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Let $D$ be the ambient Levi-Civita connection and $\nabla$ the one for the induced metric on $M$. The Gauss formula iswhere the second fundamental form $II(X,Y)=(D_{X}Y{)}^{\perp }$ is normal. The tangential part of $D$ is metric-compatible and torsion-free, so the existence and uniqueness of the Levi-Civita connection identifies it with $\nabla$.

For a normal field $\xi$, the Weingarten formula defines the shape operator in a normal direction $A_{\xi }$ and the normal connection byDifferentiate $⟨\xi ,Y⟩=0$ using metric compatibility to getThe symmetry of the second fundamental form follows from the ambient torsion-free condition:since the Lie bracket of vector fields tangent to $M$ is tangent. Thereforeso every normal-direction shape operator is self-adjoint.

The paper uses $R(X,Y)=D_{[X,Y]}-[D_X,D_Y]$, so use the Gauss equation with reversed curvature convention. Applying the Gauss formula and Weingarten formula twice givesSubtracting in the stated curvature order yieldsPair with a tangent field $W$ and use the defining relation for the shape operator in a normal direction:This is the Gauss equation in the supplied sign convention. If one defines curvature by $[D_X,D_Y]-D_{[X,Y]}$ instead, the two quadratic terms have the opposite signs.

For an orthonormal frame $X,Y$ on a surface, positive Gaussian curvature with the convention here is $K=⟨R^M(X,Y)X,Y⟩$. If the ambient space is Euclidean and $\xi$ is a unit normal, the equation becomesThe Koszul formula determines $\nabla$ from the induced metric, and hence determines its Riemann curvature tensor and $K$. Therefore the Gaussian curvature depends only on that metric, even though its expression as a product of principal curvatures appears extrinsic. This is Theorema Egregium.

An orthogonal structure on a real vector bundle $E$ of rank $r$ is a choice of local frames whose transition matrices lie in the orthogonal group $O(r)$. In such frames declareOrthogonal transitions preserve this expression, so it defines a smooth positive-definite fiber metric. Conversely, a fiber metric turns any local frame into a smooth orthonormal frame by the Gram-Schmidt process; the resulting transition matrices are orthogonal. The two constructions are inverse, giving the required equivalence.

An orthogonal local trivialization $E{|}_U\cong U\times {\mathbb{R}}^r$ identifies each fiber isometrically with standard Euclidean space. Equivalently, its coordinate frame satisfies $h(e_i,e_j)={\delta }_{ij}$. This is local and does not assert the existence of a global frame.

Every real vector bundle over a smooth manifold admits a fiber metric. Choose a trivializing open cover and local Euclidean metrics $h_i$. The partition of unity theorem for a Hausdorff second-countable smooth manifold supplies nonnegative smooth functions ${\rho }_i$, with locally finite supports contained in the respective cover members, such that ${\sum }_i{\rho }_i=1$. Setextending each weighted metric by zero off its chart. Local finiteness makes this smooth. For every nonzero fiber vector $v$, at least one positive weight gives $h(v,v)={\sum }_i{\rho }_i{h}_i(v,v)>0$. Thus $h$ is a fiber metric and provides an orthogonal structure.

One definition of an orthogonal connection is that its covariant derivative preserves the fiber metric:for all vector fields $X$ and smooth sections of a vector bundle $s,t$. Another is that its parallel transport along every smooth curve is an isometry between the endpoint fibers.

For the first implication, take parallel sections $s,t$ along a curve. Their covariant derivatives vanish, so the displayed identity makes $h(s,t)$ constant: parallel transport preserves a fibre metric. Conversely, if parallel transport preserves the inner product, choose parallel extensions of arbitrary initial fiber vectors along a curve with prescribed initial tangent $X$. Differentiating their constant inner product gives $({\nabla }_{X}h)(s,t)=0$ at the initial point. Since $X,s,t$ were arbitrary, the connection is metric-compatible.

The equivalent local description is that the connection matrix in an orthonormal frame is skew-symmetric. Writing $\nabla e_j={\sum }_i{A}^i{}_j{e}_i$ in an orthogonal trivialization, differentiate $h(e_i,e_j)={\delta }_{ij}$ to obtainConversely this matrix identity gives the metric-preservation formula by expanding arbitrary sections. It is the orthogonal group connection description of the same condition.

