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\section{Base change and the Grothendieck
complex}\label{base-change-and-the-grothendieck-complex}

\emph{Written by GPT-6.1 Sol (OpenAI), in Codex, at Ultra effort,
October 2026. Self-checked by the writing AI, GPT-6.1 Sol, at Ultra
effort. Public domain (CC0).}

Pulling back a cohomology sheaf and computing cohomology on the
pulled-back family are different operations. Flat change of the base
makes them agree. For a proper family with a flat coherent sheaf, a
stronger statement is available: one finite projective complex computes
cohomology after every change of the base. Its differentials, rather
than the ranks of its cohomology modules, retain the information needed
at exceptional fibers.

We use \href{affine-cohomology-and-serres-criterion.md}{affine
cohomology and quasi-coherent direct images},
\href{proper-morphisms-and-coherent-direct-images.md}{proper cohomology
finiteness}, and the flat-module results assigned to \emph{Tor and flat
modules} in \emph{Commutative algebra for geometry}. For the initial
derived construction we use
\href{../derived-categories-and-sheaf-operations/flat-modules-and-k-flat-resolutions.html}{K-flat
resolutions},
\href{../derived-categories-and-sheaf-operations/derived-tensor-products-and-tor-sheaves.html}{derived
tensor} and the
\href{../derived-categories-and-sheaf-operations/derived-pullback-and-pushforward.html}{pullback--pushforward
adjunction}. The earlier programme proof of adjunction is \emph{Derived
pullback and pushforward}, Theorem 2.2; Proposition 3.1 proves
composition, and Theorem 1.1 constructs derived pullback. These supply
the units, counits and composition used in Section 1. Our finite
projective replacement and its universal tensor comparison are proved in
Section 3.

\subsection{1. Constructing the
comparison}\label{constructing-the-comparison}

Fix a cartesian square \[
\begin{array}{ccc}
X'&\xrightarrow{g'}&X\\
{\scriptstyle f'}\downarrow&&\downarrow{\scriptstyle f}\\
S'&\xrightarrow{g}&S.
\end{array}
\] Let \(F\) be quasi-coherent on \(X\), and put \(F'=g'^*F\). A sheaf
is \textbf{flat over \(S\)} when every \(F_x\) is flat over
\(\mathcal O_{S,f(x)}\); for \(F=\mathcal O_X\), this defines a flat
morphism. Base change preserves this stalkwise condition. Flatness of
\(F\) and flatness of \(g\) will enter different theorems.

The canonical comparison maps are \[
\beta^q:g^*R^qf_*F\longrightarrow R^qf'_*F'.
\] Here is a derived construction that also applies to a nonflat \(g\).
Derived adjunction turns the composite \[
Lf'^*Lg^*Rf_*F
\cong Lg'^*Lf^*Rf_*F
\longrightarrow Lg'^*F
\longrightarrow g'^*F
\] into a map \(Lg^*Rf_*F\to Rf'_*F'\). The first arrow is the
pullback--pushforward counit; the last is the natural map from derived
pullback of a sheaf to its degree-zero, ordinary pullback. Commutation
of the two derived pullbacks follows from equality of the composite
morphisms and composition of derived pullback. It does not assert that
derived pullback is exact.

There is a natural cohomology edge map \[
g^*H^q(C)\longrightarrow H^q(Lg^*C)
\] for a bounded-above cohomology object \(C\). Locally, choose a
bounded-above free resolution \(P\) of its complex over \(A\), and use
the cocycle map \[
H^q(P)\otimes_A B\longrightarrow H^q(P\otimes_A B).
\] A cocycle tensored with an element of \(B\) is still a cocycle; an
original boundary becomes a boundary. Right exactness of tensor lets
this descend from the cocycle module to \(H^q(P)\otimes B\). Homotopic
comparison maps of free resolutions induce the same map, so the edge is
intrinsic and compatible with restriction. Composing it with the
cohomology map of the derived comparison defines \(\beta^q\). For the
quasi-compact, quasi-separated morphisms used below, the boundedness
needed here is the affine-cover bound from the third lesson. For flat
\(g\), the edge is simply the identification \(g^*H^q(C)=H^q(g^*C)\).

