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Use the kernel squares in an abelian category argument. If $b$ is monic and $r:X\to A$, $s:X\to K^{\prime }$ satisfy $ar=f^{\prime }s$, then $bgr=g^{\prime }ar=g^{\prime }{f}^{\prime }s=0$, so $gr=0$. The kernel in a category property gives a unique $t:X\to K$ with $ft=r$. The equation $f^{\prime }kt=aft=ar=f^{\prime }s$ and monicity of $f^{\prime }$ give $kt=s$. This proves the left square is a pullback in a category.

Now suppose the right square is a pullback, without imposing the earlier monicity hypothesis on $b$. The pair $(f^{\prime },0):K^{\prime }\to A^{\prime }{\times }_{B^{\prime }}B$ gives a unique $r:K^{\prime }\to A$ with $ar=f^{\prime }$ and $gr=0$. Factor $r=f\ell$ through the kernel. Then $f^{\prime }k\ell =af\ell =ar=f^{\prime }$, so $k\ell =1_{K^{\prime }}$. Also $rk$ and $f$ have the same two pullback projections, hence $rk=f$. Thus $f\ell k=f$ and $\ell k=1_K$. Therefore $k$ is an isomorphism. These arguments use only the relevant kernels, zero arrows and pullback properties; the abelian hypothesis supplies them.

For an affine variety $V\subseteq {\mathbb{A}}_k^N$, one construction defines ${\mathcal{O}}_V(U)$ as the functions $U\to k$ which locally have the form $p/q$, where $p,q$ are elements of the coordinate ring $k[V]$ and $q$ does not vanish on the neighbourhood in question. Restrictions are restrictions of functions, and locality gives the sheaf axioms. In particularThe second formula identifies its local rings.

In the classical convention, an algebraic variety over $k$ is an irreducible ringed space with a sheaf of $k$-valued functions, admitting a finite open cover by spaces isomorphic to affine varieties, and satisfying the separated variety condition that its diagonal is closed in $X\times X$. Its topology is Noetherian. One can instead allow reducible reduced varieties; the arguments below still work with the denominator-clearing version of the localization argument. A morphism of varieties is a continuous map whose pullback takes local regular functions to regular functions, equivalently a morphism of locally ringed $k$-spaces. The maps on local rings are local because a function nonzero at the image point stays nonzero at the source point.

An affine variety supplies its own finite affine cover, is Noetherian, and is separated: its diagonal in $V\times V$ is cut out by the differences of corresponding coordinates. A regular map between affine varieties pulls coordinate functions back to regular functions. It is continuous because the inverse image of any polynomial zero set is the zero set of the pulled-back regular functions; on affine charts such zero sets are closed. Pulling back a locally represented fraction $p/q$ gives a regular fraction wherever its denominator is nonzero. Hence a regular map is a morphism of varieties in this definition.

For a morphism $\varphi :X\to Y$ which is an isomorphism over each member $U_i$ of an open cover of $Y$, every fibre contains exactly one point. Thus $\varphi$ is bijective. Its inverse is continuous and regular on every $U_i$, since there it is the given inverse of the local isomorphism. These inverses agree on overlaps, being inverses of the same map. They glue to a global inverse morphism of varieties. Therefore being an isomorphism is local on the target.

Let $A=\Gamma (X,{\mathcal{O}}_X)$, and let $Y$ be affine with coordinate ring $B$. Given a $k$-algebra homomorphism $\alpha :B\to A$, choose a presentation $B=k[y_1,\dots ,y_N]/I$. The maplands in $Y$ because all the equations in $I$ become zero functions. Its coordinates are global regular functions, so it is a morphism on every affine chart of $X$, and therefore globally. On a target neighbourhood where a fraction $b/c$ is defined its pullback is $\alpha (b)/\alpha (c)$, proving that the induced map on global sections is exactly $\alpha$. This also proves independence of the chosen generators and uniqueness. This is the affine-target adjunction for varieties.