A $d$-dimensional embedded submanifold $F\subseteq M^m$ has the subspace topology and, near each of its points, a manifold chart identifying it with a coordinate slice ${\mathbb{R}}^d\times \{0\}$. These slice charts for an embedded submanifold give its smooth structure, and its inclusion is a smooth embedding.

A regular value $q\in N^n$ of a smooth map $f:M^m\to N^n$ is one for which $d{f}_p:T_{p}M\to T_{q}N$ is surjective at every $p\in f^{-1}(q)$. If the preimage is nonempty this forces $m\ge n$.

The inverse function theorem used here says that if $G:U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^m$ is smooth and $D{G}_p$ is an invertible matrix, then its restriction to a suitable neighbourhood of $p$ is a diffeomorphism onto an open neighbourhood of $G(p)$, with smooth inverse.

To prove the regular level set theorem, fix $p\in f^{-1}(q)$ and choose coordinates centred at $p,q$. The coordinate representative $\tilde{f}$ has a rank-$n$ Jacobian matrix. After permuting domain coordinates, its first $n$ columns form an invertible matrix. DefineIts Jacobian matrix is block triangular with invertible diagonal blocks, so the inverse function theorem makes $G$ a local coordinate change. In these coordinates, $f^{-1}(q)$ is precisely the slice with first $n$ coordinates zero. Thus it is an embedded submanifold of dimension $m-n$, withAn empty preimage is harmless; the local assertion is then vacuous.

Three equivalent formulations of orientability of a smooth manifold are: an oriented atlas whose transition Jacobian determinants are positive; a smoothly varying orientation of a vector space on each tangent space; and a nowhere-vanishing smooth top-degree differential form. An oriented atlas gives compatible oriented coordinate frames. A partition of unity glues their positive local top forms, and a nonzero top form declares a frame positive precisely when it evaluates positively on that frame. These constructions prove the equivalence.

For the orientability of a regular fibre, write $F=f^{-1}(q)$. The derivative gives a short exact sequence of vector bundlesThis is the reason the normal bundle of a regular fibre is trivial. Choose any fixed positive basis $b_1,\dots ,b_n$ of $T_{q}N$ and an ambient orientation on $M$. Declare a frame $v_1,\dots ,v_{m-n}$ of $T_{p}F$ positive whenis positive in $T_{p}M$, where $d{f}_p({\tilde{b}}_i)=b_i$. Changing a lift adds a tangent vector and leaves the determinant sign unchanged. Local smooth lifts exist because $df$ is surjective, so this rule varies smoothly and agrees on overlaps. It gives a global orientation on $F$. Only an orientation of the single vector space $T_{q}N$ was chosen; $N$ itself need not be orientable.

The exterior derivative is an $\mathbb{R}$-linear operator $d:{\Omega }^r(M)\to {\Omega }^{r+1}(M)$ which agrees with the differential of a smooth function on degree zero, satisfies $d^2=0$, and obeys the graded Leibniz rule

For the uniqueness of the exterior derivative from its axioms, first note that these axioms force locality. If $\alpha$ vanishes near $p$, choose a smooth bump function $\chi$ equal to one near $p$ and supported where $\alpha =0$. Then $0=d(\chi \alpha )=d\chi \wedge \alpha +\chi d\alpha$ gives $d\alpha (p)=0$. Thus $d\alpha (p)$ depends only on the form near $p$.

In a coordinate chart write $\alpha ={\sum }_I{a}_{I}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_r}$. Since $d(d{x}^i)=d^2{x}^i=0$, the axioms forceThis also proves existence. Define $d$ by that formula in each chart. The product rule for partial derivatives gives the graded Leibniz rule, and symmetry of mixed partial derivatives against antisymmetry of the wedge product of differential forms gives $d^2=0$. On overlaps it still agrees with the differential of every function, by the chain rule, so the preceding uniqueness argument in the second coordinate system makes the formulas agree. They therefore glue to a global exterior derivative.

For a differential one-form $\omega ={\sum }_i{a}_{i}d{x}^i$, the exterior derivative of a one-form evaluated on vector fields follows directly:In $X(\omega (Y))-Y(\omega (X))$, the additional terms are ${\sum }_i{a}_i[X(Y^i)-Y(X^i)]=\omega ([X,Y])$, where the bracket is the Lie bracket of vector fields. Hence

The de Rham cohomology iswith zero denominator in degree zero. The Poincaré lemma states that a closed differential form of positive degree on a star-shaped open subset of Euclidean space is an exact differential form; consequently every closed positive-degree form is locally exact on a smooth manifold.

