Alphabeta MathHomological Algebra
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
1 page- Chain Complexes and Homology47 results
Successive differentials squaring to zero are the load-bearing datum of the page.
16 definitions, 5 lemmas, 10 propositions, 8 theorems, 2 corollaries, 6 false statementsExamples & counterexamples →
Level 1
2 pages- Chain Homotopy and the Homotopy Category40 results
Homology remembers less than a chain complex, and this page isolates the first equivalence relation it forgets.
11 definitions, 5 lemmas, 9 propositions, 8 theorems, 2 corollaries, 5 false statementsExamples & counterexamples → - Spectral Sequences44 results
A filtered chain complex gives a sequence of increasingly restrictive cycle and boundary quotients.
17 definitions, 8 lemmas, 9 propositions, 3 theorems, 1 counterexample, 6 false statementsExamples & counterexamples →
Level 2
2 pages- Homological Gaussian Elimination9 results
This page develops Gaussian elimination for complexes over an additive category, in the form used in Khovanov homology computations: an invertible block of a differential is…
2 definitions, 1 lemma, 3 propositions, 2 theorems, 1 corollaryExamples & counterexamples → This page fixes the sign convention for mapping cones and then uses the cone and cylinder formulas exactly at chain level.
6 definitions, 4 lemmas, 11 propositions, 6 theorems, 2 corollaries, 5 false statementsExamples & counterexamples →
Level 3
1 page- Long Exact Sequences in Homology35 results
This page builds the homology connecting morphism from the lift-boundary recipe attached to a short exact sequence of complexes and shows how that construction fits the…
4 definitions, 6 lemmas, 6 propositions, 6 theorems, 8 corollaries, 5 false statementsExamples & counterexamples →
Level 4
3 pagesA double complex carries two differentials, and its total complex turns them into a single differential.
11 definitions, 9 lemmas, 7 propositions, 11 theorems, 1 corollary, 1 counterexample, 6 false statements, 1 remarkExamples & counterexamples →- Projective and Injective Resolutions51 results
This page introduces projective and injective resolutions as augmented or coaugmented exact complexes and then keeps the major structural results separate: existence…
9 definitions, 12 lemmas, 5 propositions, 13 theorems, 6 corollaries, 6 false statementsExamples & counterexamples → - Triangulated Categories46 results
This page fixes the signed Verdier convention, develops its Hom-exact and splitting consequences, and verifies the cone triangulation of K(A).
17 definitions, 4 lemmas, 13 propositions, 4 theorems, 1 corollary, 6 false statements, 1 remarkExamples & counterexamples →
Level 5
1 page- Derived Functors44 results
This page builds derived functors only relative to displayed projective or injective resolution data.
10 definitions, 6 lemmas, 13 propositions, 8 theorems, 1 corollary, 5 false statements, 1 remarkExamples & counterexamples →
Level 6
1 page- Delta Functors and Universality34 results
This page packages the long exact sequence data attached to derived functors into the abstract language of delta functors.
8 definitions, 7 lemmas, 5 propositions, 5 theorems, 3 corollaries, 5 false statements, 1 remarkExamples & counterexamples →
Level 7
1 page- Ext and Balanced Resolutions35 results
This draft compares the projective and injective resolution constructions of Ext.
5 definitions, 7 lemmas, 10 propositions, 5 theorems, 2 corollaries, 6 false statementsExamples & counterexamples →
Level 8
1 pageThis draft develops extensions, Baer addition, the Yoneda product, and homological dimensions.
14 definitions, 7 lemmas, 6 propositions, 9 theorems, 2 corollaries, 6 false statementsExamples & counterexamples →
Level 9
2 pages- Group Cohomology as a Derived Functor42 results
This page develops group (co)homology from invariants and coinvariants, then supplies the bar calculation, change-of-groups comparison, transfer, and integral cohomological dimension.
