Given a reduced local algebra T over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections R → T which induce isomorphisms Ω R/k ⊗ T → Ω T /k. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions ...

In this paper, we study non-Cohen–Macaulay Buchsbaum Stanley– Reisner rings with linear free resolution. In particular, for given integers c, d, q with c ≥ 1, 2 ≤ q ≤ d, we give an upper bound hc,d,q on the dimension of the unique non-vanishing homology H̃q−2(∆; k) of a d-dimensional Buchsbaum ring k[∆] with q-linear resolution and codimension c. Also, we discuss about existence for such Buchsba...

Let S be an unramified regular local ring of mixed characteristic two and R the integral closure in a biquadratic extension its quotient field obtained by adjoining roots sufficiently general square free elements f,g?S. S2 denote subring lifting to image Frobenius map on S/2S. When at least one f,g?S2, we characterize Cohen-Macaulayness show that admits birational small Cohen-Macaulay module. I...

For a skew version of graded (A ∞ ) hypersurface singularity A, we study the stable category maximal Cohen-Macaulay modules over A. As consequence, see that A has countably infinite Cohen–Macaulay representation type and is not noncommutative isolated singularity.

We define local cohomology on a diagram of schemes, and discuss basic properties of it. In particular, we prove the independence theorem and the flat base change for local cohomology on diagrams of schemes. We also introduce the notion of G-localness of a G-scheme. It is an equivariant analogue of localness of a scheme. As an application, we prove that Cohen–Macaulay property is inherited by th...

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over...

Given a reduced local algebra T over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections R → T which induce isomorphisms Ω R/k ⊗ T → Ω T /k. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions ...

We study rings of integral modular forms for congruence subgroups as modules over the ring full group. In many cases these are free or decompose at least into well-understood pieces. apply this to characterize which Cohen--Macaulay and prove finite generation results. These theorems based on decomposition results about vector bundles compactified moduli stack elliptic curves.

The concept of residual intersection, introduced by Artin and Nagata [l] in 1972, is a fruitful generalization of linkage as the following two examples attest. Let I be a strongly Cohen-Macaulay ideal in a CohenMacaulay local ring R. If Z satisfies the condition (G,), then Huneke [6, Proposition 4.31 has proved that the extended Rees algebra R[It, t‘1 is defined by an ideal which is obtained fr...

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $\omega$. Attached to each such ring R, when $\omega$ is an ideal, there are integers--the type the reduction number $\omega$--that provide valuable metrics express deviation R from being Gorenstein ring. In arXiv:1701.05592 and arXiv:1711.09480 we enlarged this list with degree bi-canonical degree. work extend wh...