For the suspension isomorphism for top de Rham cohomology of spheres, put $U=S^n\setminus \{\text{south pole}\}$ and $V=S^n\setminus \{\text{north pole}\}$. Both are diffeomorphic to ${\mathbb{R}}^n$, while $W=U\cap V$ deformation retracts onto the equatorial $S^{n-1}$. The Mayer--Vietoris sequence for de Rham cohomology comes from the short exact sequence of cochain complexesThe final map is surjective: a partition of unity ${\chi }_U+{\chi }_V=1$ lets a form $\eta$ on $W$ be lifted by extending ${\chi }_V\eta$ to $U$ and $-{\chi }_U\eta$ to $V$ by zero. The Poincaré lemma makes $H^{n-1}(U),H^{n-1}(V),H^n(U),H^n(V)$ vanish when $n>1$. Exactness givesIf $j:S^{n-1}\hookrightarrow W$ is the equatorial inclusion, homotopy invariance of de Rham cohomology makes $j^{\ast }$ an isomorphism. Thereforeis injective, indeed an isomorphism. Concretely, choose local primitives $d{\alpha }_U=\omega {|}_U$, $d{\alpha }_V=\omega {|}_V$, and send $[\omega ]$ to $[j^{\ast }({\alpha }_U-{\alpha }_V)]$. Their difference is closed; changing either primitive changes it by an exact form, so this agrees with the cohomological construction.

Let $S=k[X_0,\dots ,X_n]$ with its usual grading, and put $S(m{)}_d=S_{m+d}$. Define the twisting sheaf on projective space byMore explicitly, on $U_i=D_{+}(X_i)$ its sections are the homogeneous degree-$m$ part of $S_{X_i}$. They are a free rank-one module over $(S_{X_i}{)}_0=k[X_j/X_i:j\ne i]$, with generator $e_i=X_i^m$. These generators make sense for every integer $m$, since $X_i$ is invertible on this chart. Their transition rule isThe ratios are regular units on overlaps and satisfy the cocycle identity, so they glue to an invertible sheaf. In particular $\mathcal{O}(m)\otimes \mathcal{O}(r)\cong \mathcal{O}(m+r)$.

For $n\ge 1$, a global section is a compatible homogeneous degree-$m$ rational expression lying in every $S_{X_i}$. Inside the fraction field of $S$,To justify this, write a fraction in lowest terms in the unique factorization domain $S$. Membership in $S_{X_i}$ forces every irreducible divisor of its denominator to be associated with $X_i$. Since there are at least two nonassociated variables, the denominator must be a unit. ThereforeThe degree-$m$ monomials are indexed by nonnegative integers $a_0,\dots ,a_n$ with sum $m$. The stars and bars count givesHere and below, an all-integer binomial shorthand must use the counting convention $\left(\genfrac{}{}{0ex}{}{a}{n}\right)=0$ when $a<n$, including negative $a$. The polynomial extension of binomial coefficients to negative upper arguments does not give the stated dimensions. In dimension zero, ${\mathbb{P}}^0$ is one point and every ${\mathcal{O}}_{{\mathbb{P}}^0}(m)$ is trivial, so its space of global sections is $k$ for every $m$; this case is treated separately.

For a closed inclusion $i:H\hookrightarrow {\mathbb{P}}^n$, direct image sheaf formation is exact: its stalk is the original stalk at a point of $H$ and zero outside $H$. It preserves flabby sheaves, since restrictions on the target are restrictions between the corresponding open subsets of $H$. Push forward a flasque resolution of $\mathcal{F}$; exactness gives a flasque resolution of $i_{\ast }\mathcal{F}$, and its complex of global sections is the original one becauseTaking cohomology of these identical complexes proves sheaf cohomology under a closed inclusion:This argument applies even without quasi-coherence.

We derive the remaining sheaf cohomology dimensions using the given eventual vanishing, the long exact sequence in sheaf cohomology, and induction on dimension. Multiplication by the linear equation of a hyperplane $H\cong {\mathbb{P}}^{n-1}$ gives the hyperplane exact sequence for twisting sheavesExactness can be checked in each chart by quotienting its polynomial ring by the equation of $H$; multiplication is injective because that equation is a non-zero-divisor. We use the closed-inclusion cohomology identification to replace the cohomology of the final sheaf by that on $H$.