The construction is functorial in \(F\) and in successive base changes:
units, counits and pullback composition give the same map for a
composite square as for the two successive squares. On affine charts
with a Čech model, it is the cocycle comparison just described. Thus the
isomorphisms proved below concern these canonical maps, not a separately
chosen identification of abstract vector spaces.

\textbf{Proposition 1.1 (affine morphisms).} If \(f\) is affine, every
\(\beta^q\) is an isomorphism for every base change \(g\), without a
flatness assumption.

\textbf{Proof.} Higher direct images of a quasi-coherent sheaf under an
affine morphism are zero, and the same holds for \(f'\). Locally write
\(S=\operatorname{Spec}A\), \(X=\operatorname{Spec}R\),
\(S'=\operatorname{Spec}B\), and \(F=\widetilde M\). The pullback sheaf
corresponds to \[
M\otimes_R(R\otimes_A B)\cong M\otimes_A B.
\] This is precisely the degree-zero base-change map. The tensor
identity proves the assertion in degree zero, and the vanishing proves
it in higher degrees. \(\square\)

The hypothesis concerns the morphism \(f\). Affineness of \(g\) alone
does not guarantee comparison isomorphisms; a closed point of an affine
parameter scheme is already a nonflat affine base change.

\subsection{2. Flat base change, including nonseparated
schemes}\label{flat-base-change-including-nonseparated-schemes}

\textbf{Theorem 2.1 (flat base change).} If \(f\) is quasi-compact and
quasi-separated, \(F\) is quasi-coherent, and \(g\) is flat, then \[
g^*R^qf_*F\xrightarrow{\sim}R^qf'_*F'\quad(q\geq0).
\] In particular, for a flat ring map \(A\to B\) and a quasi-compact,
quasi-separated \(A\)-scheme, \[
H^q(X,F)\otimes_A B\xrightarrow{\sim}H^q(X_B,F_B).
\]

\textbf{Proof.} First suppose \(S,S'\) are affine and \(X\) is
separated. Choose a finite affine cover \(U_1,\ldots,U_r\). Its
intersections are affine, so the finite ordered complex \(C^\bullet\) of
their section modules computes cohomology. The induced cover after base
change again has affine intersections. Proposition 1.1 identifies its
section complex with \(C^\bullet\otimes_A B\). Flatness of \(B\)
preserves kernels and images and gives \[
H^q(C^\bullet)\otimes_A B\cong H^q(C^\bullet\otimes_A B).
\] The identification is the canonical cocycle map.

For a quasi-separated \(X\), a finite affine cover still exists, but its
intersections need not be affine. Each intersection \(U_I\) is
quasi-compact and separated: it is an open subscheme of one of the
affine \(U_i\). The case just proved therefore applies to every \(U_I\).
Use the Čech-to-cohomology spectral sequence of the second lesson, in
the form \[
E_1^{p,q}=\bigoplus_{|I|=p+1}H^q(U_I,F)\Longrightarrow H^{p+q}(X,F).
\] Tensoring this spectral sequence with flat \(B\) preserves its
successive kernels, images and cohomology. The comparison with the
base-changed cover is an isomorphism on every first-page term, by the
separated case. Hence it is an isomorphism on every page and on the
filtered abutments. The cover has finitely many intersections and each
has a finite affine-cover cohomological bound, so there are only
finitely many possibly nonzero terms. Convergence and comparison
therefore involve finite filtrations and prove the asserted global
identity.

Finally work locally on \(S'\), with its affine open contained over an
affine open of \(S\). The higher direct images are quasi-coherent by the
third lesson, and over this affine base they are the sheaves associated
to the cohomology modules just compared. The global identities give the
sheaf comparison locally, and canonical functoriality glues them.
\(\square\)

Notice that \(F\) need not be flat over the original base in this
theorem. Tensor is exact because the \emph{change of base} is flat. In
contrast, we next keep \(F\) flat and permit every change of base.

\subsection{3. A finite projective
replacement}\label{a-finite-projective-replacement}

We first prove the algebraic step, retaining its tensor behavior rather
than only its cohomology over the original ring.

