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A decomposition of a module into a direct sum of submodules. The index set for the collection of submodules is then called the grading set. Graded modules arise naturally in ...

Let K be a finite complex, and let phi:C_p(K)->C_p(K) be a chain map, then sum_(p)(-1)^pTr(phi,C_p(K))=sum_(p)(-1)^pTr(phi_*,H_p(K)/T_p(K)).

A theorem which states that if a Kähler form represents an integral cohomology class on a compact manifold, then it must be a projective Abelian variety.

If K is a finite complex and h:|K|->|K| is a continuous map, then Lambda(h)=sum(-1)^pTr(h_*,H_p(K)/T_p(K)) is the Lefschetz number of the map h.

The commutation operation [a,b]=ab-ba corresponding to the Lie product.

The multiplication operation corresponding to the Lie bracket.

A simplicial complex K is said to be locally finite if each vertex of K belongs only to finitely many simplices of K.

The Maclaurin-Bézout theorem says that two curves of degree n intersect in n^2 points, so two cubics intersect in nine points. This means that n(n+3)/2 points do not always ...

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

An octic curve is an algebraic curve of order eight. The pear curve is an example of an octic curve.