The dimension-zero base is a point, whose global-section functor is exact. Thus all its higher sheaf cohomology is zero. For $n=1$, the relevant long exact sequence isWhen $m\ge 0$, restriction of homogeneous polynomials to the hyperplane point is surjective: after choosing that point as $[1:0]$, the section $X_0^m$ restricts to a generator. Therefore the consecutive $H^1$ groups are isomorphic for $m\ge 0$. The given vanishing for large $m$ propagates down to $m=-1$. For $m<0$, both displayed $H^0$ groups are zero, and the sequence becomesStarting with $H^1(\mathcal{O}(-1))=0$, induction downward gives $h^1(\mathcal{O}(m))=-m-1$ for $m\le -2$. For degrees $i>1$, the point contributes no cohomology to the exact sequence, so consecutive $H^i$ groups are isomorphic for every $m$; the given eventual vanishing makes them all zero. Hence the complete positive-degree formula holds for ${\mathbb{P}}^1$.

Now let $n\ge 2$ and assume the formulas for $H\cong {\mathbb{P}}^{n-1}$. For every $m$, the restrictionis surjective: for $m\ge 0$ extend a polynomial in the hyperplane coordinates to the same polynomial on ${\mathbb{P}}^n$, and for $m<0$ both groups vanish because $H$ has positive dimension. Consequently the exact sequence gives an injectionFor $2\le i\le n-1$, the induction hypothesis says $H^{i-1}(H,{\mathcal{O}}_H(m))=0$, and gives the same injection in degree $i$. For each fixed $m$, composing finitely many injections reaches a sufficiently large twist, whose higher cohomology is zero by the given hypothesis. ThereforeFor $i>n$, the two neighbouring hyperplane groups $H^{i-1}(H,{\mathcal{O}}_H(m))$ and $H^i(H,{\mathcal{O}}_H(m))$ are zero by induction. Consecutive $H^i$ groups on ${\mathbb{P}}^n$ are then isomorphic, and eventual vanishing again makes them zero. This derives vanishing above the dimension too, without needing a special projective-space cohomology theorem.

For top cohomology the exact sequence now reduces toWrite $b_n(m)=h^n({\mathcal{O}}_{{\mathbb{P}}^n}(m))$. The induction hypothesis and eventual vanishing give the recurrenceConsecutive top groups agree for $m\ge -n+1$, so descending from a large twist gives $b_n(m)=0$ for $m\ge -n$. Below this range, iterating the recurrence gives a finite sum:The last equality follows by summing Pascal's identity. These short exact sequences also prove that every group whose dimension is being counted is finite-dimensional. Thus, for $n\ge 1$,With the stated counting convention, the latter is the requested single binomial formula. The base ${\mathbb{P}}^0$ has $h^0=1$ for all twists and no higher cohomology; it is not governed by the negative-twist $H^0$ rule for positive-dimensional projective space.

Finally calculate the canonical line bundle of a smooth variety from its transition functions. On $U_0$ write $t_j=X_j/X_0$ for $1\le j\le n$. On $U_i$, $i>0$, take the coordinates $u_j=X_j/X_i$ in increasing order of $j\ne i$. ThusIn their wedge product, $d{u}_0=-t_i^{-2}d{t}_i$ and
$d{u}_j=t_i^{-1}d{t}_j-t_j{t}_i^{-2}d{t}_i$. Every term involving a second $d{t}_i$ vanishes, so reordering the remaining differentials givesThis computation is valid in every characteristic. Multiply the ordered local generator on $U_i$ by $(-1{)}^i$. The resulting generators ${\eta }_i$ satisfy ${\eta }_i=(X_i/X_0{)}^{-n-1}{\eta }_0$ on $U_i\cap U_0$, and hence the corresponding transition rule on every overlap. They are exactly the transitions of $\mathcal{O}(-n-1)$. Therefore the canonical bundle of projective space is

For a line bundle, Serre duality statesThe intermediate groups on both sides vanish. In the two outer degrees, the top group at twist $m$ has the same dimension as the global-section group at twist $-m-n-1$:The polynomial degree on the right is nonnegative precisely when $m\le -n-1$, and its dimension is $\left(\genfrac{}{}{0ex}{}{-m-1}{n}\right)$. Interchanging $m$ and $-m-n-1$ checks the other outer degree. On ${\mathbb{P}}^0$ both sides are one-dimensional. Thus all the formulas are consistent with Serre duality, which was used only for this final consistency check, not to prove the cohomology formulas.