\textbf{Lemma 3.1 (bounded flat complexes).} Let \(A\) be Noetherian.
Let \(C^\bullet\) be a complex of flat \(A\)-modules, zero outside
\([a,b]\), with finite cohomology modules. There is a complex
\(K^\bullet\) of finite projective modules, also zero outside \([a,b]\),
and a quasi-isomorphism \[
K^\bullet\longrightarrow C^\bullet
\] that remains a quasi-isomorphism after tensoring with every
\(A\)-module.

\textbf{Proof.} We construct a bounded-above finite free resolution of
the complex, then truncate its unnecessary left tail.

Start with no terms of \(P\), and attach them in descending degrees,
beginning at \(b\). Suppose terms \(P^{j+1},\ldots,P^b\) and a chain map
\(f:P\to C\) have been constructed, with its cone \(D\) having zero
cohomology in degrees above \(j\). The cone terms and differential are
\[
D^i=C^i\oplus P^{i+1},\qquad d_D(c,p)=(d_Cc+f(p),-d_Pp).
\] Its cohomology modules are finite. This holds initially for \(C\),
and remains true for a cone of a map from a finite complex of finite
free modules: its long exact sequence expresses cohomology as extensions
of subquotients of finite modules, which are finite because \(A\) is
Noetherian.

Choose finitely many cocycles \((c_\ell,p_\ell)\) generating \(H^j(D)\).
Set \(P^j\) equal to the finite free module on these choices and define
\[
f(e_\ell)=c_\ell,\qquad d_P(e_\ell)=-p_\ell.
\] The cocycle equations give \(d_P^2=0\) and \(d_Cf=fd_P\). In the new
cone, the differential of \((0,e_\ell)\) is \((c_\ell,p_\ell)\), so
\(H^j\) is killed while higher cohomology remains zero. Finiteness of
lower cohomology is preserved by the same long exact sequence.
Continuing indefinitely to the left gives a chain map \(P\to C\), with
each \(P^j\) finite free and \(P^j=0\) for \(j>b\). Its cone is acyclic:
in any fixed degree, the construction kills the cohomology after
finitely many stages, and later stages do not change it. Thus \(P\to C\)
is a quasi-isomorphism.

We need a tensor fact about its cone. An acyclic bounded-above complex
\(T\) of flat modules remains acyclic after every tensor product. At its
last nonzero degree, the preceding differential surjects onto the last
term. The kernel is flat, because a kernel between flat middle and flat
quotient modules is flat. Repeating downward shows all cycle modules are
flat and every sequence \[
0\to Z^i(T)\to T^i\to Z^{i+1}(T)\to0
\] remains exact under tensor. These tensor sequences imply acyclicity
after tensoring. The downward induction reaches any specified degree in
finitely many steps even if the complex is unbounded to the left.
Applying it to the cone of \(P\to C\) gives \[
H^i(P\otimes_A M)\cong H^i(C\otimes_A M)=0\quad(i<a)
\] for every \(M\).

Now put \(Q=\operatorname{coker}(P^{a-1}\to P^a)\). It is finite, and
the exact sequence \[
\cdots\to P^{a-2}\to P^{a-1}\to P^a\to Q\to0
\] is a free resolution of \(Q\). Exactness uses \(H^i(P)=0\) for
\(i<a\). Therefore \[
\operatorname{Tor}_1^A(Q,M)=H^{a-1}(P\otimes_A M)=0
\] for all \(M\), since the differential through degree \(a\) is the
same in that resolution. Thus \(Q\) is flat. It is finitely presented
over Noetherian \(A\), so it is finite projective by the flat-module
criterion.

Define \(K^a=Q\), \(K^i=P^i\) for \(a<i\leq b\), and all other terms
zero. The map \(P^a\to C^a\) kills the preceding image because
\(C^{a-1}=0\); it factors through \(Q\), giving a chain map \(K\to C\).
Truncation preserves all cohomology, so this map is a quasi-isomorphism.
Its cone is a bounded acyclic complex of flat modules. The tensor fact
proves that \(K\otimes M\to C\otimes M\) is a quasi-isomorphism for
every \(M\), as required. \(\square\)

The finite replacement does not come from assuming Čech terms are
finite, nor from assuming the ring has finite global dimension.
Finiteness of the \emph{cohomology} constructs the finite free
approximation. Flatness and its bounded range then make the last
retained quotient projective.

