We generalize the Białynicki-Birula decomposition from actions of G m on smooth varieties to actions of linearly reductive group G on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks.We define the Białynicki-Birula decomposition functorially: for a fixed G-scheme X and a monoid G which partially compactifies G, the BB decomposition parameterizes G-schemes over X for which the G-action extends to the G-action. The freedom of choice of G makes the theory richer than the G m -case.Corollary 1.2. Let X be a G-scheme locally of finite type over k. Then every point of X has an affine G-stable neighbourhood.Proof. Note that X = X + by Proposition 5.7. By Theorem 1.1 the projection π X is G-equivariant and affine. The required neighbourhood of x ∈ X is the preimage of any affine open neighbourhood of π X (x) ∈ X G . This is striking, since for an arbitrary, even normal, G-scheme X the existence of affine G-stable neighbourhoods is subtle, see Brion [Bri15]. As an example of non-normal scheme: the node of the nodal curve C = P 1 /(0 = ∞) with its natural G m -action does not admit an affine G m -stable neighbourhood; here C + is equal to A 1 . Now we describe X + for an affine X. Recall that every irreducible finite dimensional Grepresentation appears in H 0
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