A quasi-coherent sheaf is a sheaf of modules which locally has a presentation by free module sheaves:where the index sets may be infinite. Equivalently, on affine neighbourhoods it is an affine module sheaf. A coherent sheaf is locally finitely generated, with the additional requirement that the kernel of every map from a finite free module sheaf is locally finitely generated. On a Noetherian scheme, these conditions are equivalent to being locally associated with a finitely generated module.

For an affine variety $X$ with coordinate ring $A$ and an $A$-module $M$, setRestriction on principal open subsets is the natural module localization map. On an arbitrary open subset, sections are families of these stalk values locally represented by a single fraction from a localization. This constructs the affine module sheaf; its global sections are $M$. Choose a free-module presentation $A^{(J)}\to A^{(I)}\to M\to 0$. Exactness of localization and its compatibility with direct sums yieldThus $\tilde{M}$ is a quasi-coherent sheaf, without a finite-generation assumption.

We now prove coherence of an affine module sheaf. If $M$ is finitely generated, take a surjection $A^r\to M$. Since $A$ is a Noetherian ring, its kernel is finitely generated, so $M$ has a finite presentation of a module. More generally, on every principal affine open, the kernel of a map $A_a^s\to M_a$ is a submodule of a finite free module over the Noetherian ring $A_a$, and is finitely generated. On sheafifying, this verifies both parts of the definition of a coherent sheaf.

Conversely, if $\tilde{M}$ is coherent, choose a finite principal-open cover $X={\bigcup }_{i}D(a_i)$ on which it has finitely many generators. Write those local generators as $m_{ij}/a_i^{r_{ij}}$. Their numerators generate a finite submodule $N={\sum }_{i,j}A{m}_{ij}$ of $M$, and $N_{a_i}=M_{a_i}$ for every $i$. To check that $M/N=0$, let $q\in M/N$. Each equality after localization gives an exponent $r_i$ such that $a_i^{r_i}q=0$. The ideal generated by the $a_i^{r_i}$ is the unit ideal: otherwise a maximal ideal containing all of them would give a point outside the principal-open cover. A linear combination of these powers equal to $1$ consequently annihilates $q$. Every $q$ is zero, proving

Next consider a morphism $\varphi :{\mathcal{G}}_1\to {\mathcal{G}}_2$ between quasi-coherent sheaves. On an affine open $V$ of the target variety, use the assumed affine module-sheaf equivalence to write ${\mathcal{G}}_j{|}_V=\widetilde{M_j}$, with $M_j=\Gamma (V,{\mathcal{G}}_j)$. The sheaf morphism is the sheafification of the module map $\psi =\Gamma (V,\varphi )$. Indeed, its action on a section $m/a^r$ of a principal open is forced by module-linearity and restriction to be $\psi (m)/a^r$. Therefore exactness of localization givesThese identities are compatible with restriction, so $(\ker \varphi ){|}_V=\widetilde{\ker \psi }$. Since this holds on an affine cover, the kernel sheaf is quasi-coherent on every variety. This proof of kernels of quasi-coherent sheaf morphisms does not assert that taking global sections is right exact.

For direct image along a morphism of affine varieties, write $Y$ with coordinate ring $A$, $X$ with coordinate ring $B$, and let $\alpha :A\to B$ be the induced ring homomorphism. If $\mathcal{F}=\tilde{M}$ on $X$, regard $M$ as an $A$-module by restriction of scalars. For $a\in A$,The last equality identifies localization with the scalar action of $a$. It identifies the direct image sheaf with the affine module sheaf of the restricted module, so $f_{\ast }\mathcal{F}$ is quasi-coherent.

Suppose $Y$ is affine and $X={\bigcup }_{i=1}^r{U}_i$ is a finite affine cover. Since a variety is separated, every $U_i\cap U_j$ is affine: it is identified with the inverse image of the closed diagonal morphism inside the affine product $U_i{\times }_k{U}_j$. Write $g_i=f{|}_{U_i}$ and $g_{ij}=f{|}_{U_i\cap U_j}$. Form the morphism of sheaves of modulesBoth sums are finite, and every summand is quasi-coherent by the affine calculation. For every open $V\subseteq Y$, the sheaf gluing axiom on $f^{-1}(V)$ identifies the kernel of this map on sections with $\mathcal{F}(f^{-1}(V))$. HenceThe kernel result proved above establishes quasi-coherence.

