Add method to compute the cohomology algebra

src/sage/algebras/commutative_dga.py

@@ -2439,6 +2439,69 @@ def extend(phi, ndegrees, ndifs, nimags, nnames):
 
         return phi
 
+    def cohomology_algebra(self, max_degree=3):
+        """
+        Compute a CDGA with trivial differential, that is isomorphic to the cohomology of
+        self up to``max_degree``
+
+        INPUT:
+
+        - ``max_degree`` -- integer (default: `3`); degree to which the result is required to
+        be isomorphic to self's cohomology.
+
+        EXAMPLES::
+
+        sage: A.<e1,e2,e3,e4,e5,e6,e7> = GradedCommutativeAlgebra(QQ)
+        sage: d = A.differential({e1:-e1*e6,e2:-e2*e6,e3:-e3*e6,e4:-e5*e6,e5:e4*e6})
+        sage: B = A.cdg_algebra(d)
+        sage: M = B.cohomology_algebra()
+        sage: M
+        Commutative Differential Graded Algebra with generators ('x0', 'x1', 'x2') in degrees (1, 1, 2) with relations [x2^2] over Rational Field with differential:
+           x0 --> 0
+           x1 --> 0
+           x2 --> 0
+        sage: M.cohomology(1)
+        Free module generated by {[x1], [x0]} over Rational Field
+        sage: B.cohomology(1)
+        Free module generated by {[e7], [e6]} over Rational Field
+        sage: M.cohomology(2)
+        Free module generated by {[x0*x1], [x2]} over Rational Field
+        sage: B.cohomology(2)
+        Free module generated by {[e6*e7], [e4*e5]} over Rational Field
+        sage: M.cohomology(3)
+        Free module generated by {[x1*x2], [x0*x2]} over Rational Field
+        sage: B.cohomology(3)
+        Free module generated by {[e4*e5*e7], [e4*e5*e6]} over Rational Field
+        """
+        DI = self.defining_ideal()
+        DR = DI.ring()
+        cohomgens = self.cohomology_generators(max_degree)
+        chgens = []
+        degrees = [g.degree() for g in self.gens()]
+        for d in cohomgens:
+            for g in cohomgens[d]:
+                degrees.append(d)
+                chgens.append(g)
+        FA = FreeAlgebra(self.base_ring(), self.ngens()+len(chgens), 'x')
+        NCrels = {}
+        for i in range(len(degrees)-1):
+            for j in range(i+1, len(degrees)):
+                if is_odd(degrees[i]*degrees[j]):
+                    NCrels[FA.gen(j)*FA.gen(i)] = -FA.gen(i)*FA.gen(j)
+        BR = FA.g_algebra(NCrels)
+        i1 = DR.hom(BR.gens()[:self.ngens()], check=False)
+        J = i1(DI) + BR.ideal([BR.gen(self.ngens()+i)-i1(chgens[i].lift()) for i in range(len(chgens))])
+        elimrels = J.twostd().elimination_ideal(BR.gens()[:self.ngens()])
+        NA = GradedCommutativeAlgebra(self.base_ring(), ['x{}'.format(i) for i in range(len(chgens))], degrees[self.ngens():])
+        NArels = []
+        for g in elimrels.gens():
+            di = g.dict()
+            ndi = {e[self.ngens():]:di[e] for e in di}
+            NArels.append(NA(ndi))
+        NB = NA.quotient(NA.ideal(NArels))
+        return NB.cdg_algebra({})
+
+
 
 
     class Element(GCAlgebra.Element):