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Haynes Miller works in algebraic topology, homotopy theory and homological algebra. His early research helped develop chromatic homotopy theory, and his solution of the Sullivan conjecture connected questions about maps of spaces with the structure of Lie groups. His lecture notes and expository writing develop the algebraic topology used to turn geometric questions into calculations with algebraic invariants.^[ref]

Identity source: math.mit.edu/directory/profile.html?pid=186

Homological algebra studies chain complexes and cochain complexes, their homology and cohomology, and the functors that compare them. The identity $d^2=0$ ensures that every boundary is a cycle, so the quotient measures cycles not explained by boundaries.

A basic motivation is failure of exactness. Tensoring the inclusion $2\mathbb{Z}\hookrightarrow \mathbb{Z}$ with $\mathbb{Z}/2\mathbb{Z}$ gives a zero map between two nonzero groups, so tensoring need not preserve injections. Tor functor measures this failure, while Ext functor measures related failures for Hom functors. Projective resolutions replace a module by a complex of easier modules and make these invariants computable.

Mark Powell studies the topology of manifolds, especially in dimensions three and four, together with homological algebra. His Glasgow profile verifies this research specialization.^[ref]

His Introduction to Differential Geometry by Mark Powell connects the local geometry of curves and surfaces with global invariants. Curvature and torsion of a curve describe how a curve bends in space, whereas the Gauss-Bonnet theorem turns integrated surface curvature and boundary terms into an Euler characteristic. These examples show how local differential measurements can constrain a whole geometric object.^[ref]

Nicholas J. Williams studies representation theory for finite-dimensional associative algebras and its connections to combinatorics. His research connects partially ordered sets and triangulations of convex polytopes with the structure of finite-dimensional algebras, including quivers and homological algebra. This interaction allows combinatorial order and geometric decomposition to encode relationships between algebraic objects.^[ref]

Identity source: nchlswllms.github.io/

Here the usual homological algebra convention is that the right-exact functor $F$ is additive. Module categories have enough projective modules, because any module is a quotient of a free module. Choose a projective resolutionApply $F$ to the unaugmented chain complex and define the left derived functors byAdditivity ensures that $F$ preserves zero maps, sums of maps and homotopy identities, so the transformed differentials still compose to zero.

For a module homomorphism $u:M\to N$, lift it to a chain map between chosen projective resolutions. In degree zero, lift through $Q_0\twoheadrightarrow N$ using projectivity of $P_0$. In each higher degree, the already constructed map lands in the next kernel; exactness of $Q_{\bullet }$ and projectivity of $P_i$ supply the next lift. Any two lifts are chain homotopic: their degree-zero difference lands in $\ker (Q_0\to N)=\mathrm{im}{d}_1^Q$, so it lifts to $h_0:P_0\to Q_1$. Inductively lift the remaining difference after subtracting the preceding homotopy term. This givesApplying $F$ preserves this identity, so the lifts induce the same homology map. Identity lifts and composable lifts show that $L_{i}F$ is a functor. Comparison lifts between two resolutions over the identity of $M$ are inverse up to chain homotopy; hence the construction is independent of the chosen resolution up to its canonical natural isomorphism.

Right exactness identifies $L_{0}F(M)=\mathrm{coker}(F(P_1)\to F(P_0))$ naturally with $F(M)$. For a projective module $P$, the length-zero resolution gives $L_{i}F(P)=0$ for $i>0$. For a short exact sequence $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$, the horseshoe lemma builds a degreewise split short exact sequence of projective resolutions. Concretely, lift the projective generators covering $M^{\prime \prime }$ into $M$ and combine them with generators covering $M^{\prime }$; repeat on the kernels. The middle projective term in each degree is the direct sum of the two outer ones. An additive functor preserves these split sequences, and lifting cycles, taking their differentials and returning to the subcomplex produces the connecting maps inThis is the natural long exact derived-functor sequence. In particular, if $0\to K\to P\to M\to 0$ has $P$ projective, then $L_{i}F(M)\cong L_{i-1}F(K)$ for $i\ge 2$, and $L_{1}F(M)=\ker (F(K)\to F(P))$. These are dimension-shifting formulas. Positive degrees can be killed by the surjection $P\to M$. If $F$ is exact, it preserves every resolution's exactness and all its positive left derived functors vanish. Conversely, if $L_{1}F$ vanishes on all modules, the displayed long exact sequence makes $F$ exact. A natural transformation between additive right-exact functors induces transformations on the left derived functors by applying it termwise.

For extension of scalars $F(M)=B{\otimes }_{A}M$, these derived functors are the Tor functors ${\mathrm{Tor}}_i^A(B,M)$. When $B=A/(a)$ and $a$ is a non-zero-divisor, there is the length-one free resolutionAlthough the construction above resolves $M$, one may instead use this resolution of $B$. To justify that calculation, tensor a free resolution of $M$ with the two-term resolution of $B$ to form a double complex. Taking homology first along the resolution of $B$ leaves $B{\otimes }_A{P}_{\bullet }$; taking it first along the resolution of $M$ leaves $M\stackrel{a}{\to }M$. The free terms preserve exactness under tensoring. The double complex is first-quadrant with finitely many terms on each total diagonal, so both filtrations compute the same total homology. This proves the required balancing directly.

The kernel and cokernel of multiplication by $a$ now give Tor for a principal non-zero-divisor quotient:These are naturally $B$-modules. For $M=A$, the degree-one group is zero; for $M=B$, multiplication by $a$ is zero and both degrees zero and one are $B$. Thus, when $B\ne 0$, degree one explicitly detects the failure of this base-change functor to be exact.

Samuel Eilenberg was a mathematician whose work connected algebraic topology, homological algebra and category theory. With Saunders Mac Lane, he introduced the language of categories, functors and natural transformations; with Norman Steenrod, he gave an axiomatic account of homology and cohomology.^[ref] This language records more than the invariants attached to spaces: a continuous map also induces a map between the corresponding algebraic objects. Tracking those induced maps makes it possible to compare constructions and explain why many apparently different versions of a theory express the same underlying structure.

Saunders Mac Lane was a mathematician whose research and books helped shape modern algebra, homological algebra and category theory. His collaboration with Samuel Eilenberg introduced categories, functors and natural transformations as a language for relationships among mathematical constructions.^[ref] One illuminating example treats a group as a category with one object and invertible arrows: composition of arrows is the group operation. This viewpoint allows the same language to describe both a familiar algebraic structure and maps between entire mathematical theories. His expository work helped make such abstract ideas usable in everyday mathematical reasoning.