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Let k be a field and let R be a quadratic standard graded
k-algebra with dim R_2<4. We construct a graded
surjective Golod homomorphism from P to R such that P
is a complete intersection of codimension at most 3. Furthermore,
we show that R is absolutely Koszul (that is, every finitely
generated R-module has finite linearity defect) if and only if
R is Koszul if and only if R is not a trivial fiber extension of
a non-Koszul and non-Artinian quadratic algebra of embedding
dimension 3. In particular, we recover earlier results on the
Koszul property of Backelin, Conca and
D'Al`i.
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