Non-normal affine monoids

We give a geometric description of the set of holes in a non-normal affine monoid $Q$. The set of holes turns out to be related to the non-trivial graded components of the local cohomology of $k[Q]$. From this, we see how various properties of $k[Q]$ like local normality and Serre's conditions $(R_1)$ and $(S_2)$ are encoded in the geometry of the holes. A combinatorial upper bound for the depth the monoid algebra $k[Q]$ is obtained and some cases where equality holds are identified. We apply this results to seminormal affine monoids.Comment: 18 pages, 3 figures. Simplified proof of the main result, shortened. An even shorter version appeared with the title "Non-normal affine monoid algebra" in manuscripta mathematic

Linear maps in minimal free resolutions of Stanley-Reisner rings

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex $\Delta$. Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of $\Delta$. Along the way, we also show that if a monomial ideal has at least one generator of degree $2$, then the linear strand of its minimal free resolution can be written using only $\pm 1$ coefficients.Comment: 7 pages, 1 figure. An error was found in the earlier version, therefore a part of the paper has been remove

Edge rings satisfying Serre's condition R_1

A combinatorial criterion for the edge ring of a finite connected graph to satisfy Serre's condition R_1 is studied