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For finite-dimensional vector spaces $V,W$ over a field $F$, a bilinear pairing $B:V\times W\to F$ is perfect when the induced linear maps $V\to W^{\ast }$ and $W\to V^{\ast }$ are isomorphisms. Equivalently, both dimensions agree and the matrix of the pairing in any two bases is invertible. The evaluation pairing between a finite-dimensional vector space and its dual space is perfect. This notion allows two different spaces; a nondegenerate bilinear form is the case $V=W$.

A commutative ring of prime characteristic $p$ is perfect in the Frobenius sense when its Frobenius endomorphism is an automorphism. It is necessarily reduced: if $a$ is nilpotent, some $p$-power iterate is zero, and injectivity gives $a=0$. This use of perfect is distinct from the homological notion of a perfect ring. The Absolute Frobenius morphism of a scheme is an isomorphism exactly when all its local rings are perfect in this sense, because it is the identity on points and is an isomorphism on every stalk.

For a closed oriented $d$-dimensional manifold over a field $F$, the cup product pairingis a perfect pairing between complementary degrees. The cap product with the fundamental class is an isomorphism by Poincare duality, and the universal coefficient theorem for cohomology over $F$ identifies the complementary cohomology with the full dual space of the resulting homology. Evaluation is therefore a perfect pairing. Taking the top-degree part of the same formula defines a nondegenerate bilinear form on the full graded cohomology. A boundary requires relative groups and Poincare-Lefschetz duality instead.

For a uniform pro-p group with $U_i=G^{p^i}$, the map $x\mapsto x^{p^a}$ transports congruences byCommutator collection gives $(yu{)}^p\equiv y^p{u}^p(\mathrm{mod}{U}_{b+2})$ for $u\in U_b$, and the power map is an isomorphism $U_b/U_{b+1}\to U_{b+1}/U_{b+2}$. Thus a nontrivial first difference moves down exactly one layer when taking a $p$th power. Iterating gives the equivalence; completeness supplies unique roots when the congruence is read in reverse.

Here $I$ is a set regarded as a discrete category. Finite limits and regular epimorphisms are coordinatewise. A well-supported object is a family of nonempty sets. With the axiom of choice, every coordinate element extends to a global point of its product. Any subobject containing all global points must therefore contain every coordinate element, and be the whole family. Thus the category is capital. Families with some empty coordinates explain why global points need not reflect arbitrary isomorphisms.

Let $f$ be a chain map between degreewise finitely generated free abelian groups. If its reduction modulo every prime number is a quasi-isomorphism, the mapping cone acyclicity criterion makes $M(f)\otimes {\mathbb{F}}_p$ acyclic. The short exact sequence of chain complexes $0\to M\stackrel{p}{\to }M\to M\otimes {\mathbb{F}}_p\to 0$ implies that multiplication by $p$ is an isomorphism on every $H_i(M)$. These are finitely generated abelian groups. Their free summands would make multiplication by $p$ fail to be surjective, and any nonzero finite summand would make it fail to be injective for a prime dividing its order. Thus $H_i(M)=0$ and $f$ is a quasi-isomorphism. Degreewise finite generation is essential to this argument.

A binary categorical product of objects of a category $A,B$ is an object $P$ with categorical morphisms $p_A:P\to A$ and $p_B:P\to B$. Its universal property is that, for every object $X$ and pair of arrows $f:X\to A$, $g:X\to B$, there is exactly one arrow $h:X\to P$ satisfyingThe arrows $p_A,p_B$ are called the product projections. Existence lets us combine two maps; uniqueness means there is no extra unspecified information in the combined map.

In the Category of sets, take $A=\{0,1\}$ and $B=\{a,b,c\}$. Their Cartesian product contains six ordered pairs, and the projection maps select the first and second coordinates. Let $X=\{u,v,w\}$, $f(u)=0$, $f(v)=1$, $f(w)=1$, and $g(u)=a$, $g(v)=b$, $g(w)=a$. The unique required map is $h(u)=(0,a)$, $h(v)=(1,b)$, $h(w)=(1,a)$. Reading either coordinate recovers the corresponding input map.

In a partially ordered set viewed as a category, the projection arrows mean $P\le A$ and $P\le B$. The universal property says that every common lower bound $X$ satisfies $X\le P$, so $P$ is the greatest lower bound. For subsets ordered by inclusion this is $A\cap B$, the intersection; it is a different construction from the Cartesian product in the Category of sets because the allowed arrows have changed.

The universal property also compares any two proposed products: each supplies a unique map to the other compatible with the projections. The two composites have the same projections as the corresponding identity morphisms, so uniqueness makes them identities. Thus a categorical product is determined up to a unique projection-preserving isomorphism, which explains why we can use whichever concrete construction is convenient.