For $f\in A$, the natural map $A_f\to \Gamma (X_f,{\mathcal{O}}_X)$ sends $a/f^N$ to that regular function. In the irreducible convention, if $f\ne 0$, $X_f$ is dense, so this map is injective: a global regular function vanishing there vanishes everywhere. Cover $X$ by finitely many affine charts $U_j$. A section $s$ on $X_f$ restricts on $U_j\cap X_f=D_{U_j}(f{|}_{U_j})$ to an element of $\Gamma (U_j,{\mathcal{O}}_X{)}_{f{|}_{U_j}}$. A common power $f^N$ clears all these finitely many denominators. The resulting regular sections on $U_j$ agree on the dense principal open in each overlap, hence agree there altogether and glue to a global section $a$. Thus $s=a/f^N$. If $f=0$, both sides are the zero ring of sections on the empty open. ConsequentlyThis is localization of global sections on a principal open. For reduced reducible varieties, equality on a principal open instead means that a sufficiently large power of $f$ annihilates the difference; finitely many charts and overlap refinements allow one common extra power. The same denominator-clearing proof then gives the stated localization isomorphism without a density assumption.

Now suppose ${\sum }_i{g}_i{f}_i=1$ in $A$ and every $X_{f_i}$ is affine. Each $A_{f_i}=\Gamma (X_{f_i},{\mathcal{O}}_X)$ is a finitely generated $k$-algebra. Select finitely many generators, writing them as $a_{ij}/f_i^{N_{ij}}$. Let $B\subseteq A$ be the $k$-subalgebra generated by all $f_i$, all $g_i$, and all $a_{ij}$. It is finitely generated, andThe inclusion from left to right is immediate, while the chosen generators give the reverse inclusion. This construction does not assume that $A$ was finitely generated in advance.

The reduced algebra $B$ is the coordinate ring of an affine variety $Y$ (irreducible when $X$ is). The map $B\hookrightarrow A$ produces $X\to Y$ by the preceding construction. The sets $D_Y(f_i)$ cover $Y$, since their functions generate the unit ideal already in $B$. Their inverse images are $X_{f_i}$, and on them the map is the isomorphism corresponding to $B_{f_i}=A_{f_i}$. The target-local argument above now provesThis is affineness from a unit-ideal principal affine cover. The printed sets are $X_{f_i}$; the missing index in the TeX aid is not a different hypothesis.

A sheaf of modules $\mathcal{F}$ over ${\mathcal{O}}_X$ is quasi-coherent when locally it admits a presentationwith arbitrary indexing sets $I,J$. On an affine chart, sheafifying the corresponding module presentation and using exactness of localization identifies its cokernel with a sheaf $\tilde{N}$. Thus one may equivalently require that locally on affine charts $\mathcal{F}$ is associated with a module. This does not impose finite generation, which belongs to the stronger coherent condition.

On the affine variety $X$ with coordinate ring $A$, put $M=\Gamma (X,\mathcal{F})$. Refine local module charts to a finite principal open cover $X={\bigcup }_{j}D(g_j)$ such that $\mathcal{F}{|}_{D(g_j)}\cong {\tilde{N}}_j$ for an $A_{g_j}$-module $N_j$. This is possible because principal opens form an open basis and $X$ is quasi-compact. On the overlap $D(g_j{g}_k)$ the sections are the corresponding further localization of $N_j$ or $N_k$. The sheaf gluing axiom gives an equalizerwhere the last map is the difference of restrictions. For any $f\in A$, localize this sequence. Exactness of localization and commutation with finite products give exactly the equalizer for the cover $D(f)={\bigcup }_{j}D(f{g}_j)$. HenceThe isomorphisms are canonical and commute with all basic-open restrictions. Since the sections of $\tilde{M}$ on these opens are $M_f$, they identify the two sheaves:Conversely $\tilde{M}$ is quasi-coherent for every $A$-module $M$, since a free presentation of $M$ gives the required sheaf presentation. Also $\Gamma (X,\tilde{M})=M$, and module homomorphisms sheafify, while a sheaf morphism is determined on every $D(f)$ by the localization of its map on global sections. This proves the affine module-sheaf equivalence.

