# homology theory for affine and projective algebraic sets?

Given $f_1,\ldots,f_r\in K[x_1,\ldots,x_m]$, resp. homogeneous $f_1,\ldots,f_r\in K[x_0,\ldots,x_m]$, is there a chain complex built from these polynomials, such that any polynomial map $\varphi\!: Z(f_1,\ldots,f_r)\!\subseteq\!\mathbb{A}^m\longrightarrow Z(g_1,\ldots,g_s)\!\subseteq\!\mathbb{A}^n$, resp. $\varphi\!: Z(f_1,\ldots,f_r)\!\subseteq\!\mathbb{P}^m\longrightarrow Z(g_1,\ldots,g_s)\!\subseteq\!\mathbb{P}^n$, induces a chain map $\varphi_\ast$?

For example, for simplicial complexes (Poincare), cell complexes, Lie algebras (Chevalley), groups (Eilenberg, MacLane), algebras (Hochschild), knots (Khovanov), etc. there already are nice functors into the category of chain complexes. I'm asking if this can also be done for algebraic sets.

It is desirable that the chain complex consists of free modules and is explicitly given (so we can use a computer), and the homology reflects geometric properties of the algebraic set. Ideally, for $K\!=\!\mathbb{R}$ this would be isomorphic to singular homology, but I am skeptic that a nice functor with this requirement exists.

• Maybe Delf's homology theory (defined using semi-algebraic sets as the basic building blocks) will be of some use to you: springer.com/us/book/9783540546153 – Misha Feb 3 '15 at 23:45
• @Misha Do you have a more specific reference (page, theorem) in that book? What is the basis of each module and what are the boundary morphisms? – Leon Feb 4 '15 at 11:47