12 definitions, 7 lemmas, 4 propositions, 13 theorems, 1 corollary, 5 false statementsExamples & counterexamples → - Tor Flatness and Global Dimension41 results
Tensor complexes first distinguish right and left modules and fix the Koszul differential.
8 definitions, 4 lemmas, 9 propositions, 12 theorems, 1 corollary, 6 false statements, 1 remarkExamples & counterexamples →
Level 10
4 pages- Derived Categories60 results
This page constructs derived categories as size-controlled roof localizations, proves their triangulated and Verdier-quotient descriptions, develops bounded projective and…
14 definitions, 12 lemmas, 13 propositions, 14 theorems, 7 false statementsExamples & counterexamples → This page develops the Eilenberg–Watts theorem for arbitrary unital rings, with left modules throughout and no commutativity assumption.
1 definition, 7 lemmas, 2 theorems, 3 corollariesExamples & counterexamples →- Graded Bimodules and Tensor Functors9 results
This page extends the published commutative grading convention to unital associative ℤ-graded algebras, graded bimodules and degree-zero maps.
3 definitions, 2 lemmas, 1 proposition, 3 theoremsExamples & counterexamples → This draft develops the stated conventions and boundary cases in manifest order.
4 definitions, 8 lemmas, 3 propositions, 7 theorems, 4 corollaries, 1 counterexample, 5 false statementsExamples & counterexamples →
Level 11
7 pagesThis page defines bounded cochain complexes of graded bimodules and their signed tensor totalization, keeping cochain signs separate from the internal grading.
1 definition, 1 lemma, 1 proposition, 3 theoremsExamples & counterexamples →This page develops the two Grothendieck groups attached to finite-dimensional modules over a finite-dimensional unital algebra over a field: the short-exact-sequence group G…
4 definitions, 2 lemmas, 8 theoremsExamples & counterexamples →A Cartan–Eilenberg resolution resolves a complex together with its cycles, boundaries and cohomology.
4 definitions, 11 lemmas, 5 propositions, 12 theorems, 3 corollaries, 6 false statements, 1 remarkExamples & counterexamples →This page develops Hochschild homology of unital algebras over a field.
4 definitions, 3 lemmas, 1 proposition, 5 theorems, 1 corollaryExamples & counterexamples →Assuming the Axiom of Choice, this page proves the Kunneth short exact sequence for nonnegative complexes of arbitrary-rank free modules over a commutative PID.
3 lemmas, 2 theoremsExamples & counterexamples →This page isolates the abstract Hard Lefschetz splitting mechanism of a bounded t-structure, the form in which the Lefschetz theorems of Hodge theory are consumed by perverse-sheaf arguments.
1 definition, 5 lemmasExamples & counterexamples →This page develops Morita theory for unital rings as a bicategorical statement, in the left-module handedness fixed for the Eilenberg-Watts expansion: a 1-cell A→ B is a…
4 definitions, 8 lemmas, 3 theorems, 1 corollaryExamples & counterexamples →
Level 12
4 pagesThis page develops the finite-module theory behind the intrinsic definition of a finite k-linear abelian category.
1 definition, 3 lemmas, 1 proposition, 4 theorems, 3 corollariesExamples & counterexamples →This page proves the graded Eilenberg–Watts theorem for graded k-algebras over a field k, with the internal shift taken seriously as coherence data.
1 definition, 7 lemmas, 1 theorem, 2 corollaries, 1 remarkExamples & counterexamples →Starting from the Hochschild chain complex of a bimodule, this page records two ways to apply Hochschild homology to a bounded cochain complex.
2 definitions, 1 lemma, 5 theoremsExamples & counterexamples →This page defines perfect complexes over a unital associative ring and their graded analogue, and computes the Grothendieck group built from distinguished triangles on them.
2 definitions, 3 lemmas, 4 theoremsExamples & counterexamples →
Level 13
1 pageFunctor categories are formed on chosen small representatives of the finite categories.
2 definitions, 5 lemmas, 2 propositions, 2 theorems, 1 corollaryExamples & counterexamples →