For a short exact sequence of quasi-coherent sheaves, the sequence of their stalks is exact. Under this equivalence the stalk at $x$ is the localization of the global-section module at ${\mathfrak{m}}_x$. Exactness of module sequences can be checked at all maximal ideals, by localization detects zero elements applied to their homology, so the global-section modules also form a short exact sequence. Therefore global sections are exact on quasi-coherent sheaves over an affine variety. This is stronger than the left exactness of the global section functor on arbitrary sheaves, and follows from localization, not from assuming the cohomology vanishing still to be proved.

For an open inclusion $j:U\hookrightarrow X$, the requested sheaf isRestriction gives $\mathcal{F}\to {}_U\mathcal{F}$. This direct image from an open restriction is not extension by zero: its stalks outside $U$ may be nonzero.

The locally vanishing principle for sheaf cohomology says that a class $\xi \in H^i(X,\mathcal{F})$, $i>0$, is killed on a suitable neighbourhood of every point. Under the stated hypothesis that basis opens and their finite intersections have zero cohomology in degrees $1,\dots ,i-1$, these neighbourhoods can be chosen in that basis so that the image of $\xi$ underis zero. In particular its restriction in $H^i(U,\mathcal{F}{|}_U)$ is zero. This is a statement about individual classes; it does not assert that every locally vanishing class is already globally zero.

Here is a noncircular proof of vanishing of quasi-coherent cohomology on an affine scheme, applied to the variety. Induct on $i>0$, simultaneously for every affine variety and quasi-coherent sheaf. Assume all lower positive degrees vanish. The principal-open basis is closed under finite intersections, so it satisfies the principle's hypothesis. Given $\xi \in H^i(X,\mathcal{F})$, choose a finite principal cover $\mathcal{U}=(D(f_j))$ on which it restricts to zero.

Applying the Čech cochain complex to a flasque resolution gives the Čech lifting below the first possible local cohomology degree comparison segment, under lower-degree vanishing on all intersectionsFor completeness, the double complex has terms ${\check{C}}^p(\mathcal{U},{\mathcal{I}}^q)$. Its augmented rows are exact because each ${\mathcal{I}}^q$ is flasque, so its total cohomology is the cohomology of global sections of the resolution. The vertical cohomology on intersections in degrees $0<q<i$ is zero. Equivalently, start with local primitives of a cocycle representing $\xi$, take their differences on pairwise overlaps, and solve successively for primitives of those differences in degrees $i-1,i-2,\dots ,1$. The last difference is a Čech $i$-cocycle with values in $\mathcal{F}$. This identifies the kernel of the restriction map with the displayed Čech group. No vanishing in degree $i$ on the intersections has been assumed.

It remains a module calculation. Write $\mathcal{F}=\tilde{M}$. The augmented Čech complex isSince the $D(f_j)$ cover $X$, the $f_j$ generate the unit ideal. This complex is exact. To see it algebraically, use alternating cochains, with repeated indices giving zero. For a cocycle $c$, choose $N$ large enough to clear every $f_j$-denominator in $f_j^N{c}_{jI}$, as well as the finitely many cocycle relations; vanishing criterion in a module localization allows a further power to clear relations which initially hold only after localization. Since the ideal $(f_j^N)$ is still the unit ideal, choose $a_j$ with ${\sum }_j{a}_j{f}_j^N=1$. Defineusing those cleared representatives in $M_{f_I}$. The cocycle identity gives $\delta b=({\sum }_j{a}_j{f}_j^N)c=c$. The same argument with the augmentation gives gluing and uniqueness in degree zero. This is exactness of the unit-ideal localization Čech complex.

Thus ${\check{H}}^i(\mathcal{U},\mathcal{F})=0$, so the class $\xi$, whose restrictions were zero, is zero. The induction starts at $i=1$, when the lower-degree condition is empty. We have proved

A monomorphism is a morphism $m:X\to Y$ such that $mu=mv$ implies $u=v$ for every pair $u,v:Z\to X$. Dually, an epimorphism $e:X\to Y$ satisfies $ue=ve\Rightarrow u=v$. A regular epimorphism is a coequalizer of some parallel pair. An isomorphism has a two-sided inverse.

For a product in a category, write ${\Delta }_A:A\to A\times A$ for its categorical diagonal. Since ${\pi }_1{\Delta }_A=1_A$, equality ${\Delta }_{A}u={\Delta }_{A}v$ gives $u={\pi }_1{\Delta }_{A}u={\pi }_1{\Delta }_{A}v=v$. Thus the diagonal is a monomorphism, indeed a split monomorphism.

