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The Gram matrix of the canonical height pairing of rational points gives a robust independence certificate. For two points, set $a=\hat{h}(P)$, $b=\hat{h}(Q)$ and $c=(\hat{h}(P+Q)-a-b)/2$. If bounds from rounded measurements imply $a>0$ and $ab-c^2>0$, the matrix is positive-definite. Any integer relation $rP+sQ=O$ would give $a{r}^2+2crs+b{s}^2=0$, forcing $r=s=0$. Thus the points generate a free abelian group of rank two. Checking an interval lower bound for the determinant, rather than just its approximate central value, makes the certificate insensitive to rounding. It certifies independence, not that the points are a basis of the entire Mordell-Weil group.

For finite sets in an abelian group and a bipartite graph $\Gamma \subseteq A\times B$, the restricted sumset is $\{a+b:(a,b)\in \Gamma \}$. It retains only sums corresponding to graph edges.

For a ring homomorphism $\theta :R\to A$, an $A$-module becomes an $R$-module by the displayed action. Its underlying abelian group stays the same.

For an abelian group $A$,For a finite direct sum of cyclic groups this gives one summand $C_{\gcd (m,n)}$ for every pair $C_m,C_n$.

A sheaf of abelian groups is a sheaf of sets whose sections on each open set form an abelian group and whose restriction maps are group homomorphisms. Its stalk of a sheaf is the corresponding group of germs. Kernels are computed on sections, whereas cokernels are obtained by sheafification of the sectionwise cokernel. An exact sequence can therefore be checked on stalks.

A Sidon set in an abelian group is a subset $A$ for which an equality $a+b=c+d$ with $a,b,c,d\in A$ forces the unordered pairs $\{a,b\}$ and $\{c,d\}$ to be equal. Equivalently, the ordered nonzero differences $a-b$ with $a\ne b$ are all distinct.

A simplicial abelian group is a simplicial object in abelian groups. Its underlying simplicial set is always a Kan complex.

The singular chain group is the free abelian group on all continuous maps ${\Delta }^q\to X$, the singular simplices. In particular it is free even when the set of simplices is infinite. This freeness makes applying $\mathrm{Hom}(C_q(X),-)$ to a short exact coefficient sequence exact.

A singular $n$-cochain with coefficients in an abelian group $A$ is a homomorphism from the singular chain group $C_n(X;\mathbb{Z})$ to $A$. It is specified by its values on all singular simplices, without a finite-support restriction. Its coboundary evaluates on the alternating sum of the faces. The resulting cochain complex computes singular cohomology.

For a topological space $X$ and abelian group $A$, singular cohomology is the cohomology of the cochain complex $C^n(X;A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),A)$, where $C_n$ is the singular chain group. Its differential is dual to the singular boundary. This theory has relative cohomology, long exact sequences and the Excision theorem. On spaces that are not locally contractible it can differ from Čech cohomology.

For $A$ in a finite abelian group and normalized Fourier transform $\widehat{1_A}$, the normalized number of sextuples satisfyingisThis follows by inserting orthogonality of complex exponentials for the displayed equation: the three variables on one side contribute $\widehat{1_A}(\gamma {)}^3$ and those on the other side contribute its complex conjugate.

For a point inclusion $i:\{x\}\hookrightarrow X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$ has sections $A$ on opens containing $x$ and zero on other opens. Its restriction maps are identities or maps to zero. If $A\ne 0$, its nonzero stalks occur at every point of the closure $\overline{\{x\}}$: exactly those points whose every neighbourhood contains $x$. Thus the familiar assertion that only the stalk at $x$ is nonzero requires $x$ to be a closed point. It is always flasque, giving the stated skyscraper sheaf cohomology.

The square-class group identifies two nonzero elements of a field when their quotient is a square. Multiplication makes it an abelian group of exponent two, hence a vector space over ${\mathbb{F}}_2$. It records quadratic extensions and the images of Kummer theory for $n=2$.

A divisible Abelian group is torsion-free precisely when division by each nonzero integer is unique. Defining rational scalar multiplication by $n((m/n)a)=ma$ identifies these groups with vector spaces over the rational numbers, including the zero space.

The torsion subgroup of an abelian group consists of all elements of finite order. For a finitely generated abelian group, it is the finite direct summand complementary to the free abelian part.

An integral intersection form on a finite-rank free abelian group $L$ is unimodular if its adjoint $x\mapsto \lambda (x,-)$ is an isomorphism to the integral dual. In an integral basis, this is equivalent to the pairing matrix having determinant $1$ or $-1$, not merely nonzero determinant. For example, projective lines generate a unimodular intersection form on the Complex projective plane, with matrix $(1)$ in its complex orientation.

Every abelian group is amenable, and amenability is preserved by finite extensions. Hence every group with an abelian finite-index subgroup is an amenable group.

The zeroth homology group is the free abelian group on the path components of $X$. In particular, if $X$ has finitely many path components, then