ExactCouples : Table of Contents

• ExactCouples -- spectral sequences by Massey's method of exact couples
• An exact couple associated to a Young tableau -- a standard filling provides a filtration by monomial ideals
• Bockstein spectral sequence -- a singly-graded spectral sequence built from the chain self-map "multiplication by p"
• canonicalFiltration -- filters a complex by its truncations
• Cellular chains as an E1 page -- A spectral sequence construction of the usual cellular differential
• 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
• Conventions and first examples -- specifics on encoding exact couples as modules for a ring
• 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)
• Elementary introduction: solving linear equations in abelian groups -- a discussion of the equation 3x+6y=0.
• Encoding diagrams as modules -- building graded modules with specified modules in certain degrees, and with specified action maps
• enforceCoupleRelations -- mods out by tautological relations satisfied by every exact couple
• evaluateInDegree -- evaluates a module in a particular degree
• evaluateInDegreeLaw (missing documentation)
• Exact couples for Tor and Ext -- building couples by applying Tor or Ext to a filtered module or a graded R[t]-module
• 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
• Functoriality for Tor and Ext couples -- induced maps between couples and spectral sequences
• Homology of a combinatorial filtration of $X^n$ -- A spectral sequence that assembles absolute homology from relative
• internalDegreeIndices -- for a ring, returns the degree-coordinates of its coefficient ring
• 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
• 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)
• Mayer-Vietoris Spectral Sequence -- A spectral sequence that assembles homology from an open cover
• oneEntry -- builds a one-by-one matrix
• Page -- for a couple ring Q, Q.Page is the page number
• 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
• Serre spectral sequence in homology -- exact couple associated to a fibration
• 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