For $A\in \mathcal{C}$, the comma category $(A\downarrow G)$ has objects $(D,u)$ with $D\in \mathcal{D}$ and $u:A\to GD$. A morphism $(D,u)\to (D^{\prime },u^{\prime })$ is $h:D\to D^{\prime }$ satisfying $Ghu=u^{\prime }$. Identities and composition are those of $\mathcal{D}$, and functoriality of $G$ verifies the condition under composition.

The Freyd general adjoint functor theorem states: if $\mathcal{D}$ is a locally small category with all small categorical limits, and $\mathcal{C}$ is locally small, then $G:\mathcal{D}\to \mathcal{C}$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition. The latter means that for each $A$ there is a set-indexed family $u_i:A\to G{D}_i$ such that every $u:A\to GD$ equals $Gh{u}_i$ for some $i$ and some $h:D_i\to D$.

For necessity, use the standard result that a right adjoint preserves categorical limits. If $F⊣G$, the singleton family containing the unit of an adjunction ${\eta }_A:A\to GFA$ is a solution set, since transposition gives $u=Gh{\eta }_A$ for a unique $h:FA\to D$.

For sufficiency, use the following standard limit fact: if $\mathcal{D}$ is complete and $G$ preserves limits, the projection $(A\downarrow G)\to \mathcal{D}$ creates small limits. Indeed, a compatible family $A\to G{D}_j$ induces a unique arrow into $G(\lim D_j)$, and this makes the underlying limit a limit in the comma category. The comma category is locally small because each of its hom-sets is a subset of a hom-set of $\mathcal{D}$. Its solution family is a weakly initial set.

We prove the remaining initial-object lemma for complete categories with a weakly initial set. In any locally small category $\mathcal{K}$ with all small limits and a weakly initial set $(K_i)$, form $W={\prod }_i{K}_i$. It is weakly initial: for any $X$, some $K_i\to X$ exists and may be composed with the projection $W\to K_i$. The empty family cannot be weakly initial in a nonempty complete category, which has a terminal object.

The set $\mathrm{End}(W)$ is small. Form a simultaneous equalizer $e:E\to W$ of every endomorphism of $W$ and $1_W$; thusThis equalizer exists by completeness, for example as the equalizer of two maps $W\rightrightarrows W^{\mathrm{End}(W)}$. The object $E$ is still weakly initial, since it maps to $W$.

Given $a,b:E\to X$, take their equalizer $j:Y\to E$. Weak initiality of $W$ gives $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, we have $ejte=e$, and cancellation of the monomorphism $e$ gives $jte=1_E$. Thus $j$ is a split epimorphism as well as a monomorphism, so it is an isomorphism. From $aj=bj$ follows $a=b$. There is at least one map $E\to X$ by weak initiality, so $E$ is initial.

Apply this lemma to every $(A\downarrow G)$ and choose its initial object $(FA,{\eta }_A)$. For $v:A\to A^{\prime }$, initiality gives the unique $Fv:FA\to F{A}^{\prime }$ satisfying $GFv{\eta }_A={\eta }_{A^{\prime }}v$. Uniqueness proves the functor laws. The same initiality gives natural bijectionsHence $F⊣G$, completing the theorem without invoking another adjoint functor theorem.

An isomorphism $f:A\to B$ is a morphism with an inverse $f^{-1}:B\to A$ satisfying $f^{-1}f=1_A$ and $f{f}^{-1}=1_B$. A groupoid is a category in which every morphism is an isomorphism.

With the given one-sided inverses,Consequently $f$ is an isomorphism with $f^{-1}=g=h$. This argument uses only the category axioms.

The minimal deterministic finite automaton is unique up to an isomorphism preserving the initial state, transitions and accepting states. We use complete deterministic finite automata over the fixed alphabet; the PDF's abbreviation FDA has this meaning. State names themselves cannot be unique.

