ExactCouples -- spectral sequences by Massey's method of exact couples

Description

An exact couple is a pair of R-modules E and A together with maps A --> E --> A and A --> A with the conditions that

im(A --> E) = ker(E --> A)

im(E --> A) = ker(A --> A)

im(A --> A) = ker(A --> E).

Every exact couple gives rise to a spectral sequence, and most spectral sequences can be constructed by means of an exact couple. Exact couples were introduced in

Massey, W. (1952). Exact Couples in Algebraic Topology (Parts I and II). Annals of Mathematics, 56(2), second series, 363-396.

This package constructs the exact couple arising from a bounded chain complex of finitely generated modules $C_*$ and an endomorphism $f : C_* \to C_*$. Then, we take A to be the homology of $C_*$, and E to be the homology of the mapping cone of $f$. The maps of the exact couple come from the usual long exact sequence, and the resulting spectral sequence converges to the homology of $C_*$. For example, several standard exact sequences for Tor and Ext are of this form; see Exact couples for Tor and Ext. Another example is the spectral sequence of a filtered simplicial complex; see filteredSimplicialComplexCouple.

Since an exact couple of R-modules is itself an R-module with some extra action maps, it can be considered a module for a larger ring with new variables acting by these extra maps. This observation allows us to encode exact couples as modules and thereby compute with them using Groebner bases and other standard Macaulay2 methods.

This encoding strategy works generally for commuting diagrams; see Encoding diagrams as modules. This style may feel unfamiliar at first. For example, the r-page of a spectral sequence is often considered as a grid of R-modules with differentials of degree {r,1-r}; in this package, this information is encoded as a module for a ring like

R[$D_r$, Degrees => {{r,1-r}} ] / $(D_r)^2$

so that the differential is encoded by the action of $D_r$. See pageModule. One benefit comes in our approach to functoriality; see Functoriality for Tor and Ext couples, restackRing and restackModule.

If this encoding makes sense to you, read Conventions and first examples for the specifics of our approach to exact couples.

A common complaint

It may seem strange or useless to compute a spectral sequence converging to $H_* C_*$, since under our assumptions it would be much easier to compute these modules directly. However, it is a misconception that the purpose of a spectral sequence is to compute its abutment.

Rather, spectral sequences enable proofs, which are easier to construct if you can look at examples.

For the serious user of this package, there is at least one common mathematical difficulty:

Even if a spectral sequence with finitely generated entries can be constructed by means of a couple, the auxiliary data is often not finitely generated. The author of this package would be interested in any systematic method of replacing such couples with finitely generated versions that recover the same spectral sequence. If you have information or ideas, please send an email!

Author

John D. Wiltshire-Gordon <jwiltshiregordon@gmail.com>

Version

This documentation describes version 0.6 of ExactCouples.

Source code

The source code from which this documentation is derived is in the file ExactCouples.m2. The auxiliary files accompanying it are in the directory ExactCouples/.

Exports

Functions and commands
- canonicalFiltration -- filters a complex by its truncations
- chainModule -- writes a chain complex of R-modules as an R[d]/d^2-module
- chainModuleHomology -- computes the d-cohomology of an R[d]/d^2-module
- contravariantExtCouple -- the exact couple obtained by applying Ext(-,Y) to a filtered module
- contravariantExtLES -- the long exact sequence in Ext induced by an inclusion in the first coordinate of Hom
- cospan -- mods out by a collection of module elements
- coupleRing -- builds a couple ring
- covariantExtCouple -- the exact couple obtained by applying Ext(W,-) to a filtered module
- covariantExtLES -- the long exact sequence in Ext induced by an inclusion in the last coordinate of Hom
- declareCouple -- initializes generating classes for an exact couple
- declareGenerators -- builds a free module and names its generators
- derivedCouple -- builds the derived couple of an exact couple
- derivedCoupleRing -- forms the ring that acts on a derived couple
- distinguishedTriangleLaw (missing documentation)
- eid (missing documentation)
- enforceCoupleRelations -- mods out by tautological relations satisfied by every exact couple
- evaluateInDegree -- evaluates a module in a particular degree
- evaluateInDegreeLaw (missing documentation)
- exactCouple -- builds an exact couple from a R[d,f]/d^2-module
- excerptCouple -- displays one of the long exact sequences in an exact couple
- excerptLES -- displays a few entries of a long exact sequence
- expectChainRing -- accepts rings of the form R[d]/d^2
- expectCoupleRing -- accepts certain rings of the form R[e_r,f_r], and installs Page, isEvenDegree, and isOddDegree
- expectExactCouple -- accepts a module if it encodes an exact couple
- expectFiltrationList -- accepts a list of modules if each includes in the next
- expectSequenceRing -- accepts rings of the form R[t]
- expectTriangleRing -- accepts certain rings of the form R[d,e,f]/(d^2, e^3)
- extensionInDegree -- places a copy of a module in a certain degree
- extensionInDegreeLaw (missing documentation)
- externalDegreeIndices -- for a ring Q, returns the degree-coordinates present in Q but not in its coefficient ring
- filteredSimplicialComplexCouple -- builds the exact couple associated to a filtered simplicial complex
- filtrationModule -- converts a filtered module to an R[t]-module
- internalDegreeIndices -- for a ring, returns the degree-coordinates of its coefficient ring
- longExactSequence -- finds the long exact sequence associated to a map of R[d]/d^2-modules
- mapToTriangleRing -- embeds a ring of the form R[d,f]/d^2 in its triangle ring R[d,e,f]/(d^2,e^3)
- oneEntry -- builds a one-by-one matrix
- pageModule -- gives a page of a spectral sequence as a module for R[d]/d^2 where d is the differential
- plotPages -- displays a few pages of a spectral sequence
- restackModule -- restacks the ring that acts on a module
- restackRing -- changes the order in which variables were adjoined
- sequenceModule -- builds a graded R[t]-module from a sequence of maps
- structureMap -- computes the action of a ring element on a particular degree
- toChainComplex -- converts a module for R[d]/d^2 to a chain complex
- TorCouple -- the exact couple obtained by applying Tor(W,-) to a filtered module
- TorLES -- the long exact sequence in Tor induced by an inclusion in the second coordinate
- triangleRing -- builds a triangle ring
Symbols
- isEvenDegree -- for a couple ring Q, Q.isEvenDegree returns true on page-degrees of Q
- isOddDegree -- for a couple ring Q, Q.isOddDegree returns true on auxiliary-degrees of Q
- Page -- for a couple ring Q, Q.Page is the page number

For the programmer

The object ExactCouples is a package.