Questions: Graded Rings and Modules

Question 1

In the polynomial ring k[x, y] graded by total degree, which of the following is a homogeneous element?

• A. x² + y (sum of degree 2 and degree 1 terms)
• B. x²y + 3xy² - y³ (all terms have total degree 3)
• C. 1 + x + x² (terms of degrees 0, 1, and 2)
• D. x²y + x (terms of degrees 3 and 1)

Answer: (B)

A homogeneous element lies entirely in one graded component — all its terms have the same total degree. The polynomial x²y + 3xy² - y³ has every term of total degree 3, so it is a homogeneous element of degree 3. The other options mix terms of different degrees. Homogeneous elements are the building blocks of graded algebra: every element decomposes uniquely as a sum of homogeneous components.

Question 2

An ideal I in a graded ring R is called homogeneous if it is generated by homogeneous elements. Which equivalent characterization is also correct?

• A. I is homogeneous if and only if it contains at least one homogeneous element
• B. I is homogeneous if and only if for every f ∈ I, each homogeneous component of f also lies in I
• C. I is homogeneous if and only if R/I is an integral domain
• D. I is homogeneous if and only if I = R₊ = R₁ ⊕ R₂ ⊕ ···

Answer: (B)

A homogeneous ideal has the property that it 'respects the grading': if f ∈ I and f = f₀ + f₁ + ··· + fₙ is its decomposition into homogeneous components, then each fᵢ ∈ I. This is strictly stronger than merely containing some homogeneous element (every nonzero ideal does). The ideal R₊ is one specific homogeneous ideal (the irrelevant ideal), not the only one. This characterization is essential for working with projective varieties, where only homogeneous ideals define geometric objects.

Question 3

The polynomial ring k[x₁, ..., xₙ] admits a grading by total degree where k[x₁, ..., xₙ]_d is the vector space of homogeneous polynomials of degree d.

• T. True
• F. False

Answer: True

This is the standard grading. The degree-d component consists of all k-linear combinations of monomials x₁^{a₁}···xₙ^{aₙ} with a₁ + ··· + aₙ = d. It is a k-vector space of dimension C(n+d-1, d). The product of a degree-d polynomial and a degree-e polynomial has degree d+e, confirming that RₘRₙ ⊆ Rₘ₊ₙ. This grading encodes the geometry of projective space.

Question 4

Every ideal in a graded ring is homogeneous.

• T. True
• F. False

Answer: False

In k[x, y] with the standard grading, the ideal (x + 1) is not homogeneous: x + 1 has components x (degree 1) and 1 (degree 0), and the ideal does not contain 1 (otherwise it would be the whole ring) or x separately. Homogeneous ideals are a special class — they are exactly the ideals that can be generated by homogeneous elements. In projective algebraic geometry, only homogeneous ideals define projective varieties.

Question 5

Explain why the irrelevant ideal R₊ = R₁ ⊕ R₂ ⊕ ··· in a graded ring R = k ⊕ R₁ ⊕ R₂ ⊕ ··· is called 'irrelevant' and what role it plays.

Model answer: The ideal R₊ is called irrelevant because in projective geometry, the vanishing set of all positive-degree homogeneous polynomials is the empty set — it defines no points in projective space. A homogeneous polynomial f(x₀, ..., xₙ) vanishes at a projective point [a₀ : ··· : aₙ] if and only if f(a₀, ..., aₙ) = 0 for every representative, and the only simultaneous zero of all homogeneous polynomials of positive degree is the origin (0, ..., 0), which does not represent a projective point.

Despite being 'irrelevant' geometrically, R₊ is algebraically important. It is the unique homogeneous maximal ideal in a graded polynomial ring over a field. In the theory of Hilbert functions, R₊ is the ideal whose powers define the filtration used to measure growth of graded components. The distinction between 'relevant' prime ideals (which define actual projective varieties) and the irrelevant ideal is fundamental to projective algebraic geometry.