\textbf{Theorem 3.2 (the Grothendieck complex).} Let \(A\) be
Noetherian, \(f:X\to\operatorname{Spec}A\) proper, and \(F\) coherent
and flat over \(A\). There is a bounded complex \(K\) of finite
projective \(A\)-modules, in nonnegative degrees, such that for every
\(A\)-algebra \(B\), \[
H^q(K\otimes_A B)\cong H^q(X_B,F_B),
\] functorially in \(B\). More generally
\(H^q(K\otimes_A M)\cong H^q(X,F\otimes_A M)\) for every \(A\)-module
\(M\).

\textbf{Proof.} Properness makes \(X\) quasi-compact and separated. Take
a finite affine cover with \(r\) members. Its ordered Čech complex
\(C\), zero outside \([0,r-1]\), computes \(F\)-cohomology. Every term
is \(A\)-flat: for any injection of \(A\)-modules, tensoring the sheaf
preserves injectivity at all stalks, and taking sections on an affine
intersection preserves exactness of quasi-coherent sheaves. This is the
section-module flatness argument already used for high twists in the
preceding lesson. Its cohomology is finite by proper finiteness. Apply
Lemma 3.1 to obtain \(K\to C\).

Every affine intersection stays affine after a base change. Its section
module for \(F_B\) is the original one tensored with \(B\), so the
complex after base change is \(C\otimes_A B\). It computes
\(H^q(X_B,F_B)\). The universal tensor quasi-isomorphism of Lemma 3.1
identifies its cohomology with that of \(K\otimes_A B\). For an
arbitrary module \(M\), the same affine correspondence identifies
\(C\otimes_A M\) with the Čech complex of \(F\otimes_A M\). All maps
arise from a fixed chain map and affine tensor identifications, so are
compatible with homomorphisms of modules and algebras. \(\square\)

In derived language, a bounded complex of finite projective modules
represents a \textbf{perfect} object. Thus the theorem proves that
\(R\Gamma(X,F)\) is perfect and \[
R\Gamma(X,F)\otimes_A^L B\cong R\Gamma(X_B,F_B)
\] for every \(B\). This is stronger than flat base change. It does not
imply that \(H^q(X,F)\otimes B\to H^q(X_B,F_B)\) is always an
isomorphism: taking cohomology of a tensor product is still distinct
from tensoring its cohomology. The finite complex makes that distinction
computable.

\subsection{4. Euler constancy and explicit
models}\label{euler-constancy-and-explicit-models}

\textbf{Corollary 4.1.} Under Theorem 3.2, the fiber Euler
characteristic \(s\mapsto\chi(X_s,F_s)\) is locally constant on
\(\operatorname{Spec}A\).

\textbf{Proof.} The complex \(K\otimes_A\kappa(s)\) is a finite complex
of finite-dimensional vector spaces computing fiber cohomology. Image
cancellation gives \[
\chi(X_s,F_s)=\sum_i(-1)^i\dim_{\kappa(s)}(K^i\otimes_A\kappa(s))
=\sum_i(-1)^i\operatorname{rank}_s K^i.
\] Each finite projective module has locally constant rank. There are
finitely many of them, so their alternating sum is locally constant.
\(\square\)

For \(\mathcal O\) on \(\mathbf P^1_A\), use \(K=A[0]\). The
projective-space computation gives \(H^0=B\) and all higher cohomology
zero for every \(A\)-algebra \(B\). More generally \(\mathcal O(d)\) has
the explicit free complex concentrated in degree zero for \(d\geq0\), in
degree one for \(d\leq-2\), and the zero complex for \(d=-1\). This
includes \(\mathcal O(-2)\) represented by \(A\) in degree one.

For the family of conics \(xy=tz^2\) over \(A=k[t]\), the monic
homogeneous equation makes the family flat, as proved in the preceding
lesson. Over every \(A\)-algebra \(B\), multiplication by the equation
remains injective in the polynomial ring: its leading coefficient is
one. The sequence on \(\mathbf P^2_B\) \[
0\to\mathcal O(-2)\to\mathcal O\to\mathcal O_{\mathcal C_B}\to0
\] and projective-space cohomology give a Grothendieck complex \(A[0]\)
for the structure sheaf. Twisting once replaces \(\mathcal O(-2)\) by
\(\mathcal O(-1)\) and \(\mathcal O\) by \(\mathcal O(1)\), giving
\(A^3[0]\) for \(\mathcal O_{\mathcal C}(1)\). These computations work
on the reducible special fiber as well.

