Michal Hrbek -  Czech Academy of Sciences Prague
Product-complete tilting complexes

Abstract: We consider tilting complexes in the derived module category
D(Mod-R) of a ring R which are product-complete in the sense that
Add(T) is a definable subcategory of D(Mod-R). Using the theory of
contramodules, we show that the cotilting heart associated to a
product-complete tilting complex is locally coherent and locally
coperfect. If R is commutative noetherian, this implies in particular
that the cotilting t-structure restricts to the bounded derived
category D^b(mod-R). If R is a homomorphic image of a Cohen-Macaulay
ring of finite Krull dimension, we show that the codimension function
on the Zariski spectrum Spec(R) always induces a product-complete
tilting complex. We use this to show two new results in commutative
algebra: 1) if R is Cohen-Macaulay then the class of R-modules of
finite projective dimension is of finite type, 2) a local commutative
noetherian ring is CM-excellent if and only if there is a triangle
equivalence D^b(Mod-R) \cong D^b(A) for some locally coherent and
locally coperfect Grothendieck category A. This is a report on ongoing
joint works with Giovanna Le Gros and Lorenzo Martini.