In a pointed category with categorical kernels and categorical cokernels, where every monomorphism is normal, every $h$ factors through $i=\ker (\mathrm{coker}h)$ as $h=ie$, with $e$ a pseudo-epimorphism. If $te=0$, write $k=\ker t$. Then $e$ factors through $k$, and the monic $ik$ is a categorical kernel of some $q$ by normality. Since $qh=0$, the cokernel of $h$ forces $qi=0$. Hence $i$ factors through $ik$, forcing $k$ to be an isomorphism and therefore $t=0$. This establishes the factorization without assuming the stronger abelian category axioms.

A homomorphism between finitely generated one-variable Iwasawa modules is a pseudo-isomorphism if its kernel and cokernel are finite. Finite errors do not change the Iwasawa-module rank or the characteristic ideal of a torsion module. They can still change its integral structure, so a pseudo-isomorphism is weaker than an isomorphism.

Over an algebraically closed field, choose a point $P$ on the smooth projective curve. The Riemann-Roch theorem gives $\dim L(P)=2$: the complementary canonical divisor term has negative degree. A nonconstant function in $L(P)$ has exactly one simple pole, so it defines a degree-one morphism of algebraic curves to the projective line. A degree-one map between smooth projective curves is an isomorphism. The existence of a rational point is essential over a field which is not algebraically closed.

A finitely generated module $M$ over a commutative ring $A$ is reflexive when its evaluation homomorphism $M\to M^{\ast \ast }$ to the double dual module is an isomorphism.

For a based Serre fibration with fibre $F$ over the basepoint, lift a based cube in $B$ from its constant bottom face, keeping the side faces fixed. Its final face lies in $F$, giving surjectivity of the relative map. Homotopies in $B$ lift; two lifts of the same cube are relatively homotopic by lifting with both side liftings prescribed. This gives injectivity. For $n\ge 2$ this is a group isomorphism; at $n=1$ it is initially a bijection of pointed sets, unless the relative set is endowed with the group law transported through the fibration.

If $gf$ has the left lifting property against monomorphisms, so does $g$. Precompose a lifting square for $g$ with $f$ and lift the composite. To check the remaining triangle, cancel the monic side of the square, rather than cancelling $f$. In particular, a monic right factor of a lifting-only strong morphism is an isomorphism.

For a possibly reducible algebraic variety $X$, take pairs $(U,f)$ with $U$ a dense open subset and $f$ a regular function on $U$. Identify pairs agreeing on a common dense open subset. Addition and multiplication on intersections make these classes a ring. For an affine variety with reduced coordinate ring $A$, this ring is its total quotient ring. For an irreducible variety, it is the function field. An isomorphism of dense open subsets induces an isomorphism of these rings.

Over an algebraically closed field, a smooth four-variable projective quadratic equation can be written $x_0{x}_3-x_1{x}_2=0$, including in characteristic two. Its quadratic form splits into two hyperbolic planes: choose an isotropic vector and its paired vector, adjust the latter to be isotropic, and repeat on the orthogonal complement. In characteristic two, smoothness forces the alternating polar form in dimension four to have full rank, since a positive-dimensional radical has a nonzero isotropic vector over the algebraically closed field and would give a singular point. The Segre embedding $([s_0:s_1],[t_0:t_1])\mapsto [s_0{t}_0:s_0{t}_1:s_1{t}_0:s_1{t}_1]$ then gives the isomorphism. Each matrix of coordinates has rank one, which supplies the inverse pair of projective factors.

In the covariant functor category $[\mathbb{N},\mathbf{Set}]$, the classifier at $n$ consists of thresholds $k\ge n$ and the empty threshold $\infty$. Its transition to $n+1$ sends $k$ to $\max (k,n+1)$. The displayed maps commute with transitions and are all surjective, so define an epimorphism. They identify $n$ and $n+1$, so the endomorphism is not an isomorphism.

In the nested partial unary operation category, suppose the lower-operation diagram $A\rightrightarrows B\stackrel{q}{\to }Q$ has a split coequalizer with $qs=1$, $ft=1$ and $gt=sq$. The section $s$ takes eligible fixed points to eligible fixed points, so the displayed next operation is defined on exactly the required domain. For eligible $x\in B$, the equations giveThus $q$ preserves the next operation. Every factored morphism preserves it by evaluating at $sz$, giving creation of the specified coequalizer. Together with reflection of isomorphisms, the Beck monadicity theorem proves one-step monadicity.

A stable letter is the new generator of an HNN extension which implements an isomorphism between two specified subgroups by conjugation. In the convention $ta{t}^{-1}=\varphi (a)$, it moves elements of the source subgroup $A$ to the target subgroup $B$; using $t^{-1}at=\varphi (a)$ reverses the naming convention.

Cover the sphere by complements of its two poles. Each open set is contractible, and their intersection deformation retracts to the equatorial sphere. The Poincaré lemma and homotopy invariance of de Rham cohomology make the two neighboring groups in the Mayer--Vietoris sequence for de Rham cohomology zero, so its connecting map is an isomorphism. Its inverse takes a top-degree form to the difference of its local primitives on the equator, up to the connecting-map sign convention.