A more revealing example is the rank-two extension on
\(\mathbf P^1_{k[t]}\) \[
0\to\mathcal O(-2)\to E\to\mathcal O\to0
\] with extension class \(t\). The projective-line calculations and its
connecting map give the model \[
K=[\,A\xrightarrow{t}A\,]\quad\text{in degrees }0,1.
\] Indeed the cohomology triangle for the extension is the fiber of
multiplication by \(t\) between the two copies of \(A\), which is this
two-term complex; choosing the generator of \(H^1(\mathcal O(-2))\)
fixes that sign. Tensoring with any \(B\) replaces the differential by
multiplication by the image of \(t\). Over the original integral ring,
\(H^0(E)=0\) and \(H^1(E)=A/(t)\). At the zero fiber the differential is
zero and both fiber groups are \(k\). Thus \(\beta^0\) is the map
\(0\to k\), while \(\beta^1\) is an isomorphism. The Euler
characteristic is zero everywhere. Retaining the differential recovers
the missing fiber sections.

\subsection{5. Exercises with solutions}\label{exercises-with-solutions}

\textbf{Exercise 5.1 (easy: affine comparison).} For an affine morphism
and an arbitrary change of base, write the degree-zero comparison
explicitly and explain the higher-degree statement.

\textbf{Solution.} With \(A\to R\), \(A\to B\), and \(M\) an
\(R\)-module, the map is \(M\otimes_A B\to M\otimes_R(R\otimes_A B)\),
taking \(m\otimes b\) to \(m\otimes(1\otimes b)\). The inverse takes
\(m\otimes(r\otimes b)\) to \(rm\otimes b\). Higher source and target
sheaves are zero by affine vanishing. Neither flatness of \(M\) nor
flatness of \(B\) is needed.

\textbf{Exercise 5.2 (easy: flat Čech terms).} Prove flatness over \(A\)
of the section modules on affine intersections when \(F\) is flat over
\(A\). Identify where separatedness is used in Theorem 3.2.

\textbf{Solution.} Every stalk \(F_x\) is flat over its parameter local
ring and therefore over \(A\), since localization of \(A\) is flat.
Tensoring an injection of \(A\)-modules with \(F\) is thus stalkwise
injective. The resulting sheaves are quasi-coherent, and their sections
on an affine open form an exact sequence. Hence tensoring the affine
section module preserves injections, proving flatness. Separatedness
ensures that the finite cover's intersections are affine; without it,
the single section complex need not compute higher cohomology.

\textbf{Exercise 5.3 (medium: the alternating ranks).} Derive Euler
constancy from a Grothendieck complex, and decide whether it forces its
cohomology ranks to be locally constant.

\textbf{Solution.} On each fiber, boundaries cancel in the alternating
dimension sum, leaving the alternating ranks of the finitely many
projective terms. Those ranks are locally constant, which proves Euler
constancy. Cohomology ranks need not be: for \([A\xrightarrow{t}A]\),
both are zero away from \(t=0\) and both are one there. The alternating
difference stays zero.

\textbf{Exercise 5.4 (medium: the conic hyperplane bundle).} Compute a
Grothendieck complex for \(\mathcal O(1)\) on \(xy=tz^2\), and check its
behavior under the nonflat base change \(k[t]\to k[t]/(t^2)\).

\textbf{Solution.} The twisted hypersurface sequence is
\(0\to\mathcal O(-1)\to\mathcal O(1)\to\mathcal O_{\mathcal C}(1)\to0\).
On \(\mathbf P^2_B\), \(\mathcal O(-1)\) has no cohomology and
\(\mathcal O(1)\) has only \(H^0=B^3\), for every \(B\). The monic
equation preserves injectivity even when \(B\) has nilpotents. Thus
\(K=A^3[0]\) works, and over \(B=A/(t^2)\) it gives \(B^3\) in degree
zero and zero otherwise. This is a genuine arbitrary-base-change
calculation, not an application of the flat-base-change theorem.