For words $u,v\in {\Sigma }^{\ast }$, introduce the Myhill-Nerode equivalenceIt is an equivalence relation, and appending the same letter to equivalent words preserves it: a suffix $w$ after $ua$ is the suffix $aw$ after $u$. Since $L$ is a regular language, take any recognizing deterministic finite automaton. Words reaching the same state are equivalent, since every further suffix gives the same computation. Thus ${\sim }_L$ has finite index.

The canonical residual automaton has one state $[u]$ for each class, initial state $[ϵ]$, transition $[u]\stackrel{a}{\to }[ua]$, and accepting states those with $u\in L$. These choices are well-defined by Myhill-Nerode equivalence. Induction on the input length shows that reading $u$ reaches $[u]$, so the deterministic finite automaton recognizes $L$, and all its states are accessible states of a deterministic finite automaton. The same state can be described by the left quotient of a formal languageTwo states are different precisely when some suffix distinguishes their acceptance behavior.

Every recognizing deterministic finite automaton has at least as many states as there are classes: pick a representative from each class; two different representatives cannot reach the same state. The canonical residual automaton attains this bound, and therefore is a minimal deterministic finite automaton.

For uniqueness, let $\mathcal{A}$ be any minimal deterministic finite automaton. All its states are accessible, since removing inaccessible states leaves a complete recognizing deterministic finite automaton with fewer states. Map a state reached by $u$ to $[u]$. This is well-defined because two words reaching the same state are equivalent. It is surjective because every class has a representative. Both sets have the minimal number of states, so it is a bijection. It preserves the initial state and transitions by construction, and preserves accepting states by taking the empty word as the suffix. This is the required isomorphism with the canonical residual automaton, provingThe empty word is included throughout, and empty or universal regular languages have the corresponding one-state complete deterministic finite automata.

Every countable structure in a countable first-order language has a Scott sentence whose countable models are precisely its isomorphic copies. In particular two countable structures with the same countable infinitary logic sentences are isomorphic. Countability of both structures is essential to the back-and-forth conclusion; the sentence need not exclude uncountable models.

Let $M$ be a nonempty countable structure in a countable first-order language. The countable infinitary logic $L_{{\omega }_1,\omega }$ permits countable logical conjunctions and logical disjunctions, but only finite strings of quantifiers and finitely many free variables in each formula. We construct a Scott formula ${\varphi }_{\bar{a}}^{\alpha }(\bar{x})$ for every finite tuple $\bar{a}$ from $M$ and every countable ordinal $\alpha$.

At stage zero, let ${\varphi }_{\bar{a}}^0$ be the conjunction of all atomic formulas true of $\bar{a}$ and the negations of all atomic formulas false of $\bar{a}$. Include atomic formulas involving arbitrary terms and constants, not just relation symbols applied directly to variables. There are only countably many such formulas. This complete atomic description ensures that matching tuples determine a partial isomorphism of structures.

At successors putAt a nonzero limit ordinal $\lambda <{\omega }_1$, put ${\varphi }_{\bar{a}}^{\lambda }={\bigwedge }_{\beta <\lambda }{\varphi }_{\bar{a}}^{\beta }$. All these are formulas of countable infinitary logic: each indexing set is countable, and the free variables are only the fixed finite tuple. Rename bound variables when necessary. Transfinite induction also shows $M\models {\varphi }_{\bar{a}}^{\alpha }(\bar{a})$.

For tuples of the same length within $M$, define $\bar{a}{\equiv }_{\alpha }\bar{b}$ by $M\models {\varphi }_{\bar{a}}^{\alpha }(\bar{b})$. Induction identifies this with the usual symmetric back-and-forth method equivalence: the tuples have the same atomic description at stage zero, and at a successor every one-element extension on either side has a matching extension at the previous stage. Thus these are decreasing equivalence relations, simultaneously for every finite tuple length.

There are only countably many pairs of finite tuples in $M$. A pair can cease to be equivalent at most once. The supremum of the first separation stages of all pairs that separate below ${\omega }_1$ is a countable ordinal. Choose a countable $\alpha$ at least that supremum. No pair can first separate at $\alpha +1$, soThis justifies stabilization without assuming that all tuples stabilize at one predetermined finite stage.

