a linear algebra question


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Let K field, let V be a K-vector space, let

(V(i)) with i in I

be a family of vector subspaces of V, let

(a(i,j)) with i, j in I

be a family of vectors of V satisfying

a(i,j) + a(j,k) = a(i,k) mod V(i) +V(j) +V(k)

for all i, j, k in I. Is there always a family

(x(i)) with i in I

of vectors of V satisfying

x(j) - x(i) = a(i,j) mod V(i) + V(j)

for all i, j in I ?

Variation: K is a ring (with 1), V is a K-module, the V(i) are submodules.

.

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