Locally blueprinted space

Definition

A blueprinted space is a topological space $X$ together with a sheaf $\mathcal O_X$ of blueprints. It is locally blueprinted if every stalk $\mathcal O_{X,x}$ is a local blueprint, meaning that it has a unique maximal ideal $\mathfrak m_x$ .

A morphism

(f,f^\#):(X,\mathcal O_X)\longrightarrow(Y,\mathcal O_Y)

of locally blueprinted spaces consists of a continuous map $f:X\to Y$ and a morphism of sheaves

f^\#:\mathcal O_Y\longrightarrow f_*\mathcal O_X

such that the induced map on every stalk

f_x^\#:\mathcal O_{Y,f(x)}\longrightarrow\mathcal O_{X,x}

is local: it sends the maximal ideal of the source into the maximal ideal of the target.

The resulting category is commonly denoted $\mathbf{LocBlprSp}$ . Blue schemes form the full subcategory of locally blueprinted spaces that admit affine open covers by blueprint spectra.

References

Oliver Lorscheid, The geometry of blueprints, Part I: Algebraic background and scheme theory, §3.1.