Now form the following Scott sentence, where the case $n=0$ has no displayed variables or quantifiers:It is a sentence of countable infinitary logic, because the family of all finite tuples is countable. The stabilization above and the truth of ${\varphi }_{\varnothing }^{\alpha }$ show $M\models {\sigma }_M$.

Suppose a countable structure $N$ satisfies ${\sigma }_M$. Start with the empty matching tuples, and maintain $N\models {\varphi }_{\bar{a}}^{\alpha }(\bar{b})$. The corresponding conjunct of ${\sigma }_M$ upgrades this to $N\models {\varphi }_{\bar{a}}^{\alpha +1}(\bar{b})$. The existential conjuncts extend the match by any specified element of $M$. The universal-disjunction conjunct extends it by any specified element of $N$. The atomic formula information makes a new element on one side match a new element on the other, and makes repeated elements agree with their earlier matches.

Enumerate both structures and alternate these two extension steps, including the least element not yet covered at each step. The union is a bijection preserving and reflecting every atomic formula, hence an isomorphism; for function symbols, eventually include both a tuple and the value of its function term to see explicitly that the function is preserved. For finite structures the same construction stops when both are covered. Conversely every isomorphic copy of $M$ satisfies ${\sigma }_M$, since isomorphisms preserve formulas of countable infinitary logic by induction on their construction. ThereforeIf countable $M,N$ have the same $L_{{\omega }_1,\omega }$ sentences, $N$ satisfies this Scott sentence of $M$ and is isomorphic to $M$. This proves Scott isomorphism theorem.

The opening change-of-rings tensor quotient sends $m{\otimes }_{R}n$ to $m{\otimes }_{A}n$. The target pairing is bilinear and $R$-balanced because $\theta (r)m{\otimes }_{A}n=m{\otimes }_A\theta (r)n$. On the source, let $A$ act through the first factor; commutativity makes that action compatible with the $R$-balancing relations, and the map is $A$-linear. It is surjective: the target is the quotient imposing the additional relations $am{\otimes }_{R}n=m{\otimes }_{R}an$ for every $a\in A$. The construction commutes with module homomorphisms in both variables.

The pullback-direct-image adjunction unit is obtained locally by pulling a section $h\in \mathcal{H}(V)$ to ${\varphi }^{-1}V$ and sending it to $1\otimes h$ in ${\varphi }^{\ast }\mathcal{H}({\varphi }^{-1}V)$. These maps respect restrictions and the ${\mathcal{O}}_Y$-actions, hence defineFor $\mathcal{H}={\mathcal{O}}_Y$, multiplication identifies ${\varphi }^{\ast }{\mathcal{O}}_Y$ with ${\mathcal{O}}_X$, and ${\eta }_{{\mathcal{O}}_Y}$ is precisely the structure morphism ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$.

To construct the direct-image tensor comparison, on $V\subseteq Y$ send $s\otimes t$, with $s\in \mathcal{F}({\varphi }^{-1}V)$ and $t\in \mathcal{G}({\varphi }^{-1}V)$, to its tensor section of $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$. This pairing is ${\mathcal{O}}_Y(V)$-balanced through ${\varphi }^{\#}$ and is compatible with restrictions. The universal property of the tensor product of modules and sheafification therefore giveApply this with $\mathcal{G}={\varphi }^{\ast }\mathcal{H}$ after tensoring the unit ${\eta }_{\mathcal{H}}$ with ${\varphi }_{\ast }\mathcal{F}$. The composite is the map in the projection formula for sheaves. If $\mathcal{H}{|}_V\cong {\mathcal{O}}_V^r$, then ${\varphi }^{\ast }\mathcal{H}{|}_{{\varphi }^{-1}V}\cong {\mathcal{O}}_{{\varphi }^{-1}V}^r$, and the composite identifies withthe identity on the $r$ components. Thus it is an isomorphism for a locally free sheaf of finite rank. Neither $\mathcal{F}$ nor $\mathcal{G}$ was assumed quasi-coherent.

