Assume that A is a cochain DG polynomial algebra such that its underlying graded algebra A # is a polynomial algebra generated by n degree 1 elements. We determine the DG Krull dimension, the global dimension, the ghost dimension and the Rouquier dimension of A.
introductionLet k is an algebraically closed field of characteristic 0. This paper deals with DG polynomial k-algebras, which are connected cochain DG algebras whose underlying graded algebras are polynomial algebras generated by degree one elements. In [MGYC], the differential structures and homological properties of DG polynomial algebras are systematically studied. However, we still know very little about various homological invariants of DG polynomial algebras.In dimension theory for rings, it is well known that that the Krull dimension and the global dimension of the polynomial ring k[x 1 , · · · , x n ] are both equal to n. The DG Krull dimension and the global dimension for DG algebras have been introduced in [BSW] and [MW3], respectively. It is natural for one to be intensely curious about the DG Krull dimension and the global dimension of DG polynomial rings. In group theory, the terminology 'class' is used to measure the shortest length of a filtration with sub-quotients of certain type. Avramov-Buchweitz-Iyengar [ABI] introduced free class, projective class and flat class for differential modules over a commutative ring. Inspired from their work, the first author and Wu [MW3] introduced the notion of 'DG free class' for semi-free DG modules over a DG algebra A. In brief, the DG free class of a semi-free DG A-module F is the shortest length of all strictly increasing semi-free filtrations. For a more general DG module M , the first author and Wu [MW3] introduced the invariant 'cone length', which is the minimal DG free class of all its semi-free resolutions. The studies of these invariants for DG modules can be traced back to Carlsson's work in 1980s. In [Car], Carlsson studied 'free class' of solvable free DG modules over a graded polynomial ring R in n variables of positive degree. By [Car, Theorem 16], one sees that the cone length of any totally finite DG R-module M satisfies the inequality cl R M ≤ n. Note that the invariant 'cl R M ' was denoted by 'l(M )' in [Car]. The invariant 'cone length' of a DG Amodule plays a similar role in DG homological algebra as the 'projective dimension' of a module over a ring does in classic homological ring theory (cf.[MW3]). The left (resp. right) global dimension of a connected DG algebra A is defined to be the supremum of the set of cone lengths of all DG A-modules (resp. A op -modules). The difficulty in the studies of the DG Krull dimension and the global dimension of DG polynomial rings comes from the fact that these two invariants are determined by a combination of the graded algebra structure and the differential system. Due to the classifications of DG polynomial algebras in [MGYC], we show the following theorem (see Theorem 4.2 and Theorem 4.4):
