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The frame bundle of an $n$-dimensional smooth manifold has fiber over $x$ consisting of linear isomorphisms $u:{\mathbb{R}}^n\to T_{x}M$. Right composition by $GL(n,\mathbb{R})$ makes it a principal bundle. A global frame of a vector bundle is a global section of a fiber bundle and trivializes the tangent bundle. A metric restricts the allowed frames to the orthonormal frame bundle.

A subset $X$ is a free basis of a group $G$ when the natural map from the free group on $X$ to $G$ is an isomorphism. Equivalently, $X$ generates $G$ and no nonempty freely reduced word in $X\cup X^{-1}$ equals the identity. A generating set is not automatically a free basis.

For a free module $F={⨁}_{i\in J}A{e}_i$, the natural map $U{\otimes }_{A}F\to {⨁}_{i\in J}U$ sends $u\otimes {\sum }_i{a}_i{e}_i$ to $(a_{i}u{)}_i$ and is an isomorphism, with inverse sending the tuple $(u_i)$ to ${\sum }_i{u}_i\otimes e_i$. For any injection $U\hookrightarrow V$, its tensor product of modules with $F$ is therefore the coordinatewise injection ${⨁}_{J}U\hookrightarrow {⨁}_{J}V$. Thus every free module is flat, including those of infinite rank.

Write the suspension of an $(n-2)$-connected based CW complex as two cones meeting in the original space. Each cone pair is $(n-1)$-connected because its relative groups shift the original homotopy groups by one. Homotopy excision theorem compares the cone pair with the suspension relative to the other cone, giving the suspension homomorphism as an isomorphism for $i<2n-2$ and a surjection at $i=2n-2$ in the indexing ${\pi }_{i-1}X\to {\pi }_i\Sigma X$.

An integral Hodge isometry between complex K3 surfaces which carries some Kähler class to a Kähler class is the pullback of a unique isomorphism of the surfaces. A Hodge isometry without that cone condition need not itself be geometric: reflection in an effective $(-2)$-curve reverses its class and cannot be induced by an isomorphism.

Given schemes $X_i$ and isomorphisms between designated open subschemes satisfying inverse and cocycle conditions, identify the underlying open spaces and glue their structure sheaves. The resulting locally ringed space is locally one of the $X_i$, hence a scheme. The same construction works for formal schemes and topological structure sheaves. Compatible morphisms out of the charts glue uniquely.

// Target: ringed-space.bigb

For a finite free module $M$ of rank $n$, this scheme represents rank-$r$ locally free module quotients of $M$ after base change, with quotient maps identified up to commuting isomorphism. It carries a universal exact sequence $0\to \mathcal{S}\to M\otimes \mathcal{O}\to \mathcal{Q}\to 0$, of ranks $n-r,n,r$. Its standard affine charts of the quotient Grassmannian prove representability over any base ring. Over a field, taking kernels gives the usual Grassmannian of $(n-r)$-planes.

// Target: ringed-space.bigb

A group isomorphism is a bijective group homomorphism $\phi :G\to H$. It matches the entire group operation: $\phi (ab)=\phi (a)\phi (b)$, with each product taken in its own group. Its inverse is automatically a group homomorphism. A group automorphism is the special case $H=G$.^[ref]

The target is part of the claim. The doubling example under isomorphism is a group isomorphism $d:\mathbb{Z}\to 2\mathbb{Z}$, between the additive integers and the additive even integers. Exactly the same rule $d(k)=2k$, declared as $d:\mathbb{Z}\to \mathbb{Z}$, is an injective endomorphism that misses $1$ and every other odd integer, so is not a group automorphism. Changing the declared target changes surjectivity, even though it does not change the formula or its compatibility with addition.

An isomorphism lets a calculation in one model solve the same problem in another. The example under group identifies addition modulo four with function composition of rotations of a square via $k\mapsto r^k$. Through this group isomorphism, adding exponents corresponds exactly to composing motions. More generally, a group isomorphism preserves order of a group element, carries subgroups to subgroups, and carries a generating set of a group to a generating set of a group. Having equally many elements is insufficient: the permutation in group automorphism already fails compatibility on two inputs.

Why does the inverse preserve the group operation? Write $u=\phi (a)$ and $v=\phi (b)$, possible because of surjectivity. ThenThis proves the inverse condition directly. Analogous calculations work for each operation of a ring and for addition and scalar multiplication in a vector space. The argument relies on preserving those operations; it supplies no general continuity claim for the inverse of a continuous bijection between topological spaces.

For a compact Kähler manifold of complex dimension $n$, wedging with the $(n-k)$th power of its Kähler class is the displayed isomorphism for $0\le k\le n$. The Hard Lefschetz isomorphism on Dolbeault cohomology and Hodge decomposition theorem for compact Kähler manifolds give the result by summing the bidegrees. In particular, $L:H^k\to H^{k+2}$ is injective when $k<n$: if $L\alpha =0$, then $L^{n-k}\alpha =0$. Hence $b_k\le b_{k+2}$ below the middle dimension.