\textbf{Exercise 5.5 (hard: truncate a free approximation).} For a
bounded flat complex with finite cohomology over a Noetherian ring,
explain why a finite free approximation may have an infinite left tail,
and prove that its truncation term at the original lower bound is
projective.

\textbf{Solution.} Successively killing the cone's top cohomology
constructs finite free terms, but new kernel classes can appear one
degree lower at each step. The ring need not have finite global
dimension, so this process need not stop. Let the original flat complex
start in degree \(a\), and let \(P\) be the resulting bounded-above free
approximation. Its acyclic flat cone remains acyclic after tensoring, by
the cycle induction of Lemma 3.1. Thus \(H^{a-1}(P\otimes M)=0\) for
every \(M\). The free tail through \(P^a\) resolves
\(Q=\operatorname{coker}(P^{a-1}\to P^a)\), so this equality is
\(\operatorname{Tor}_1(Q,M)=0\). It implies flatness of \(Q\); finite
generation and Noetherianness give finite presentation, hence
projectivity. Replacing the whole left tail by \(Q\) retains the
cohomology and universal tensor comparison. This is the complete
truncation mechanism, not a finite global-dimension argument.

\textbf{Exercise 5.6 (challenging: why the flat-sheaf hypothesis
matters).} Take the proper identity map
\(X=\operatorname{Spec}k[t]\to\operatorname{Spec}k[t]\) and
\(F=\widetilde{k[t]/(t)}\). Show that no bounded finite projective
complex can compute its ordinary fiber cohomology for all base changes
in the manner of Theorem 3.2, although \(R\Gamma(X,F)\) itself is
perfect.

\textbf{Solution.} Its fiber is zero away from \(t=0\), and at zero it
is \(k\), with no positive cohomology because the fibers are affine. A
hypothetical complex would therefore have fiber Euler characteristic
zero generically and one at zero. Its alternating term ranks would be
locally constant, contradicting connectedness of
\(\operatorname{Spec}k[t]\). Meanwhile \(k[t]/(t)\) has the finite free
resolution \([A\xrightarrow{t}A]\) in degrees \(-1,0\), so it is
perfect. Tensoring that resolution with \(k\) creates a nonzero
degree-\(-1\) Tor group. Derived base change remembers this group;
ordinary pullback of \(F\) does not. Thus perfectness alone does not
remove the need for flatness in the universal ordinary-family
comparison.

\subsection{References and the next use of the
complex}\label{references-and-the-next-use-of-the-complex}

\begin{itemize}
\tightlist
\item
  \textbf{{[}Stacks{]}} The Stacks project authors, \emph{The Stacks
  project}, in its AI Integrated Stacks Project edition: affine
  comparison
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-affine-base-change}{Tag
  02KG}; flat comparison
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-flat-base-change-cohomology}{Tag
  02KH}; base-change Čech complexes
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-base-change-complex}{Tag
  01XM}; proper flat perfect direct images
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html\#coherent-lemma-perfect-direct-image}{Tag
  07VK}. The construction and finite replacement needed here are proved
  in Sections 2--3.
\item
  The exact earlier programme foundation is
  \href{../derived-categories-and-sheaf-operations/derived-pullback-and-pushforward.html}{\emph{Derived
  pullback and pushforward}, Theorems 1.1 and 2.2 and Proposition 3.1},
  proving pullback, adjunction and composition. The corresponding
  supplementary open adjunction is
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cohomology.html\#cohomology-lemma-adjoint}{Tag
  079W}, using
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html\#derived-lemma-derived-adjoint-functors}{derived
  adjoint functors} and its
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html\#derived-lemma-pre-derived-adjoint-functors-general}{general
  localization proof}. Composition of derived pullbacks is
  \href{https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cohomology.html\#cohomology-lemma-derived-pullback-composition}{cohomology,
  lemma-derived-pullback-composition}.
\item
  These open reference texts retain GNU FDL 1.2. This exposition, its
  finite replacement proof and exercise solutions are independently
  written CC0. The next lesson studies ranks of the differentials to
  prove semicontinuity and the precise criterion for a cohomology
  base-change map to be an isomorphism.
\end{itemize}
\end{document}