  In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3]
(arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342
[math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with
a fixed $B$-field background $(Y,\alpha_B)$ along the line of the
Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the
current work. We focus first on the gauge-field-twist effect of $B$-field to
the Chan-Paton module on D-branes. Basic properties of the moduli space of
D-branes, as morphisms from Azumaya schemes with a twisted fundamental module
to $(Y,\alpha_B)$, are given. For holomorphic D-strings, we prove a
valuation-criterion property of this moduli space. The setting is then extended
to take into account also the deformation-quantization-type noncommutative
geometry effect of $B$-field to both the D-brane world-volume and the
superstring target-space(-time) $Y$. This brings the notion of twisted ${\cal
D}$-modules that are realizable as twisted locally-free coherent modules with a
flat connection into the study. We use this to realize the notion of both the
classical and the quantum spectral covers as morphisms from Azumaya schemes
with a fundamental module (with a flat connection in the latter case) in a very
special situation. The 3rd theme (subtitled "Sharp vs.
Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3]
(arXiv:hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately
"Dijkgraaf-Holland-Su{\l}kowski-Vafa vs. Polchinski-Grothendieck") is to be
read with the related sections in [D-H-S-V] (arXiv:0709.4446 [hep-th]) and
[D-H-S] (arXiv:0810.4157 [hep-th]) of Dijkgraaf, Hollands, Su{\l}kowski, and
Vafa.