Finally, for arbitrary $Y$, take an affine neighbourhood $V\subseteq Y$ and apply the preceding argument to $f^{-1}(V)\to V$. The open subset $f^{-1}(V)$ is again a variety and has a finite affine cover, because varieties have Noetherian Zariski topology. The restriction of $f_{\ast }\mathcal{F}$ to $V$ is exactly the direct image for this restricted morphism. ThusThe finiteness and separation conditions are the relevant hypotheses, not affineness of the original source or target. More generally, quasi-coherence of direct image under a quasi-compact quasi-separated morphism is valid. If one permits nonseparated finite-type prevarieties, each overlap in the argument has a finite affine cover; replacing its summand by the sum of direct images from these charts gives the same kernel, since agreement can be tested on a cover. This also proves the assertion in that convention. It does not extend by this argument to arbitrary spaces with an unrestricted infinite cover.

Let $X$ be an affine variety over an algebraically closed field $k$, with reduced coordinate ring $A=k[X]$. On an open subset $U$, define ${\mathcal{O}}_X(U)$ to be the functions $h:U\to k$ which, near every point, have an expression $a/b$ with $a,b\in A$ and $b$ nowhere zero on that neighbourhood. Restriction is ordinary restriction of functions. Compatible local functions glue uniquely and remain locally quotients, so this defines the structure sheaf. On the principal open subset $D(f)$ it gives $A_f$, and at $P$ it gives the local ring $A_{{\mathfrak{m}}_P}$. The principal-open identification can also be seen by identifying $D(f)$ with the affine algebraic set obtained by adjoining a coordinate $t$ and the equation $tf=1$; its coordinate ring is $A[t]/(tf-1)\cong A_f$. This construction permits several irreducible components; no division by a function vanishing identically on a component is assumed.

An abstract algebraic variety is a locally ringed space locally isomorphic to such affine models, with a finite affine cover; we use the usual separated convention. Reducible varieties are allowed here. The separation condition says that the diagonal morphism is a closed immersion. Two varieties are related by a birational map with an inverse when some dense open subset of one is isomorphic as a locally ringed space to a dense open subset of the other.

Define the ring of rational functions on a reduced variety by pairs $(U,h)$, where $U\subseteq X$ is dense open and $h\in {\mathcal{O}}_X(U)$, identifying two pairs when their restrictions agree on a common dense open subset. Finite intersections of dense open subsets are dense, so addition and multiplication on these intersections are well defined. For an irreducible variety this gives the usual function field; for a reducible variety it need not be a field. Restriction to any dense open subset $V$ induces an isomorphismIndeed, intersecting domains with $V$ defines the restriction, and a dense open subset of $V$ is also dense open in $X$, which defines its inverse. An isomorphism between dense opens therefore induces an isomorphism of their rational rings. Thus birational varieties have isomorphic rings of rational functions, also in the reducible case.

For the finite-cover assertion, write the closed complement of $U$ as $V(I)$, where $I=(f_1,\dots ,f_r)$ is an ideal of the Noetherian ring $A$. ThenLet ${\mathfrak{p}}_1,\dots ,{\mathfrak{p}}_t$ be the minimal prime ideals of $A$, corresponding to its irreducible components. Since $U$ is dense, it meets every component, so $I$ is contained in none of these primes. Prime avoidance supplies $s\in I$ outside their union. Consequently $D(s)\subseteq U$ meets a dense open subset of every component and is dense in $X$. Add $D(s)$ to the finite cover if necessary. This proves that a finite principal-open cover can be chosen with a dense member. It does not say that every principal-open cover already has such a member: for $X=V(xy)$, the dense open $X\setminus \{(0,0)\}=D(x)\cup D(y)$ has neither displayed member dense, although adding $D(x+y)$ supplies a dense member.

