This article investigates how classical homological invariants extend from module categories to a triangulated category $\mathcal {T}$ with a compact silting generator M. We construct sup-projective, inf-injective, and sup-flat resolutions for objects in $\mathcal {T}$ and use them to establish basic properties of the corresponding homological dimensions. We then introduce the global dimension $\mathrm {gldim}\mathcal {T}$ and prove that it coincides with the global dimension of the heart $\mathcal {H}_M=M^{\bot _{\neq 0}}$ and with the $\mathrm {Add}(M)$-dimension of $\mathcal {T}^{\mathrm {b}}$ when M is tilting. These results provide a unified framework for homological dimension theory in triangulated categories and reveal how silting and tilting objects encode essential homological information. As applications, we study homological properties, such as Gorensteinness and regularity for $\mathcal {T}$ with a focus on the derived category over rings and dg rings.

Given a presilting object in a triangulated category, we find necessary and sufficient conditions for the existence of a complement. This is done both for classic (pre)silting objects and for large (pre)silting objects. The key technique is the study of associated co-t-structures. As a consequence of our techniques we recover some known cases of the existence of complements, including for derived categories of some hereditary abelian categories and for silting-discrete algebras. Moreover, we also show that a finite-dimensional algebra is silting discrete if and only if every bounded large silting complex is equivalent to a compact one.

We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.

We investigate symmetry of the silting quiver of a given algebra which is induced by an anti-automorphism of the algebra. In particular, one shows that if there is a primitive idempotent fixed by the anti-automorphism, then the 2-silting quiver ($=$ the support $\tau$-tilting quiver) has a bisection. Consequently, in that case, we obtain that the cardinality of the 2-silting quiver is an even number (if it is finite).

We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$-tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$-tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$-tilting-finite algebra is representation-finite.