Let $Y$ be the affine variety with coordinate ring $A$. Its principal opens $D(f_i)$ cover $Y$ because ${\sum }_i{g}_i{f}_i=1$. On $X_{f_i}$, the localization of global sections on a principal open identifies its coordinate ring with $A_{f_i}$, giving an isomorphismOn an overlap the two maps are induced by the same elements of $A$, or equivalently by the same identification with $A_{f_i{f}_j}$, so they agree. Glue them to $\varphi :X\to Y$. The inverses agree on the overlaps as well and glue to its inverse. At a closed point $P$, this is the map corresponding to the evaluation ring homomorphism $A\to k$, $a\mapsto a(P)$. This proves the cohomological criterion for affineness by an explicit global isomorphism.

For $U_{i_0\dots i_q}=U_{i_0}\cap \cdots \cap U_{i_q}$, the Čech cochain complex has groupsTerms in ${\delta }^2$ cancel in pairs, and Čech cohomology is $\ker \delta /\mathrm{im}\delta$. A degree-zero cocycle is exactly a family of compatible local sections. The sheaf gluing axiom gives their unique global section, proving ${\check{H}}^0(\mathcal{U},\mathcal{F})\cong \mathcal{F}(X)$. If the cover has affine finite intersections, a quasi-coherent sheaf has no higher cohomology on those intersections by vanishing of quasi-coherent cohomology on an affine scheme. The acyclic cover theorem then identifies all Čech groups with sheaf cohomology. In particular, a finite affine open cover of a separated variety has this property.

For the sheaf of units of the structure sheaf, a multiplicative degree-one cocycle is a family $g_{ij}\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$ satisfying $g_{ij}{g}_{jk}=g_{ik}$, with $g_{ji}=g_{ij}^{-1}$. It glues trivial rank-one free modules into an invertible sheaf. Changing the local frames multiplies $g_{ij}$ by a coboundary, and two sets of transition data give isomorphic line bundles precisely when their cocycles differ this way. Tensoring line bundles multiplies their cocycles. Thus line bundles trivialized by an open cover give the group isomorphismFor the remaining arguments, work over the algebraically closed ground field. On the irreducible variety $V$, put $\mathcal{Q}={\mathcal{K}}^{\ast }/{\mathcal{O}}_V^{\ast }$, a quotient of sheaves of abelian groups. A global section of $\mathcal{Q}$ is locally represented by rational functions $f_i$ whose ratios are regular units. The corresponding unit cocycle $f_j/f_i$ defines an invertible sheaf. A single global rational function has trivial cocycle. Conversely, every line bundle has a nonzero rational section: choose a nonzero vector in its one-dimensional fibre at the generic point and express it in local frames. This supplies such local $f_i$. If the associated line bundle is trivial, changing frames makes all $f_i$ restrictions of one rational function. Thereforeis exact. This is the Cartier-divisor description of the Picard group. The sheaf of nonzero rational functions on an irreducible variety is flasque, since all restrictions between nonempty open sets are the identity on $k(V{)}^{\ast }$. Apply the long exact sequence in sheaf cohomology to $1\to {\mathcal{O}}_V^{\ast }\to {\mathcal{K}}^{\ast }\to \mathcal{Q}\to 1$. Since $H^1(V,{\mathcal{K}}^{\ast })=0$, its connecting map has exactly the cokernel just computed, provingFinally the Segre description of a smooth quadric surface identifies $V$ with ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The two rulings of a smooth quadric surface have classes $F_1,F_2$ generating $\mathrm{Pic}(V)\cong {\mathbb{Z}}^2$. For completeness, remove one line in each ruling: the remaining chart is the affine plane, with factorial coordinate ring $k[s,t]$ and trivial divisor class group. The localization sequence for the divisor class group makes $F_1,F_2$ generators, and their degrees on the two ruling lines prove independence. The hyperplane class, and hence the conic $C$, has bidegree $(1,1)$. By Picard-group localization on a smooth variety, the Picard group of a smooth affine quadric surface isExplicitly, the restriction of ${\mathcal{O}}_V(1,0)$ is nontrivial: if it were trivial on $U$, its rational trivialization would have divisor supported on $C$, forcing $(1,0)$ to be an integer multiple of $(1,1)$, which is impossible.

