# ag.algebraic geometry – When is cohomology of a finitely presented dg-algebra computable? Given a smooth affine variety $$X$$ defined over $$mathbb{Q}$$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $$Omega_X^0toOmega_X^1todots$$ of differential forms on $$X$$ with polynomial coefficients. If $$X$$ is given by equations $$f_1,ldots,f_minmathbb{Q}(x_1,ldots,x_n)$$ then the algebra of differential forms has the following explicit presentation:
$$Omega_X^* = mathbb{Q}(x_1,ldots,x_n,d x_1,ldots,d x_n)/(f_1,ldots,f_m,d f_1,ldots, d f_m),$$
and the differential $$d$$ is defined in the obvious way. So the problem of computing the cohomology of $$X$$ is reduced to the problem of computing the cohomology of the explicitly presented (super-)commutative dg algebra above.

It is not clear if there is a general algorithm for computing the cohomology of a finitely presented commutative dg algebra. I suspect there is none. On the other hand, the de Rham complex can be interpreted as the complex representing the derived push-forward $$R^* pi_* O_X$$, where $$pi:mathbb{Q}^nto mathrm{point}$$ is the projection. There is an algorithm to compute this kind of push-forwards, it basically relies on the notion of a complex of D-modules with holonomic cohomology: the D-module $$O_X$$ is holonomic, and the push-forward of such a complex is again such a complex. The push-forward is computed by successively applying the push-forwards via
$$mathbb{Q}^ntomathbb{Q}^{n-1}tomathbb{Q}^{n-2}todotsto mathrm{point}.$$

For details, see this paper and other papers by these authors and references therein:

Oaku, Toshinori; Takayama, Nobuki, An algorithm for de Rham cohomology groups of the complement of an affine variety via (D)-module computation, J. Pure Appl. Algebra 139, No. 1-3, 201-233 (1999). ZBL0960.14008.

The question is: what makes the dg algebra $$Omega^*_X$$ so special that its cohomology can be computed? Is there a notion of a ‘nice’ dg algebra so that 1) all dg-algebras of the form $$Omega^*_X$$ are nice and 2) there is an algorithm for computing the cohomology of a nice dg-algebra? Posted on Categories Articles