We improve the homology stability range for the 3rd integral homology of symplectic groups over commutative local rings with infinite residue field. As an application, we show that for local commutative rings containing an infinite field of characteristic not 2 the symbol map from Milnor-Witt K-theory to higher Grothendieck-Witt groups is an isomorphism in degrees ≤ 3. Contents1. Introduction 1 2. The complex of non-degenerate unimodular sequences 3 3. The Homology spectral sequence and its E 1 -page 9 4. Triviality of d r p,q for q even 13 5. A formula for d 2 0,2n+1 14 6. Surjectivity of d 2 0,5 19 7. Localising homology groups 25 8. Homology stability 27 9. The Hurewicz map 30 10. The KO-degree map 31 Appendix A. The Pfaffian of some matrices 38

Let R be a commutative Noetherian ring of dimension d, M a commutative cancellative torsionfree monoid of rank r and P a finitely generated projective R[M ]-module of rank t.((4) Assume R is a uni-branched affine algebra over an algebraically closed field and d = 1.

We show that for any commutative Noetherian regular ring R containing Q, the map K 1 (R)) is an isomorphism. This answers a question of Gubeladze. We also compute the higher K-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher K-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.In particular, the analogue of homotopy invariance extends to K-theory of monoid algebras over regular rings in degree up to zero. However, Srinivas [31] showed that this property no longer holds in higher degrees, by showing that SK 1 (k[M ]) = 0, where k is any algebraically closed field of characteristic different from two and k[M ] is the monoid algebra k[x 2 1 , x 1 x 2 , x 2 2 ] ⊂ k[x 1 , x 2 ]. Gubeladze [12] gave a different and more algebraic proof of Srinivas' result with no condition on the ground field k. He further showed that the above monoid algebra is not 2010 Mathematics Subject Classification. Primary 19D50; Secondary 13F15, 14F35.

We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R-module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.

We show, for a finitely generated partially cancellative torsion-free commutative monoid M , that K i (R) ∼ = K i (R[M ]) whenever i ≤ −d and R is a quasi-excellent Q-algebra of Krull dimension d ≥ 1. In particular, K i (R[M ]) = 0 for i < −d. This is a generalization of Weibel's K-dimension conjecture to monoid algebras. We show that this generalization fails for X[M ] if X is not an affine scheme. We also show that the Levine-Weibel Chow group of 0-cycles CH LW 0 (k[M ]) vanishes for any finitely generated commutative partially cancellative monoid M if k is an algebraically closed field. 0.1. Weibel's conjecture for monoid algebras. Recall that a famous conjecture of Weibel [36] asserts that if R is a commutative Noetherian ring of Krull dimension d, then K −d (R) ≃ K −d (R[t 1 , · · · , t n ]) and K i (R[t 1 , · · · , t n ]) = 0 for i < −d and n ≥ 0. An affirmative answer to this conjecture was obtained recently by Kerz, Strunk and Tamme [22]. For Noetherian rings containing Q, this was earlier solved by Cortiñas, Haesemeyer, Schlichting and Weibel [3] (see also [10], [23] and [38] for older results in positive characteristics).The main technical tool that goes into the proof of Weibel's conjecture in [22] is a pro-cdhdescent theorem for algebraic K-theory. However, the final step in the proof of the conjecture 2010 Mathematics Subject Classification. Primary 19D50; Secondary 13F15, 14F35.