A homomorphism is a map between algebraic structures preserving their specified operations. For a single binary operation, this means $f(a\ast b)=f(a)\star f(b)$, where the source and target may use different operation symbols. A group homomorphism preserves the group operation; a ring homomorphism preserves addition and multiplication, with the identity convention specified; a module homomorphism preserves addition and scalar multiplication.

Reduction modulo $5$ is a ring homomorphism $\mathbb{Z}\to \mathbb{Z}/5\mathbb{Z}$: adding or multiplying first and then reducing gives the same residue as reducing first and then operating. For example, $12\cdot 14=168\equiv 3(\mathrm{mod}5)$, while the reduced operands give $2\cdot 4=8\equiv 3(\mathrm{mod}5)$. This lets a complicated integer calculation be projected into simpler finite arithmetic. It also explains why a polynomial equation over the integers imposes necessary modular arithmetic conditions on any integer solution.

A homomorphism can lose information: $0$ and $5$ have the same image under that reduction. An isomorphism loses none of the structure because it has an operation-preserving inverse. Preservation of the operation, rather than preservation of arbitrary labels or sizes, is the essential condition.

For a based CW triad with connected intersection $A$, if $(B,A)$ is $p$-connected and $(C,A)$ is $q$-connected, $p,q\ge 1$, the comparison map is an isomorphism for $i<p+q$ and a surjection for $i=p+q$. This connectivity form of the Blakers-Massey theorem controls the failure of ordinary relative homotopy to satisfy unrestricted excision.

A homotopy sphere is a closed manifold with the homotopy type of a sphere of the same dimension. In dimensions at least two, a closed simply connected integral homology sphere is a homotopy sphere: successive applications of the Hurewicz theorem produce a map from the sphere inducing an isomorphism on integral homology, and the homological Whitehead theorem makes it a homotopy equivalence. The topological generalized Poincare theorem identifies its topological type, while prescribed smooth structures require a separate question.

For a morphism in an abelian category, its coimage $\mathrm{coker}(\ker f)$ is canonically isomorphic to its image $\ker (\mathrm{coker}f)$. This yields an epimorphism followed by a monomorphism, unique up to unique compatible isomorphism. Every epi-mono factorization has this same middle object.

An invertible module over a commutative ring $A$ is a finitely generated module that is a projective module and whose localization at a prime ideal is free of rank one at every prime. Its dual module $L^{\ast }={\mathrm{Hom}}_A(L,A)$ is also finite projective. Evaluation $L{\otimes }_A{L}^{\ast }\to A$ is an isomorphism after every prime localization, hence is an isomorphism globally by localization detects zero elements. The Picard group of a ring consists of the isomorphism classes of these modules under the tensor product of modules.

The Grothendieck group of finitely generated projective right $A$-modules under direct sum. An element is a formal difference $[P]-[Q]$. Equality of two positive classes means $P\oplus T\cong Q\oplus T$ for some projective $T$; adding a complement of $T$ converts this into an isomorphism after a common finite free summand. This stabilization criterion is essential in patching arguments.

In a commutative diagram of two kernel rows, a monic arrow between the final objects makes the square between their kernels and middle objects a pullback in a category. The kernel universal property factors any categorical cone uniquely. Conversely, if the square between middle and final objects is a pullback, the induced arrow between their kernels in a category is an isomorphism, since both kernels have the same universal property after pullback.

A line bundle trivial on each open set is determined by unit-valued transition functions on pairwise intersections. Their cocycle identities make the local trivial bundles glue; changing frames changes the cocycle by a coboundary. The tensor product of sheaves multiplies the transition functions, giving the displayed group isomorphism with Čech cohomology.

On a trivialized fiber bundle over $U$, assume the fiber has finite-dimensional total cohomology over a field. The Künneth theorem identifies the total-space cohomology with a finite free graded module over $H^{\ast }(U)$. Classes restricting to a basis on every fiber give an invertible change of generators: its degree-zero coefficient blocks are invertible on each path component, while its positive-base-degree terms strictly decrease fiber degree and are therefore nilpotent. Inverting the diagonal blocks and then using a finite geometric series proves that multiplication by the chosen classes is an isomorphism over $U$.

An invertible maximal-rank minor in the Jacobian matrix eliminates generators and gives a local surjection from ${\mathcal{O}}_X^n$ to the Kähler differential sheaf. For an integral smooth variety the map is an isomorphism at the generic point. Its kernel is a submodule of a free module over an integral domain, so it is a torsion-free module; vanishing generically forces it to be zero. This gives local free frames for differentials.