On an affine chart $\mathrm{Spec}A\subseteq V$, the module of Kähler differentials ${\Omega }_{A/k}$ is generated by symbols $da$ subject to $k$-linearity and $d(ab)=adb+bda$. It represents $k$-derivations. These modules commute with localization, so their associated quasi-coherent sheaves glue to the Kähler differential sheaf ${\Omega }_{V/k}^1$. It is coherent: if $A=k[x_1,\dots ,x_N]/I$ and $I=(f_1,\dots ,f_s)$, then it has the finite presentation of a moduleFor a closed point $P$, put $R={\mathcal{O}}_{V,P}$ with maximal ideal $\mathfrak{m}$ and residue field $k$. The Zariski tangent space is ${\mathrm{Hom}}_k(\mathfrak{m}/{\mathfrak{m}}^2,k)$, equivalently ${\mathrm{Der}}_k(R,k)$. A derivation kills constants and ${\mathfrak{m}}^2$, so it factors through $a\mapsto a-a(P)$; conversely every linear functional on $\mathfrak{m}/{\mathfrak{m}}^2$ defines such a derivation by the product rule. The universal property of Kähler differentials therefore identifies this space withIn particular, ${\Omega }_{V,P}^1/{\mathfrak{m}}_P{\Omega }_{V,P}^1\cong {\mathfrak{m}}_P/{\mathfrak{m}}_P^2$, the algebraic cotangent space.

A point is a smooth point of a variety when its local ring is a regular local ring; over this algebraically closed field this says ${\dim }_k{T}_{V,P}=n=\dim V$. Tensor the finite presentation of a module above with $k(P)$. The tangent space is the kernel of the Jacobian matrix $J(P)$, hence has dimension $N-\mathrm{rank}J(P)$. This proves the Jacobian criterionFor a smooth irreducible variety $V$, choose at each point an invertible $(N-n)$-minor and shrink the affine chart so that it remains invertible. Its relations eliminate $N-n$ of the generators of ${\Omega }_{A/k}$, yielding a surjection $A^n\to {\Omega }_{A/k}$. At the generic point, its target has dimension $n$: over a perfect ground field, a separating transcendence basis of the function field has differentials forming a basis. Equivalently this follows from the assumed density of the smooth locus. The kernel therefore becomes zero over the fraction field of the integral domain $A$. As a submodule of $A^n$, it is a torsion-free module, so it is already zero. Thus these maps give local isomorphisms with ${\mathcal{O}}_V^n$, proving local freeness of differentials on a smooth variety with rank $n$.

For an affine chart $\mathrm{Spec}A\subseteq V$, write $W\cap \mathrm{Spec}A=\mathrm{Spec}(A/I)$ and $\mathcal{M}=\tilde{M}$. The restriction of a module sheaf to a closed subvariety is $\widetilde{M{\otimes }_{A}A/I}=\widetilde{M/IM}$. The Conormal exact sequence for Kähler differentials isIt follows from the generators and relations: passing to $A/I$ imposes precisely the additional relations $df=0$ for $f\in I$. The first map is well defined because $d(I^2)$ lies in $I{\Omega }_{A/k}$. Glue these exact module sequences, using exactness of localization, to obtainIn the final assertion, interpret locally principal subvariety as a proper local hypersurface. Its ideal on each chart is $(f)$ with $f\ne 0$. Since $V$ is an irreducible variety and reduced, $f$ is a non-zero-divisor, and $A/(f)\to (f)/(f^2)$, $\bar{a}\mapsto af$, is an isomorphism. These local rank-one descriptions make the conormal sheaf ${\mathcal{I}}_W/{\mathcal{I}}_W^2$ invertible.

Because $W$ is not contained in the singular locus of $V$, there is a dense open subset of $W$ where both $V$ and $W$ are smooth varieties. At a closed point there, $\dim W=\dim V-1$ by the Krull principal ideal theorem. The tangent description $T_{W,P}=\ker (df:T_{V,P}\to k)$ then forces $df\ne 0$. Hence the conormal map is injective at the generic point of $W$. Its kernel is a subsheaf of a line bundle on the integral variety $W$, so it is a torsion-free sheaf; a torsion-free sheaf with zero generic fibre is zero. This proves conormal injectivity for a generically smooth Cartier divisor. If zero equations were allowed in the phrase locally principal, $W=V$ would be a counterexample to invertibility; the proper-hypersurface convention is essential.