Double complexes and spectral sequences

Let horizontal and vertical differentials raise their respective indices and square to zero. In this lesson they commute with each other. Define total degree as the sum of the two indices, and on the component in column $p$ use the total differential $d_h+(-1)^pd_v$. The cross terms cancel because a horizontal step changes the sign used by the following vertical step. Some sources instead make the two arrows anticommute and use their unsigned sum; translating conventions is necessary before importing a formula.

In the first quadrant, each fixed total degree has finitely many components, so its direct-sum totalization equals its product totalization. Outside this setting, infinitely many components can lie on one diagonal, and direct sum and product need not coincide. The choice can alter cohomology. A diagram extending infinitely in two directions therefore requires an explicit totalization and a convergence argument rather than an appeal to the finite rectangle one happens to have drawn.

Filter the total complex decreasingly by retaining columns with index at least $p$. The associated graded differential is vertical, so the first page is vertical cohomology, with the harmless column sign understood. The first-page differential is induced by the horizontal arrow; taking its cohomology produces the second page. Later differentials are not usually original arrows in the diagram. They encode the obstruction remaining after a representative has been corrected through several filtration levels.

With cohomological indexing, a page-$r$ differential has bidegree $(r,1-r)$ and hence raises total degree by one. The next page is the cohomology of the current page under that differential. A group can disappear either by supporting a nonzero differential or by being hit by one. Seeing zero arrows on the second page does not by itself show collapse: higher-page arrows can still connect positions permitted by their bidegrees unless the location of the nonzero terms excludes them.

For this first-quadrant total complex, the column filtration is finite in each total degree. The associated spectral sequence converges to total cohomology with its induced finite filtration. Its stable page identifies the successive quotients of that filtration, not an automatic splitting of the resulting module. This degreewise bounded condition is a sufficient convergence hypothesis; it is more precise than saying vaguely that the picture is bounded somewhere. Unbounded filtrations can involve completion and limit obstructions requiring additional theorems.

Even a completely collapsed spectral sequence can leave extension problems. A group with two successive quotients each isomorphic to the integers modulo two could be the cyclic group of order four or the direct sum of two cyclic groups of order two. Their associated graded groups agree, but their element orders differ. Over a field, finite filtrations split as vector spaces after choices; that special fact does not transfer to arbitrary modules, nor does it give canonical splittings or preserve additional multiplicative structure.