For clarity, the zero divisors in this reduced Noetherian ring are exactly ${\bigcup }_i{\mathfrak{p}}_i$. If $ab=0$ and $a$ avoids every ${\mathfrak{p}}_i$, reduction in the domain $A/{\mathfrak{p}}_i$ forces $b\in {\mathfrak{p}}_i$ for all $i$, hence $b=0$. Conversely, if $a\in {\mathfrak{p}}_i$, choose for every $j\ne i$ an element $b_j\in {\mathfrak{p}}_j\setminus {\mathfrak{p}}_i$, possible because distinct minimal primes are incomparable. Their product $b$ is not in ${\mathfrak{p}}_i$ but belongs to all the other primes. Then $b\ne 0$ and $ab\in {\bigcap }_j{\mathfrak{p}}_j=0$. For a single minimal prime, reducedness makes that prime zero and this argument uses $b=1$. It follows that

The total quotient ring $S^{-1}A$, where $S$ consists of these non-zero-divisors, maps to $\mathrm{Rat}(X)$ byThe equivalence relation for localization makes this a well-defined $k$-algebra homomorphism. For surjectivity, a rational class represented on $U$ restricts to the dense principal open $D(s)\subseteq U$ constructed above. Its section there belongs to $A_s$, so it is $a/s^r$ and lies in the image. For injectivity, if such a fraction is zero on a dense open subset, its numerator vanishes on that dense subset. The numerator's zero set is closed, so it vanishes everywhere on $X$ and is zero in the reduced coordinate ring. ThusIn particular $A\to S^{-1}A$ is canonically injective: $a/1=0$ implies $sa=0$ for some non-zero-divisor $s$, hence $a=0$.

Finally let $Y$ be the reduced closed union of exactly those irreducible components of $X$ which contain $P$. Removing all the other components gives an open neighbourhood of $P$ on which $X$ and $Y$ coincide, including their structure sheaves. Hence ${\mathcal{O}}_{X,P}\cong {\mathcal{O}}_{Y,P}$. Write $B=k[Y]$ and let ${\mathfrak{n}}_P$ be its maximal ideal at $P$. Every minimal prime of $B$ is contained in ${\mathfrak{n}}_P$, because every component of $Y$ contains $P$. If $b\notin {\mathfrak{n}}_P$, it avoids each such prime and is a non-zero-divisor. Therefore the map of localizationsis defined and injective: if its value is zero, a non-zero-divisor annihilates $a$, so $a=0$. We obtain the desired local ring embeds in rational functions on incident components:At an intersection point one must retain all incident components. For example, the local ring of $V(xy)$ at the origin has two nonzero germs $x,y$ with $xy=0$; it cannot embed into the function field of just one of its axes.

For a based space $(X,x_0)$ and $k\ge 1$, the homotopy group ${\pi }_k(X,x_0)$ is the set of based homotopy classes of maps $(I^k,\partial I^k)\to (X,x_0)$, equivalently based maps $S^k\to X$. Concatenating along one coordinate defines its group operation; reversing that coordinate gives the inverse. The group is abelian for $k\ge 2$. The symbol ${\pi }_0(X)$ records path components and is generally only a pointed set. An Eilenberg–MacLane space $K(G,n)$ is a connected CW space with ${\pi }_n=G$ and all other positive homotopy groups zero; for $n>1$ the group $G$ must be abelian.

Projection onto coordinates givesa map to a product is exactly a pair of maps, and a based homotopy to the product is exactly a pair of based homotopies. These identifications respect concatenation. Thus $K(G,n)\times K(H,n)$ has exactly the homotopy groups required of $K(G\times H,n)$. Taking CW models and using the allowed uniqueness of Eilenberg–MacLane spaces gives

Let $p$ be prime. Take $S^{\infty }$ in complex coordinates, with $C_p$ acting by scalar multiplication by a primitive $p$th root of unity. This is a free action. Every map from a finite-dimensional sphere lands in a finite stage and becomes null-homotopic in a sufficiently higher sphere, so $S^{\infty }$ is weakly contractible; the Whitehead theorem for CW complexes makes it contractible. Therefore the infinite lens space $S^{\infty }/C_p$ has contractible universal cover, fundamental group $C_p$, and no higher homotopy: it is a model of $K(C_p,1)$.

