the commutative diagram of sites displayed above as the proof of Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} shows that the functor $R\epsilon_*$ gives the equivalences $D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_{X_0})$ and $D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_{Y_0})$.


spaces-perfect-remark-how-to

How to choose the collection $\mathcal{B}$ in Lemma \ref{lemma-induction-principle-separated}? Here are some examples: \begin{enumerate} \item If $X$ is quasi-compact and separated, then we can choose $\mathcal{B}$ to be the set of quasi-compact and separated objects of $X_{spaces, \etale}$. Then $X \in \mathcal{B}$ and $\mathcal{B}$ satisfies (1), (2), and (3)(a). With this choice of $\mathcal{B}$ Lemma \ref{lemma-induction-principle-separated} reproduces Lemma \ref{lemma-induction-principle}. \item If $X$ is quasi-compact with affine diagonal, then we can choose $\mathcal{B}$ to be the set of objects of $X_{spaces, \etale}$ which are quasi-compact and have affine diagonal. Again $X \in \mathcal{B}$ and $\mathcal{B}$ satisfies (1), (2), and (3)(a). \item If $X$ is quasi-compact and quasi-separated, then the smallest subset $\mathcal{B}$ which contains $X$ and satisfies (1), (2), and (3)(a) is given by the rule $W \in \mathcal{B}$ if and only if either $W$ is a quasi-compact open subspace of $X$, or $W$ is a quasi-compact open of an affine object of $X_{spaces, \etale}$. \end{enumerate}


spaces-perfect-remark-addendum

The proof of Lemma \ref{lemma-lift-map-from-perfect-complex-with-support} shows that $$ R|_W = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] $$ for some $m \geq 0$ and $n_j \geq 0$. Thus the highest degree cohomology sheaf of $R|_W$ equals that of $P$. By repeating the construction for the map $P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] \to R|_W$, taking cones, and using induction we can achieve equality of cohomology sheaves of $R|_W$ and $P$ above any given degree.


spaces-perfect-remark-pullback-generator

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $E \in D_\QCoh(\mathcal{O}_Y)$ be a generator (see Theorem \ref{theorem-bondal-van-den-Bergh}). Then the following are equivalent \begin{enumerate} \item for $K \in D_\QCoh(\mathcal{O}_X)$ we have $Rf_*K = 0$ if and only if $K = 0$, \item $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ reflects isomorphisms, and \item $Lf^*E$ is a generator for $D_\QCoh(\mathcal{O}_X)$. \end{enumerate} The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that $Lf^*$ is the left adjoint to $Rf_*$. These conditions hold if $f$ is affine (Lemma \ref{lemma-affine-morphism}) or if $f$ is an open immersion, or if $f$ is a composition of such.


spaces-perfect-remark-classical-generator

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G$ be a perfect object of $D(\mathcal{O}_X)$ which is a generator for $D_\QCoh(\mathcal{O}_X)$. By Theorem \ref{theorem-bondal-van-den-Bergh} there is at least one of these. Combining Lemma \ref{lemma-quasi-coherence-direct-sums} with Proposition \ref{proposition-compact-is-perfect} and with Derived Categories, Proposition \ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf}(\mathcal{O}_X)$.


spaces-perfect-remark-classical-generator-with-support

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. Let $G$ be a perfect object of $D_{\QCoh, T}(\mathcal{O}_X)$ which is a generator for $D_{\QCoh, T}(\mathcal{O}_X)$. By Lemma \ref{lemma-generator-with-support} there is at least one of these. Combining the fact that $D_{\QCoh, T}(\mathcal{O}_X)$ has direct sums with Lemma \ref{lemma-compact-is-perfect-with-support} and with Derived Categories, Proposition \ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf, T}(\mathcal{O}_X)$.