Here is the group cohomology computation, including the ring structure. For the cyclic group generated by $g$, the standard free resolution alternates the maps $g-1$ and $N=1+g+\cdots +g^{p-1}$. Applying ${\mathrm{Hom}}_{\mathbb{Z}[C_p]}(-,{\mathbb{F}}_p)$ makes both maps zero, so each cohomology degree is one-dimensional. The period-two identity chain map represents a class $b$ in degree two, and cup multiplication by this periodicity class is an isomorphism from each degree to the degree two higher. For odd $p$, a nonzero degree-one class $a$ has $a^2=0$ by graded commutativity, while $b^j$ and $a{b}^j$ are nonzero by periodicity. For $p=2$ the model is homotopy equivalent to ${\mathbb{RP}}^{\infty }$, whose ring is the infinite version of that computed in question 2. Consequently the mod-p cohomology ring of a cyclic group isThe standard results used are the contractible-cover description of a $K(G,1)$, its identification of singular cohomology with group cohomology, and the cyclic periodic resolution with cup-periodicity just stated.

The product equivalence and the assumed Künneth theorem over the field giveThe tensor product is graded: for odd $p$, $a_1{a}_2=-a_2{a}_1$ and each $a_i^2=0$. The dimensions in degree $d$ are $d+1$: for $p=2$ count $t_1^i{t}_2^{d-i}$; for odd $p$ the generating series is $(1+t{)}^2/(1-t^2{)}^2=(1-t{)}^{-2}$.

If an $n$-dimensional CW complex has exactly one cell in dimension $n+1$, then $C^{n+1}(X;{\mathbb{F}}_p)={\mathbb{F}}_p$. Cellular cohomology is the quotient of the kernel of its outgoing coboundary by the image of the incoming one, a quotient of a subspace of this one-dimensional vector space. ThusThis one-cell bound on next-degree cohomology is valid even with infinitely many higher cells; the printed finite-per-dimension hypothesis is more than it needs.

Now a free finite-group action makes $S^n\to M=S^n/G$ a covering. For $n>1$ the sphere is simply connected, so ${\pi }_1(M)=G$, ${\pi }_i(M)=0$ for $1<i<n$, and ${\pi }_n(M)=\mathbb{Z}$. The covering also identifies every higher homotopy group with that of $S^n$. Use the homotopy-group effect of attaching higher cells: attaching $(i+1)$-cells, $i\ge 2$, preserves homotopy groups below $i$ and quotients ${\pi }_i$ by the subgroup generated by the attaching classes and their fundamental-group translates. Attach a single $(n+1)$-cell along a generator of ${\pi }_n(M)$; its translates do not require additional cells since the original generator already generates the underlying group. This kills ${\pi }_n$ without changing ${\pi }_1$ or the lower groups.

Successively in each higher degree attach cells representing generators of the remaining homotopy group. In the union every higher homotopy group vanishes, since a map and its relevant homotopy have compact domain and are handled at a finite stage. The result is a $K(G,1)$ with exactly one $(n+1)$-cell. No finiteness assumption on the number of later cells is needed for the preceding cochain bound. If desired, one can retain finitely many per degree using the Serre finiteness theorem for homotopy groups: a simply connected CW complex of finite type has finitely generated homotopy groups. At each finite stage the universal cover is simply connected and finite type, since $G$ is finite; choosing finitely many generators then preserves finite type inductively. The first stage needs just the stated ${\pi }_n(S^n)=\mathbb{Z}$.

If $G=C_p\times C_p$, the resulting $K(G,1)$ would have $\dim H^{n+1}\le 1$ by its single cell, but its already computed cohomology has dimension $n+2>1$ in that degree. This contradiction provesThis is the free sphere actions exclude elementary abelian subgroups of rank two obstruction.

For a locally compact space, compactly supported cohomology iswith transition maps induced by enlarging supports. In ${\mathbb{R}}^n$ closed balls are cofinal; their complements retract onto $S^{n-1}$. The relative cohomology long exact sequence and contractibility of ${\mathbb{R}}^n$ giveThis is compactly supported cohomology of Euclidean space; for $n=0$ it is simply the cohomology of a point.

For an oriented $n$-manifold without boundary, Poincare duality in its compact-support form states that capping with the locally finite fundamental class gives $H_c^q(M;\mathbb{Z})\cong H_{n-q}(M;\mathbb{Z})$. If $M$ is closed, the ordinary fundamental class $[M]\in H_n(M;\mathbb{Z})$ gives $H^q(M;\mathbb{Z})\cong H_{n-q}(M;\mathbb{Z})$ and a perfect complementary-degree pairing over $\mathbb{Q}$.